exponentially weighted moving average control charts for...

Research ArticleExponentially Weighted Moving Average Control Charts for theProcess Mean Using Exponential Ratio Type Estimator

Hina Khan 1 Saleh Farooq1 Muhammad Aslam 2 and Masood Amjad Khan1

1Department of Statistics GC University Lahore 54000 Pakistan2Department of Statistics Faculty of Science King Abdulaziz University Jeddah 21551 Saudi Arabia

Correspondence should be addressed to Hina Khan hinakhangcuedupk

Received 18 April 2018 Accepted 3 July 2018 Published 1 October 2018

Academic Editor Luis A Gil-Alana

Copyright copy 2018 Hina Khan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This study proposes EWMA-type control charts by considering some auxiliary informationThe ratio estimation technique for themean with ranked set sampling design is used in designing the control structure of the proposed chartsWe have developed EWMAcontrol charts using two exponential ratio-type estimators based on ranked set sampling for the process mean to obtain specificARLs being suitable when small process shifts are of interest

1 Introduction

To ensure the quality standards of products every industrymust develop some mechanism by adopting suitable statisti-cal quality control techniques and proceduresThe absence ofa well-structured quality control system or a wrong choice ofquality control methods may affect the ability of producingquality products Control chart is an important statisticaltechnique in statistical process control (SPC) that visuallyhighlights the presence of special causes in a productionprocess that causes deviations in quality standards

It is generally observed that some variation always existsin the data A control chart can easily detect or identifywhether the variation is usual or unexpected for productionprocess because something special or unusual is happeningA control chart most widely uses statistical process controltool in determining the state of a manufacturing or a businessprocess (whether in control or out of control) Control chartsare typically used to observe changes in a process over timeThemain purpose of using control charting technique in SPCis to attain and sustain continuous improvement in the qualityand production of products by keenly observing changes inthe manufacturing process However the decision regardingthe selection of suitable control chart depends on the type ofdata In spite of the wide application and popularity it hasbeen observed that the traditional Shewhart control charts

can only be helpful when the process suffers an out-of-controlsituation due to the presence of assignable causes resulting inlarge shifts in the processThe reason is that Shewhart controlchart only considers the information provided by the lastsample and it does not pay any attention to the informationabout the process contained in the rest of the samples Thisdemerit makes Shewhart control chart insensitive towardssmall shift in the production processThus this classical chartis not a good choice for SPC where the process seems to be incontrol state and assignable causes do not create disturbancesin the process on considerably large scale

The exponentially weighted moving average chart awell-known control charting technique is sensitive to thedetection of control signals while small or moderate shiftsoccur in the production process EWMA chart was firstintroduced by Roberts (1959) and it has gradually achieveda significant place in SPC A lot of innovations and designshave been introduced in the structure of EWMA controlcharts for monitoring process mean and dispersion by theresearchers in different fields For example Haq et al [1]have developed newEWMAcontrol charts for controlling theproceduremean dispersion using the ideas of ordered doubleranked set sampling (ODRSS) and ordered imperfect doubleranked set sampling in the new designed EWMA controlcharts Abbasi et al [2] constructed a set of EWMA controlcharts based on large range of dispersion estimates used

HindawiJournal of Probability and StatisticsVolume 2018 Article ID 9413939 15 pageshttpsdoiorg10115520189413939

2 Journal of Probability and Statistics

for managing procedure dispersion based on normal anda range non-normal distribution The investigated EWMAdispersion control charts were based on an extensive varietyof dispersion estimators Abbas et al [3] recommended a newEWMA-type control chart that uses a single auxiliary variableknown as 119872119883119864119882119872119860 control chart For the estimation ofmean in defining the control constitution of the designedcontrol chart regression estimator was used Haq [4] recom-mended an enhancedmean deviation exponentially weightedmoving average (IMDEWMA) control chart based on rankset sampling to control process dispersion Haq et al [5]studied the effect of measurements errors on the detectionability of EWMA control charts for controlling processmean under ranked set sampling (RSS) median ranked setsampling (MRSS) imperfect ranked set sampling (IRSS) andimperfect median ranked set sampling (IMRSS) schemesAzam et al [6] presented repetitive exponentially weightedmoving average (EWMA) control chart using regressionestimator based on ranked set sampling (RSS) design toexamine and detect the changes in the manufacturing pro-cess They studied the detection ability of the proposedcontrol chart for monitoring shifts in the process meanRidwan et al [7] developed an EWMA scheme using ratioestimator to increase the effectiveness of typical EWMAchartin monitoring the location parameter

Although in efficiency comparison the EWMA charts arefound parallel to the Shewhart control charts these are saidto be best alternatives to the Shewhart charts in monitoringsmall shifts in the process mean because they provide quickalarms when small shifts are introduced in the process

2 Methodology

This study proposes EWMA-type control charts by consid-ering some auxiliary information provided by an auxiliaryvariable The ratio estimation technique for the mean withranked set sampling design is used in designing the controlstructure of the proposed charts Here we developed EWMAcontrol charts using Bahl and Tuteja (1991) exponential ratio-type estimator and Khan et al [8] exponential ratio-typeestimator based on ranked set sampling design for theprocess mean to obtain specific ARLs being suitable whensmall process shifts are of interest

21 Ranked Set Sampling (RSS) The idea of RSS was firstgiven by Mcintyre (1952) The mechanism of RSS design isbased on the ranking of sampling units within the samplesand it makes the collection of actual measurements ofsampling units more feasible and reliable as compared toSRS design with respect to simplicity time cost or othercomplicated factors

This sampling technique obtains samples from a popu-lation in such a way that the extent of information coversthe entire range of observations in the population In thisway ranked set sample becomes more representative thanthe simple random sample obtained by identical number ofobservations from the same population [9] The structure ofRSS design is simply described as follows Firstly n2 sampling

units are identified from a large or infinite population Theseunits are further assigned randomly to the n equal sizedsamples After that ranking is imposed to the units withineach sample with respect to some characteristics of interestor study variable A mixture of mechanisms can be used toacquire this ranking comprising visual comparison expertjudgment or the use of auxiliary variables however it cannotinclude actual measurements of the characteristics of intereston the selected units [10]

Let y11y12 y13 1199101119899 11991021 11991022 119910231199101198991 1199101198992 1199101198993 119910119899119899be lsquonrsquo independent simple random samples each of size nImplement the RSS process to these n samples to obtain aranked set sample (RSS) of size t defined by y1(1n) y2(2n)y3(3n) yt(nn)

Now RSS estimator for 119910 is defined as

119910119877119878119878 = 1119905119899119905sum119894=1

119899sum119894=1

[119910119894(119894119899)] (1)

with mean

120583119910 = 119864 (119910119877119878119878) = 119884 (2)

and variance

var (119910119877119878119878) = 1205902119899 minus 11198992119899sum119894=1

(120583(119894) minus 120583119910)2 (3)

var (119910119877119878119878) = var (119910119878119877119878) minus 11198992119899sum119894=1

(120583(119894) minus 120583119910)2 (4)

This estimator is unbiased and is also more precise thansimple random sampling estimator 11991011987811987711987822 Ratio Estimator In sampling surveys while estimat-ing the population parameters role of auxiliary variableis very significant to increase its efficiency Many authorshave improved the precision of their estimators throughthe use of auxiliary variable Auxiliary variable which ishighly correlated to the variable of interest actually providesinformation intended to improve the efficiency of the esti-mator of population parameter of the study or main variable[11]

If 119910119894 (119894 = 1 2 3 119873) is the observations for thevariable of interest 119910 from a finite population and if 119909119894 (119894 =1 2 3 119873) is the observations of auxiliary variable119909whoseinformation is completely known and is highly correlated tothe variable under consideration then the traditional ratioestimator for 119884 is defined as

yR = 119910(119883119909 ) (5)

The complete information of parameters of the auxiliaryvariable 119909 such as 119862119909 (coefficient of variation) and 1205732(119909)(coefficient of kurtosis) basically help to increase efficiencyof the estimator of the study variable (Yadav et al 2014)

Journal of Probability and Statistics 3

3 Proposed Repetitive ExponentiallyWeighted Moving Average (EWMA) ControlCharts Based on Ranked Set Sampling

Here we proposed two repetitive exponentially weightedmoving average (EWMA) control charts using exponentialratio-type estimators suggested by Bahl and Tuteja (1991)and Khan et al [8] based on ranked set sampling (RSS)design EWMA control charts using these ratio estimatorsare developed to observe process mean by obtaining specificARLs usingMonte Carlo simulation and being suitable whensmall shifts are of interest How to calculate probability ofdeclaring the process out of control when shift is introducedin the process is also elaborated here

31 Proposed Repetitive EWMA-RSS Control Charts UsingExponential Ratio-Type Estimators As it is already men-tioned the assumed study variable y is difficult to measuredirectly but it is easy to measure with the correspondingauxiliary variable x Let 119910119877119878119878119894(119894119899)119905 be the sample mean under RSSresultant of 119909119894(119894119899)119905 at time t (t = 1 2 3 4 ) then Bahl andTuteja (1991) exponential ratio-type estimator and Khan etal [8] exponential ratio-type estimator for mean under RSSrespectively are defined as

1198721198771198781198781199101 = 119910119894(119894119899)119905 [exp(119883 minus 119909119894(119894119899)119905119883 + 119909119894(119894119899)119905)] (6)

with mean and variance (by using (4))

E (1198721198771198781198781199101 ) = 119884 = 120583119910var (1198721198771198781198781199101 ) = 1205951198842 [1198622119910 + 141198622119909 (1 minus 119862)]

minus 11198992119899sum119894=1

[120583(119894) minus 120583119910]2 there4 120595 = 119873 minus 119899119873119899(7)

where constant lsquoCrsquo = 120588(119862119910119862119909)1198721198771198781198781199102 = 119910119894(119894119899)119905 exp[[Υ(

1198831ℎ minus 119909119894(119894119899)1199051ℎ1198831ℎ + (119886 minus 1) 119909119894(119894119899)1199051ℎ)]] (8)

Then the mean and the variance of this estimator areE (1198721198771198781198781199102 ) = 119884= 120583119910

var (1198721198771198781198781199102 ) = 1205951198842 [1198622119910 minus 11986221198622119909] minus 11198992119899sum119894=1

[120583(119894) minus 120583119910]2

there4120595 = 119873 minus 119899119873119899(9)

where constant C = 120588(119862119910119862119909)32 Structure of the Proposed Repetitive EWMA-RSS ControlCharts Here a sequence119872119877119878119878119896119894 based on119872119877119878119878119910119896 (where k =12)using recurrence formula is described as

119872119877119878119878119896119894 = 120582119872119877119878119878119910119896 + (1ndash120582)119872119877119878119878119896(119894minus1) (10)

where 0 lt 120582 le 1

Therefore 119872119877119878119878119896119894 (k = 12) is the statistic of EWMA-RSScontrol chart based on Bahl and Tuteja (1991) and Khan et al[8] exponential ratio-type estimators respectively for meanand its initiatory value is119872119877119878119878119896(0) = 1205830Thus mean and varianceof this EWMA statistic are

E (119872119877119878119878119896119894 ) = 120583119910 where k = 1 2var (119872119877119878119878119896119894 ) = 1205822 minus 120582var (119872119877119878119878119910119896 ) [1ndash (1 minus 120582)2119894]

(11)

For large number of times the variance of119872119877119878119878119896119894 becomes

var (119872119877119878119878119896119894 ) = 1205822 minus 120582var (119872119877119878119878119910119896 ) (12)

Thus variances of the statistic of EWMA-RSS control chartbased on Bahl and Tuteja (1991) and Khan et al [8] exponen-tial ratio-type estimators are respectively described as

var (1198721198771198781198781119894 ) = 1205822 minus 120582 [1205951198842 1198622119910 + 141198622119909 (1 minus 119862)minus 11198992

119899sum119894=1

(120583(119894) minus 120583119910)2] (13)

var (1198721198771198781198782119894 ) = 1205822 minus 120582 [1205951198842 [1198622119910 minus 11986221198622119909]minus 11198992

119899sum119894=1

(120583(119894) minus 120583119910)2](14)

Control Limits for the Proposed Repetitive EWMA-RSSControlCharts Control Limits for the proposed repetitive EWMA-RSS control charts for k = 12 are defined as

LCL1 = 120583119910 minus 1198711119904119889 (119872119877119878119878119896119894 )LCL2 = 120583119910 minus 1198712119904119889 (119872119877119878119878119896119894 )UCL1 = 120583119910 + 1198711119904119889 (119872119877119878119878119896119894 )UCL2 = 120583119910 + 1198712119904119889 (119872119877119878119878119896119894 )

(15)

where 1198711 and 1198712 are the control constant multipliers Thevalues of 1198711 and 1198712 are chosen such that the in-control ARLsof the proposed repetitive EWMA-RSS control chart reach aspecific level of decided value of target ARL of the process

Now the probability of reporting the process as out ofcontrol when actually the process is in control is obtained as

1198751199001199001199061199051 = 119875 (119872119877119878119878119896119894 lt 1198711198621198711) + 119875 (119872119877119878119878119896119894 gt 1198801198621198711) (16)

1198751199001199001199061199051 = 119875[119872119877119878119878119896119894 minus 120583119910120590 lt LCL1 minus 120583119910120590 ]

+ 119875[119872119877119878119878119896119894 minus 120583119910120590 gt UCL1 minus 120583119910120590 ] (17)


After simplification the above equation becomes

1198751199001199001199061199051 = 119875 (119885 lt minus1198711) + 119875 (119885 gt 1198711) (18)

The term Oslash() is cumulative distribution function of thestandard normal distribution

1198751199001199001199061199051 = Oslash (minus1198711) + 1 minusOslash (1198711)1198751199001199001199061199051 = 2 [1 minusOslash (1198711)] (19)

The probability of repetition 119875119900119877119864119875 for the planned controlchart is given as follows

119875119900119877119864119875 = 119875 1198801198621198712 lt 119872119877119878119878119896119894 lt 1198801198621198711+ 119875 1198711198621198711 lt 119872119877119878119878119896119894 lt 1198711198621198712 (20)

After simplification it becomes

119875119900119877119864119875 = 119875 1198712 lt Z lt 1198711 + 119875 minus1198711 lt Z lt minus1198712119875119900119877119864119875 = Oslash (1198711) minusOslash (1198712) + Oslash (minus1198712) minusOslash (minus1198711)119875119900119877119864119875 = 2 [Oslash (1198711) minusOslash (1198712)]

(21)

so

119875119900119900119906119905 = 11987511990011990011990611990511 minus 119875119900119877119864119875 (22)

119860119877119871119900 = 1119875119900119900119906119905 (23)

For Shifted Process If our mean is shift

1205831 = 1205830 + 119891120590 there4 1205830 = 1205831199101205831 = 120583119910 + 1198911205901205831= 120583119910 + 119891120590 sim N(120583119910 + 119891120590 1205822 minus 120582var (119910119896119878119877119878))

(24)

where k = 12 now the probability of reporting the processas out of control when the shift will be introduced in theprocess mean is described as

11987511199001199061199051 = 119875 (119872119877119878119878119896119894 lt 1198711198621198711 | 1205831)+ 119875 (119872119877119878119878119896119894 gt 1198801198621198711 | 1205831) where k = 1 2 (25)


11987511199001199061199051 = 119875(119885 lt minus1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))

+ 119875(119885 gt 1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))

11987511199001199061199051 = Oslash(minus1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894)) + 1

minusOslash(1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))

(26)

Similarly the probability of repetition 1198751119877119864119875 for the proposedcontrol chart when the shift will be introduced in the processmean is given as

1198751119877119864119875 = 1198751198801198621198712 lt 119872119877119878119878119896119894 lt 11988011986211987111205831 + 1198751198711198621198711 lt 119872119877119878119878119896119894 lt 11987111986211987121205831

(27)


1198751119877119864119875= 119875[1198712 minus 119891120590119910119904119889 (119872119877119878119878119896119894 ) lt 119885 lt 1198711 minus

119891120590119910119904119889 (119872119877119878119878119896119894 )]+ 119875[minus1198711 minus 119891120590119910119904119889 (119872119877119878119878

119896119894) lt 119885 lt minus1198712 minus

119891120590119910119904119889 (119872119877119878119878119896119894)]

(28)

and then

1198751119877119864119875 = Oslash(1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))

minusOslash(1198712 minus 119891120590119910119904119889 (119872119877119878119878119896119894 ))+Oslash(minus1198712 minus 119891120590119910119904119889 (119872119877119878119878

119896119894))

minusOslash(minus1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894 ))

(29)

so

1198751119900119906119905 = 119875111990011990611990511 minus 1198751119877119864119875 (30)

1198601198771198711 = 11198751119900119906119905 (31)

4 Results and Interpretation

The proposed repetitive EWMA control charts are designedfor the purpose of monitoring the shifts in the processmean using Bahl and Tuteja (1991) and Khan et al [8]exponential ratio-type estimators under ranked set samplingRSS We used ARLs as a performance criterion to comparethe performance efficiency of repetitive EWMA controlcharts Here the ARLs tables and graphs are constructedfor frequently used values of in-control ARLs such as 500and 370 for the repetitive EWMA control charts Differentvalues of the smoothing constant 120582 (0 lt 120582 le 1) are takenwith an interval of 01 to construct the ARLs tables Threedifferent datasets having different levels of correlation 120588 ie025 050 and 090 between the auxiliary variable and thevariable of interest are used to assess the performance ofthe proposed repetitive EWMA-RSS control charts Here theout-of-control ARLs are evaluated for the shifted process in


Table 1 ARLs for proposed repetitive EWMA-RSS control chart using Bahl and Tuteja (1991) exponential ratio-type estimator when 119903119900 is500

shift 1198961 = 30958 1198962 = 2339 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 42580 41209 32223002 28338 25761 13787003 17249 14808 5859004 10370 8495 2654005 6324 4982 1293010 761 550 939015 191 151 101020 112 105 100025 101 100 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100


shift 1198961 = 30958 1198962 = 2339 120582 = 02f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 46417 45648 40022002 37217 35205 23732003 27482 24883 13027004 19557 17011 7201005 13770 11566 4096010 2649 1971 415015 678 481 131020 248 187 102025 139 119 100030 109 104 100035 102 100 100040 100 100 10005 100 100 100

mean while maintaining the in-control ARLs at the samelevel and compared with each other The term 119903119900 depicts theconsidered target values of in-control ARLs By consideringthe subgroup size 119899 = 10 the values of the control constants(1198961015840119904) are determined for frequently used target values ofaverage run lengths such as 500 and 370 Finally ARLs basedon different values of correlation 120588 are obtained for the smalland moderate shifts in the process average

The ARLs tables of repetitive control charts that arearranged according to the various random small shifts from0 to 05 are given in Tables 1ndash12

In Table 1 out-of-control ARLs are calculated by usingrepetitive EWMA-RSS control chart using Bahl and Tuteja(1991) exponential ratio-type estimator for various smallshifts under different levels of 120588 by taking 120582 = 01 We

observed that for the fixed value of smoothing constant 120582the values of out-of-control ARLs decrease as the value of 120588increases For example when 119903119900 = 500 and shift is 001 thenthe value of out-of-control ARL for 120588 = 025 is 42580 butwhen 120588 = 090 the value of out-of-control ARL is 32223and when 119903119900 = 500 and shift is 002 then the value of out-of-control ARL for 120588 = 025 is 28338 but when 120588 = 090the value of out-of-control ARL is 13787 In the same wayTables 2 and 3 show a decreasing tendency in ARLs as thevalue of 120588 increases Also the trend of ARLs shows that theARLs decrease rapidly with the increasing values of shiftWhenwe comparedTables 1ndash3 an increasing trend inARLs isobserved as the value of smoothing constant 120582 increases withthe interval of 01 but at the same time a quick decreasingtrend in ARLs is also found when the value of 120588 increases



shift 1198961= 30958 119896

2= 2339 120582 = 03

f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 47840 47324 43391002 41312 39769 29994003 33388 31058 18933004 25947 23321 11732005 19780 17226 7338010 5070 3932 952015 1521 1101 230020 553 393 121025 254 191 103030 153 128 100035 118 108 100040 106 102 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 31751 30789 24408002 21608 19739 10891003 13478 11657 4840004 8308 6876 2306005 5201 4155 1188010 733 539 151015 208 165 102020 119 109 100025 102 101 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 02f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 34426 33892 29954002 27970 26539 18259003 20989 19099 10219004 15189 13301 5880005 10878 9216 3457010 2302 1751 424015 662 485 142020 265 203 104025 151 128 100030 115 106 100035 104 101 100040 101 100 10005 100 100 100



shift 1198961 = 30025 1198962 = 26008 120582 = 03f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 35410 35054 32319002 30861 27975 22805003 25242 23571 14727004 19873 17957 9341005 15354 13461 5986010 4224 3327 899015 1380 1026 248020 551 405 129025 272 207 105030 167 138 100035 126 113 100040 109 104 10005 101 100 100

Table 7 ARLs for proposed repetitive EWMA-RSS control chart using Khan et al [8] exponential ratio-type estimator when 119903119900 is 500shift 1198961 = 30958 1198962 = 2339 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 39336 37456 17762002 22636 19883 4157003 12112 9958 1181004 6561 5120 423005 3670 2746 202010 361 257 100015 123 110 100020 101 100 100025 100 100 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100








Moreover in Tables 4ndash6 for 119903119900 = 370 the same trends inARLs are observed as for 119903119900 = 500 Similarly Tables 7ndash12for the proposed repetitive EWMA-RSS control chart basedon Khan et al [8] exponential ratio-type estimator show arapid decreasing pattern in the values of ARLs as the level ofcorrelation increases and besides this an increasing trend inARLs values is also observed in these tables while the valueof smoothing constant 120582 increases with the interval of 01 Asper these tables compared in terms of the values of smoothingconstant 120582 an increasing trend in ARLs is observed as thevalue of smoothing constant120582 increases with the gap of 01 forthe proposed repetitive EWMA-RSS control charts under twodifferent exponential ratio-type estimators Itmeans that withsmall value of 120582 it detects the smaller shifts in the processquickly as compared to the large values of 120582 or we can saythat the small value of 120582 is good to detect smaller shifts in theprocess mean

41 Performance Efficiency Comparison between the ProposedRepetitive EWMA-RSS Control Charts based on Ratio Estima-tors The efficiency comparison between the performancesof the proposed repetitive EWMA-RSS charts based onexponential ratio-type estimators developed by Bahl andTuteja (1991) and Khan et al [8] is made here on thebasis of ARLs The ARLs tables are set up for comparingthe performance of the proposed repetitive EWMA controlcharts based on Bahl and Tuteja (BT) and Khan et al (HK)exponential ratio-type estimators under ranked set sampling

In each given table (Tables 13ndash18) the values of ARLs forthe proposed repetitive EWMA-RSS charts based on BT andHK ratio estimators are arranged according to the variousrandom small shifts from 0 to 05 for a specified value of120588 at the value of smoothing constant 120582 = 01 The values of1198711 119886119899119889 1198712 determined for 119903119900 = 500 370 are also specified inthe tables

From Tables 12ndash18 it is observed that the repetitiveEWMA-RSS control chart using Khan et al (HK) estimator

Table 13 ARLs for proposed repetitive EWMA-RSS control chartsusing exponential ratio-type estimators for 120588 = 025 when 119903119900 is 500at 120582 = 01

Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 42580 39336002 28338 22636003 17249 12112004 10370 6561005 6324 3670010 761 361015 191 123020 112 101025 101 100030 100 100035 100 100040 100 10005 100 100

outperforms the proposed repetitive EWMA-RSS controlchart based on Bahl and Tuteja (BT) by detecting small shiftsmuch earlier as it produces considerably smaller values ofARLs for different small shifts 0 to 05 For example inTable 13 the ARL value produced by EWMA-RSS chart usingKhan et al [8] exponential ratio-type estimator is 39336 forshift 001 which is much smaller than the values of ARLsproduced by EWMA-RSS chart using Bahl and Tuteja (1991)exponential ratio-type estimator for the same shift 001

42 ARLs Graphs for the Proposed Repetitive EWMA-RSSControl Charts ARLs graphs for the proposed repetitiveEWMA-RSS control charts based on Bahl and Tuteja (1991)and Khan et al [8] exponential ratio-type estimators based



Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 41209 37456002 25761 19883003 14808 9958004 8495 5120005 4982 2746010 550 257015 151 110020 105 100025 100 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 32223 17762002 13787 4157003 5859 1181004 2654 423005 1293 202010 939 100015 101 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

on ranked set sampling (for Tables 1ndash12) are given inAppendix A Figures 1ndash12 show that under both proposedcharts the ARLs at rho = 090 are decreased sharply from theconsidered target value as compared to the ARLs at differentvalues of rho (in all the cases) but for the increasing valueof 120582 the gradual decreasing pattern in ARLs is observed inthe graphs Thus the graphical behavior of ARLs also givesan idea about the detected small shifts of mean in the processsooner for larger value of rho (120588) and smaller value of (120582)

ARLs graphs for performance efficiency comparisonbetween the proposed repetitive EWMA-RSS control charts(for Tables 13ndash18) are provided in Appendix B Figures 13ndash18show that the proposed EWMA-RSS control chart based on


Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 31751 27470002 21608 17455003 13478 9629004 8308 5385005 5201 3120010 733 374015 208 132020 119 103025 102 100030 100 100035 100 100040 100 10005 100 100

Table 17 Average run lengths for proposed repetitive EWMA-RSScontrol charts using exponential ratio-type estimators for 120588 = 050when 119903119900 is 370 at 120582 = 01

Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 30789 18739002 19739 11429003 11657 7995004 6876 4263005 4155 2380010 539 275015 165 116020 109 101025 101 100030 100 100035 100 100040 100 10005 100 100

Khan et al [8] exponential ratio-type estimator performsmore efficiently generally in all the cases than the proposedrepetitive EWMA-RSS control chart based onBahl andTuteja(1991) exponential ratio-type estimator by detecting shifts inthe processmean sooner and faster than the rest of the charts

43 Industrial Application Here we have used the industrialdata of Brinell hardness and tensile strength for the realbivariate process considered by Chen (1994)Wang and Chen(1998) and Sultan (1986) The variable of interest Brinellhardness is denoted by Y and the variable tensile strengthis considered as an auxiliary variable denoted by X with


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50

Figure 1 ARLs for proposed repetitive EWMA-RSS control chart using Bahl and Tuteja (1991) ratio estimator when 119903119900 = 500 at 120582 = 01

ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


average value120583 = 50Data set of size 25 for both variables is asfollows

Y 143 200 168 181 148 178 162 215 161 141 175 187 187 186 172 182 177 204 178 196 160 183 179 194 181X 342 570 475 534 478 515 459 591 484 473 573 585 582 570 494 572 506 551 509 579 455 539 512 575 185 (32)

By using this data we prepared the proposed repetitiveEWMA-RSS control chart based on Singh and Tailor [12]ratio-type estimator as this control chart has been foundrelatively more efficient among all the proposed charts in ourstudy

For the above data we have 120588 = 050 and by consideringthe target value of ARL (119903119900) 500 at 120582 = 01 we havedetermined the values of 1198961 119886119899119889 1198962 ie 1198961 = 309 1198962 =233 After computation the outer and inner control limits ofthe proposed repetitive EWMA-RSS control chart (Figure 19)using Khan et al [8] exponential ratio-type estimator for theabove data are

1198711198621198711 = 5004131198801198621198711 = 5155861198711198621198712 = 5022791198801198621198712 = 513721(33)

Since the above industrial data lies within inner and outercontrol limits and no point exists beyond the control limitsthe process is in control

5 Conclusion

Both the proposed repetitive EWMA-RSS charts designedusing Bahl and Tuteja (1991) and Khan et al [8] exponential



Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 24408 13859002 10891 3505003 4840 1094004 2306 432005 1188 219010 151 100015 102 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50

Figure 3 ARLs for proposed repetitive EWMA-RSS control chartusing Bahl and Tuteja (1991) estimator when 119903119900 = 500 at 120582 = 03

ratio-type estimators based on ranked set sampling RSS areexamined individually and collectively in order to observetheir performance efficiencies on the basis of ARLs TheARLs of both the proposed repetitive EWMA-RSS controlcharts have been evaluated using R codes The two pro-posed repetitive EWMA-RSS control charts show sensitivitytowards small or gradual changes in the process by the choiceof the weighting factor (120582) and the level of correlation (120588)The ARLs tables for the small random shifts in the processmean are set up by considering preconsidered target valuesof in-control ARLs ie 500 and 370 Both the proposedrepetitive EWMA-RSS control charts proved efficient becausethey detect shifts in the process mean earlier and quicker athigh level of correlation as compared to the other levels ofcorrelation between the study and the auxiliary variables

While examining the proposed repetitive EWMA-RSScontrol charts independently it is observed that in all cases

ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50

Figure 4 For proposed repetitive EWMA-RSS control chart usingBahl and Tuteja (1991) ratio estimator when 119903119900 = 370 at 120582 = 01

ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50

Figure 5 ARLs for proposed repetitive EWMA-RSS control chartusing Bahl and Tuteja (1991) ratio estimator when 119903119900 = 370 at 120582 =02

as the value of 120588 increases the pattern of out-of-control ARLstends to decrease rapidly and when the values of smoothingconstant 120582 increase this leads to the increasing trend in thevalues of out-of-control ARLs This can also be observedthrough the graphs of ARLs that is when 120588 increases theARLs curves rapidly decrease downward

While comparing the proposed repetitive EWMA-RSScontrol charts based on Bahl and Tuteja (1991) and Khanet al [8] exponential ratio-type estimators collectively it isrevealed that in all cases the proposed repetitive EWMA-RSS control chart using Khan et al [8] exponential ratio-typeestimator outperformed the proposed repetitive EWMA-RSScontrol chart using Bahl and Tuteja (1991) exponential ratio-type estimator chart by means of detecting small shifts in theprocess mean much earlier

Both the proposed repetitive EWMA-RSS control chartsbased on exponential ratio-type estimators have provencapable ofmonitoring processmean efficiently by keeping the

Journal of Probability and Statistics 13A

RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50

Figure 7 ARLs for proposed repetitive EWMA-RSS control chartusing Khan et al [8] ratio estimator when 119903119900 = 500 at 120582 = 01

ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


14 Journal of Probability and StatisticsA

RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50

Figure 13 ARLs for proposed repetitive EWMA-RSS control chartsusing ratio estimators for 120588 = 025 when 119903119900 = 500 at 120582 = 01

ARL

s

Shi (f)

ARLs for BT

ARLs for HKARLs for BTARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BTARLs for HKARLs for BTARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


50

505

51

515

52

1 2 3 4 5

EWM

A

Observations

EWMA StatistcsLCL1LCL2

UCL1UCL2

Figure 19 Proposed repetitive EWMA-RSS control chart for Indus-trial data

process on target through quick detection of the small shiftsin the process mean Hence it is recommended that thesecontrol charts must be used for future research work in orderto get proficient results that would help to ensure the qualityproducts

Appendix

A ARLs Graphs for the ProposedRepetitive EWMA-RSS Control ChartsBased on Exponential Ratio-typeEstimators (for Tables 1ndash12)

See Figures 1ndash12

B ARLs Graphs for PerformanceEfficiency Comparison between theProposed Repetitive EWMA-RSSControl Charts (for Tables 19ndash24)


Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A Haq J Brown and E Moltchanova ldquoNew ExponentiallyWeighted Moving Average Control Charts for MonitoringProcess Mean and Process Dispersionrdquo Quality and ReliabilityEngineering International 2014

[2] S A Abbasi M Riaz A Miller S Ahmad and H Z NazirldquoEWMA Dispersion Control Charts for Normal and Non-normal Processesrdquo Quality and Reliability Engineering Interna-tional 2014

[3] N Abbas M Riaz and R J M M Does ldquoAn EWMA-Type control chart for monitoring the process mean usingauxiliary informationrdquo Communications in StatisticsmdashTheoryand Methods vol 43 no 16 pp 3485ndash3498 2014

[4] A Haq ldquoAn improved mean deviation exponentially weightedmoving average control chart to monitor process dispersionunder ranked sets samplingrdquo Journal of Statistical Computationand Simulation vol 84 no 9 pp 2011ndash2024 2014

[5] AHaq J Brown E Moltchanova andA I Al-Omari ldquoEffect ofmeasurement error on exponentially weighted moving averagecontrol charts under ranked set sampling schemesrdquo Journal ofStatistical Computation and Simulation vol 85 no 6 pp 1224ndash1246 2015

[6] M Azam H Naz M Sattam Aldosari and M Aslam ldquoTheEWMA control chart for regression estimator based on rankedset repetitive samplingrdquo Quality - Access to Success vol 18 no160 pp 108ndash114 2017

[7] R A Sanusi M Riaz N A Adegoke and M Xie ldquoAnEWMAmonitoring scheme with a single auxiliary variable forindustrial processesrdquo Computers amp Industrial Engineering vol114 pp 1ndash10 2017

[8] HKhanA SanaullahMAKhan andA F Siddiqi ldquoImprovedExponential Ratio Type Estimators for Estimating Popula-tion Means regarding full Information in ServeySamplingrdquoSciInt(Lahore) vol 26 no 5 pp 1897ndash1902 2014

[9] DAWolfe ldquoRanked Set Sampling Its Relevance and Impact onStatistical Inferencerdquo International Scholarly Research NetworkISRN Probability and Statistics pp 1ndash32 2012

[10] S Demir and H Singh ldquoAn application of the regressionestimates to ranked set samplingrdquo Hacettepe Bulletin of NaturalSciences and Engineering Series B vol 29 pp 93ndash101 2000

[11] S K Yadav S S Mishra andA K Shukla ldquoDeveloping efficientratio and product type exponential estimators of populationmean under two phase sampling for stratificationrdquo Journal ofOperational Research vol 5 no 2 pp 21ndash28 2015

[12] H P Singh and R Tailor ldquoUse of known correlation coefficientin estimating the finite population meanrdquo Statistics in Transi-tion vol 6 pp 555ndash560 2003

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for managing procedure dispersion based on normal anda range non-normal distribution The investigated EWMAdispersion control charts were based on an extensive varietyof dispersion estimators Abbas et al [3] recommended a newEWMA-type control chart that uses a single auxiliary variableknown as 119872119883119864119882119872119860 control chart For the estimation ofmean in defining the control constitution of the designedcontrol chart regression estimator was used Haq [4] recom-mended an enhancedmean deviation exponentially weightedmoving average (IMDEWMA) control chart based on rankset sampling to control process dispersion Haq et al [5]studied the effect of measurements errors on the detectionability of EWMA control charts for controlling processmean under ranked set sampling (RSS) median ranked setsampling (MRSS) imperfect ranked set sampling (IRSS) andimperfect median ranked set sampling (IMRSS) schemesAzam et al [6] presented repetitive exponentially weightedmoving average (EWMA) control chart using regressionestimator based on ranked set sampling (RSS) design toexamine and detect the changes in the manufacturing pro-cess They studied the detection ability of the proposedcontrol chart for monitoring shifts in the process meanRidwan et al [7] developed an EWMA scheme using ratioestimator to increase the effectiveness of typical EWMAchartin monitoring the location parameter

Although in efficiency comparison the EWMA charts arefound parallel to the Shewhart control charts these are saidto be best alternatives to the Shewhart charts in monitoringsmall shifts in the process mean because they provide quickalarms when small shifts are introduced in the process

2 Methodology

This study proposes EWMA-type control charts by consid-ering some auxiliary information provided by an auxiliaryvariable The ratio estimation technique for the mean withranked set sampling design is used in designing the controlstructure of the proposed charts Here we developed EWMAcontrol charts using Bahl and Tuteja (1991) exponential ratio-type estimator and Khan et al [8] exponential ratio-typeestimator based on ranked set sampling design for theprocess mean to obtain specific ARLs being suitable whensmall process shifts are of interest

21 Ranked Set Sampling (RSS) The idea of RSS was firstgiven by Mcintyre (1952) The mechanism of RSS design isbased on the ranking of sampling units within the samplesand it makes the collection of actual measurements ofsampling units more feasible and reliable as compared toSRS design with respect to simplicity time cost or othercomplicated factors

This sampling technique obtains samples from a popu-lation in such a way that the extent of information coversthe entire range of observations in the population In thisway ranked set sample becomes more representative thanthe simple random sample obtained by identical number ofobservations from the same population [9] The structure ofRSS design is simply described as follows Firstly n2 sampling

units are identified from a large or infinite population Theseunits are further assigned randomly to the n equal sizedsamples After that ranking is imposed to the units withineach sample with respect to some characteristics of interestor study variable A mixture of mechanisms can be used toacquire this ranking comprising visual comparison expertjudgment or the use of auxiliary variables however it cannotinclude actual measurements of the characteristics of intereston the selected units [10]

Let y11y12 y13 1199101119899 11991021 11991022 119910231199101198991 1199101198992 1199101198993 119910119899119899be lsquonrsquo independent simple random samples each of size nImplement the RSS process to these n samples to obtain aranked set sample (RSS) of size t defined by y1(1n) y2(2n)y3(3n) yt(nn)

Now RSS estimator for 119910 is defined as

119910119877119878119878 = 1119905119899119905sum119894=1

119899sum119894=1

[119910119894(119894119899)] (1)

with mean

120583119910 = 119864 (119910119877119878119878) = 119884 (2)

and variance

var (119910119877119878119878) = 1205902119899 minus 11198992119899sum119894=1

(120583(119894) minus 120583119910)2 (3)

var (119910119877119878119878) = var (119910119878119877119878) minus 11198992119899sum119894=1

(120583(119894) minus 120583119910)2 (4)

This estimator is unbiased and is also more precise thansimple random sampling estimator 11991011987811987711987822 Ratio Estimator In sampling surveys while estimat-ing the population parameters role of auxiliary variableis very significant to increase its efficiency Many authorshave improved the precision of their estimators throughthe use of auxiliary variable Auxiliary variable which ishighly correlated to the variable of interest actually providesinformation intended to improve the efficiency of the esti-mator of population parameter of the study or main variable[11]

If 119910119894 (119894 = 1 2 3 119873) is the observations for thevariable of interest 119910 from a finite population and if 119909119894 (119894 =1 2 3 119873) is the observations of auxiliary variable119909whoseinformation is completely known and is highly correlated tothe variable under consideration then the traditional ratioestimator for 119884 is defined as

yR = 119910(119883119909 ) (5)

The complete information of parameters of the auxiliaryvariable 119909 such as 119862119909 (coefficient of variation) and 1205732(119909)(coefficient of kurtosis) basically help to increase efficiencyof the estimator of the study variable (Yadav et al 2014)





1198721198771198781198781199101 = 119910119894(119894119899)119905 [exp(119883 minus 119909119894(119894119899)119905119883 + 119909119894(119894119899)119905)] (6)


E (1198721198771198781198781199101 ) = 119884 = 120583119910var (1198721198771198781198781199101 ) = 1205951198842 [1198622119910 + 141198622119909 (1 minus 119862)]

minus 11198992119899sum119894=1



1198831ℎ minus 119909119894(119894119899)1199051ℎ1198831ℎ + (119886 minus 1) 119909119894(119894119899)1199051ℎ)]] (8)



[120583(119894) minus 120583119910]2

there4120595 = 119873 minus 119899119873119899(9)


119872119877119878119878119896119894 = 120582119872119877119878119878119910119896 + (1ndash120582)119872119877119878119878119896(119894minus1) (10)




(11)


var (119872119877119878119878119896119894 ) = 1205822 minus 120582var (119872119877119878119878119910119896 ) (12)



119899sum119894=1

(120583(119894) minus 120583119910)2] (13)


119899sum119894=1

(120583(119894) minus 120583119910)2](14)



(15)



1198751199001199001199061199051 = 119875 (119872119877119878119878119896119894 lt 1198711198621198711) + 119875 (119872119877119878119878119896119894 gt 1198801198621198711) (16)





1198751199001199001199061199051 = 119875 (119885 lt minus1198711) + 119875 (119885 gt 1198711) (18)




119875119900119877119864119875 = 119875 1198801198621198712 lt 119872119877119878119878119896119894 lt 1198801198621198711+ 119875 1198711198621198711 lt 119872119877119878119878119896119894 lt 1198711198621198712 (20)



(21)

so

119875119900119900119906119905 = 11987511990011990011990611990511 minus 119875119900119877119864119875 (22)

119860119877119871119900 = 1119875119900119900119906119905 (23)



(24)


11987511199001199061199051 = 119875 (119872119877119878119878119896119894 lt 1198711198621198711 | 1205831)+ 119875 (119872119877119878119878119896119894 gt 1198801198621198711 | 1205831) where k = 1 2 (25)


11987511199001199061199051 = 119875(119885 lt minus1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))

+ 119875(119885 gt 1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))



(26)


1198751119877119864119875 = 1198751198801198621198712 lt 119872119877119878119878119896119894 lt 11988011986211987111205831 + 1198751198711198621198711 lt 119872119877119878119878119896119894 lt 11987111986211987121205831

(27)


1198751119877119864119875= 119875[1198712 minus 119891120590119910119904119889 (119872119877119878119878119896119894 ) lt 119885 lt 1198711 minus

119891120590119910119904119889 (119872119877119878119878119896119894 )]+ 119875[minus1198711 minus 119891120590119910119904119889 (119872119877119878119878

119896119894) lt 119885 lt minus1198712 minus

119891120590119910119904119889 (119872119877119878119878119896119894)]

(28)

and then

1198751119877119864119875 = Oslash(1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))


119896119894))


(29)

so

1198751119900119906119905 = 119875111990011990611990511 minus 1198751119877119864119875 (30)

1198601198771198711 = 11198751119900119906119905 (31)





shift 1198961 = 30958 1198962 = 2339 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 42580 41209 32223002 28338 25761 13787003 17249 14808 5859004 10370 8495 2654005 6324 4982 1293010 761 550 939015 191 151 101020 112 105 100025 101 100 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100


shift 1198961 = 30958 1198962 = 2339 120582 = 02f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 46417 45648 40022002 37217 35205 23732003 27482 24883 13027004 19557 17011 7201005 13770 11566 4096010 2649 1971 415015 678 481 131020 248 187 102025 139 119 100030 109 104 100035 102 100 100040 100 100 10005 100 100 100







shift 1198961= 30958 119896

2= 2339 120582 = 03

f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 47840 47324 43391002 41312 39769 29994003 33388 31058 18933004 25947 23321 11732005 19780 17226 7338010 5070 3932 952015 1521 1101 230020 553 393 121025 254 191 103030 153 128 100035 118 108 100040 106 102 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 31751 30789 24408002 21608 19739 10891003 13478 11657 4840004 8308 6876 2306005 5201 4155 1188010 733 539 151015 208 165 102020 119 109 100025 102 101 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 02f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 34426 33892 29954002 27970 26539 18259003 20989 19099 10219004 15189 13301 5880005 10878 9216 3457010 2302 1751 424015 662 485 142020 265 203 104025 151 128 100030 115 106 100035 104 101 100040 101 100 10005 100 100 100



shift 1198961 = 30025 1198962 = 26008 120582 = 03f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 35410 35054 32319002 30861 27975 22805003 25242 23571 14727004 19873 17957 9341005 15354 13461 5986010 4224 3327 899015 1380 1026 248020 551 405 129025 272 207 105030 167 138 100035 126 113 100040 109 104 10005 101 100 100














Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 42580 39336002 28338 22636003 17249 12112004 10370 6561005 6324 3670010 761 361015 191 123020 112 101025 101 100030 100 100035 100 100040 100 10005 100 100





Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 41209 37456002 25761 19883003 14808 9958004 8495 5120005 4982 2746010 550 257015 151 110020 105 100025 100 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 32223 17762002 13787 4157003 5859 1181004 2654 423005 1293 202010 939 100015 101 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 31751 27470002 21608 17455003 13478 9629004 8308 5385005 5201 3120010 733 374015 208 132020 119 103025 102 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 30789 18739002 19739 11429003 11657 7995004 6876 4263005 4155 2380010 539 275015 165 116020 109 101025 101 100030 100 100035 100 100040 100 10005 100 100




ARL

s

Shi (f)

ARLs at

ARLs at

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roh=025

roh=050

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ARLs at

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ARLs at

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Y 143 200 168 181 148 178 162 215 161 141 175 187 187 186 172 182 177 204 178 196 160 183 179 194 181X 342 570 475 534 478 515 459 591 484 473 573 585 582 570 494 572 506 551 509 579 455 539 512 575 185 (32)



1198711198621198711 = 5004131198801198621198711 = 5155861198711198621198712 = 5022791198801198621198712 = 513721(33)


5 Conclusion




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 24408 13859002 10891 3505003 4840 1094004 2306 432005 1188 219010 151 100015 102 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

ARL

s

Shi (f)

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1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018












1198721198771198781198781199101 = 119910119894(119894119899)119905 [exp(119883 minus 119909119894(119894119899)119905119883 + 119909119894(119894119899)119905)] (6)


E (1198721198771198781198781199101 ) = 119884 = 120583119910var (1198721198771198781198781199101 ) = 1205951198842 [1198622119910 + 141198622119909 (1 minus 119862)]

minus 11198992119899sum119894=1



1198831ℎ minus 119909119894(119894119899)1199051ℎ1198831ℎ + (119886 minus 1) 119909119894(119894119899)1199051ℎ)]] (8)



[120583(119894) minus 120583119910]2

there4120595 = 119873 minus 119899119873119899(9)


119872119877119878119878119896119894 = 120582119872119877119878119878119910119896 + (1ndash120582)119872119877119878119878119896(119894minus1) (10)




(11)


var (119872119877119878119878119896119894 ) = 1205822 minus 120582var (119872119877119878119878119910119896 ) (12)



119899sum119894=1

(120583(119894) minus 120583119910)2] (13)


119899sum119894=1

(120583(119894) minus 120583119910)2](14)



(15)



1198751199001199001199061199051 = 119875 (119872119877119878119878119896119894 lt 1198711198621198711) + 119875 (119872119877119878119878119896119894 gt 1198801198621198711) (16)





1198751199001199001199061199051 = 119875 (119885 lt minus1198711) + 119875 (119885 gt 1198711) (18)




119875119900119877119864119875 = 119875 1198801198621198712 lt 119872119877119878119878119896119894 lt 1198801198621198711+ 119875 1198711198621198711 lt 119872119877119878119878119896119894 lt 1198711198621198712 (20)



(21)

so

119875119900119900119906119905 = 11987511990011990011990611990511 minus 119875119900119877119864119875 (22)

119860119877119871119900 = 1119875119900119900119906119905 (23)



(24)


11987511199001199061199051 = 119875 (119872119877119878119878119896119894 lt 1198711198621198711 | 1205831)+ 119875 (119872119877119878119878119896119894 gt 1198801198621198711 | 1205831) where k = 1 2 (25)


11987511199001199061199051 = 119875(119885 lt minus1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))

+ 119875(119885 gt 1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))



(26)


1198751119877119864119875 = 1198751198801198621198712 lt 119872119877119878119878119896119894 lt 11988011986211987111205831 + 1198751198711198621198711 lt 119872119877119878119878119896119894 lt 11987111986211987121205831

(27)


1198751119877119864119875= 119875[1198712 minus 119891120590119910119904119889 (119872119877119878119878119896119894 ) lt 119885 lt 1198711 minus

119891120590119910119904119889 (119872119877119878119878119896119894 )]+ 119875[minus1198711 minus 119891120590119910119904119889 (119872119877119878119878

119896119894) lt 119885 lt minus1198712 minus

119891120590119910119904119889 (119872119877119878119878119896119894)]

(28)

and then

1198751119877119864119875 = Oslash(1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))


119896119894))


(29)

so

1198751119900119906119905 = 119875111990011990611990511 minus 1198751119877119864119875 (30)

1198601198771198711 = 11198751119900119906119905 (31)





shift 1198961 = 30958 1198962 = 2339 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 42580 41209 32223002 28338 25761 13787003 17249 14808 5859004 10370 8495 2654005 6324 4982 1293010 761 550 939015 191 151 101020 112 105 100025 101 100 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100


shift 1198961 = 30958 1198962 = 2339 120582 = 02f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 46417 45648 40022002 37217 35205 23732003 27482 24883 13027004 19557 17011 7201005 13770 11566 4096010 2649 1971 415015 678 481 131020 248 187 102025 139 119 100030 109 104 100035 102 100 100040 100 100 10005 100 100 100







shift 1198961= 30958 119896

2= 2339 120582 = 03

f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 47840 47324 43391002 41312 39769 29994003 33388 31058 18933004 25947 23321 11732005 19780 17226 7338010 5070 3932 952015 1521 1101 230020 553 393 121025 254 191 103030 153 128 100035 118 108 100040 106 102 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 31751 30789 24408002 21608 19739 10891003 13478 11657 4840004 8308 6876 2306005 5201 4155 1188010 733 539 151015 208 165 102020 119 109 100025 102 101 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 02f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 34426 33892 29954002 27970 26539 18259003 20989 19099 10219004 15189 13301 5880005 10878 9216 3457010 2302 1751 424015 662 485 142020 265 203 104025 151 128 100030 115 106 100035 104 101 100040 101 100 10005 100 100 100



shift 1198961 = 30025 1198962 = 26008 120582 = 03f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 35410 35054 32319002 30861 27975 22805003 25242 23571 14727004 19873 17957 9341005 15354 13461 5986010 4224 3327 899015 1380 1026 248020 551 405 129025 272 207 105030 167 138 100035 126 113 100040 109 104 10005 101 100 100














Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 42580 39336002 28338 22636003 17249 12112004 10370 6561005 6324 3670010 761 361015 191 123020 112 101025 101 100030 100 100035 100 100040 100 10005 100 100





Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 41209 37456002 25761 19883003 14808 9958004 8495 5120005 4982 2746010 550 257015 151 110020 105 100025 100 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 32223 17762002 13787 4157003 5859 1181004 2654 423005 1293 202010 939 100015 101 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 31751 27470002 21608 17455003 13478 9629004 8308 5385005 5201 3120010 733 374015 208 132020 119 103025 102 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 30789 18739002 19739 11429003 11657 7995004 6876 4263005 4155 2380010 539 275015 165 116020 109 101025 101 100030 100 100035 100 100040 100 10005 100 100




ARL

s

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ARLs at

ARLs at

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Y 143 200 168 181 148 178 162 215 161 141 175 187 187 186 172 182 177 204 178 196 160 183 179 194 181X 342 570 475 534 478 515 459 591 484 473 573 585 582 570 494 572 506 551 509 579 455 539 512 575 185 (32)



1198711198621198711 = 5004131198801198621198711 = 5155861198711198621198712 = 5022791198801198621198712 = 513721(33)


5 Conclusion




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 24408 13859002 10891 3505003 4840 1094004 2306 432005 1188 219010 151 100015 102 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

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ARLs at

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515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018










1198751199001199001199061199051 = 119875 (119885 lt minus1198711) + 119875 (119885 gt 1198711) (18)




119875119900119877119864119875 = 119875 1198801198621198712 lt 119872119877119878119878119896119894 lt 1198801198621198711+ 119875 1198711198621198711 lt 119872119877119878119878119896119894 lt 1198711198621198712 (20)



(21)

so

119875119900119900119906119905 = 11987511990011990011990611990511 minus 119875119900119877119864119875 (22)

119860119877119871119900 = 1119875119900119900119906119905 (23)



(24)


11987511199001199061199051 = 119875 (119872119877119878119878119896119894 lt 1198711198621198711 | 1205831)+ 119875 (119872119877119878119878119896119894 gt 1198801198621198711 | 1205831) where k = 1 2 (25)


11987511199001199061199051 = 119875(119885 lt minus1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))

+ 119875(119885 gt 1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))



(26)


1198751119877119864119875 = 1198751198801198621198712 lt 119872119877119878119878119896119894 lt 11988011986211987111205831 + 1198751198711198621198711 lt 119872119877119878119878119896119894 lt 11987111986211987121205831

(27)


1198751119877119864119875= 119875[1198712 minus 119891120590119910119904119889 (119872119877119878119878119896119894 ) lt 119885 lt 1198711 minus

119891120590119910119904119889 (119872119877119878119878119896119894 )]+ 119875[minus1198711 minus 119891120590119910119904119889 (119872119877119878119878

119896119894) lt 119885 lt minus1198712 minus

119891120590119910119904119889 (119872119877119878119878119896119894)]

(28)

and then

1198751119877119864119875 = Oslash(1198711 minus 119891120590119910119904119889 (119872119877119878119878119896119894))


119896119894))


(29)

so

1198751119900119906119905 = 119875111990011990611990511 minus 1198751119877119864119875 (30)

1198601198771198711 = 11198751119900119906119905 (31)





shift 1198961 = 30958 1198962 = 2339 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 42580 41209 32223002 28338 25761 13787003 17249 14808 5859004 10370 8495 2654005 6324 4982 1293010 761 550 939015 191 151 101020 112 105 100025 101 100 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100


shift 1198961 = 30958 1198962 = 2339 120582 = 02f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 46417 45648 40022002 37217 35205 23732003 27482 24883 13027004 19557 17011 7201005 13770 11566 4096010 2649 1971 415015 678 481 131020 248 187 102025 139 119 100030 109 104 100035 102 100 100040 100 100 10005 100 100 100







shift 1198961= 30958 119896

2= 2339 120582 = 03

f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 47840 47324 43391002 41312 39769 29994003 33388 31058 18933004 25947 23321 11732005 19780 17226 7338010 5070 3932 952015 1521 1101 230020 553 393 121025 254 191 103030 153 128 100035 118 108 100040 106 102 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 31751 30789 24408002 21608 19739 10891003 13478 11657 4840004 8308 6876 2306005 5201 4155 1188010 733 539 151015 208 165 102020 119 109 100025 102 101 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 02f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 34426 33892 29954002 27970 26539 18259003 20989 19099 10219004 15189 13301 5880005 10878 9216 3457010 2302 1751 424015 662 485 142020 265 203 104025 151 128 100030 115 106 100035 104 101 100040 101 100 10005 100 100 100



shift 1198961 = 30025 1198962 = 26008 120582 = 03f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 35410 35054 32319002 30861 27975 22805003 25242 23571 14727004 19873 17957 9341005 15354 13461 5986010 4224 3327 899015 1380 1026 248020 551 405 129025 272 207 105030 167 138 100035 126 113 100040 109 104 10005 101 100 100














Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 42580 39336002 28338 22636003 17249 12112004 10370 6561005 6324 3670010 761 361015 191 123020 112 101025 101 100030 100 100035 100 100040 100 10005 100 100





Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 41209 37456002 25761 19883003 14808 9958004 8495 5120005 4982 2746010 550 257015 151 110020 105 100025 100 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 32223 17762002 13787 4157003 5859 1181004 2654 423005 1293 202010 939 100015 101 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 31751 27470002 21608 17455003 13478 9629004 8308 5385005 5201 3120010 733 374015 208 132020 119 103025 102 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 30789 18739002 19739 11429003 11657 7995004 6876 4263005 4155 2380010 539 275015 165 116020 109 101025 101 100030 100 100035 100 100040 100 10005 100 100




ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

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ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

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00

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Y 143 200 168 181 148 178 162 215 161 141 175 187 187 186 172 182 177 204 178 196 160 183 179 194 181X 342 570 475 534 478 515 459 591 484 473 573 585 582 570 494 572 506 551 509 579 455 539 512 575 185 (32)



1198711198621198711 = 5004131198801198621198711 = 5155861198711198621198712 = 5022791198801198621198712 = 513721(33)


5 Conclusion




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 24408 13859002 10891 3505003 4840 1094004 2306 432005 1188 219010 151 100015 102 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

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20000

10000

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00

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ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

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ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

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01

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RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

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ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

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ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

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ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

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ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

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ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

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RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

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00

00

01

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ARL

s

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

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ARL

s

Shi (f)

ARLs for BT


60000

50000

40000

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s

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s

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ARL

s

Shi (f)

ARLs for BT


40000

30000

20000

10000

00

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01

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RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

03

04

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50

505

51

515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018










shift 1198961 = 30958 1198962 = 2339 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 42580 41209 32223002 28338 25761 13787003 17249 14808 5859004 10370 8495 2654005 6324 4982 1293010 761 550 939015 191 151 101020 112 105 100025 101 100 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100


shift 1198961 = 30958 1198962 = 2339 120582 = 02f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 46417 45648 40022002 37217 35205 23732003 27482 24883 13027004 19557 17011 7201005 13770 11566 4096010 2649 1971 415015 678 481 131020 248 187 102025 139 119 100030 109 104 100035 102 100 100040 100 100 10005 100 100 100







shift 1198961= 30958 119896

2= 2339 120582 = 03

f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 47840 47324 43391002 41312 39769 29994003 33388 31058 18933004 25947 23321 11732005 19780 17226 7338010 5070 3932 952015 1521 1101 230020 553 393 121025 254 191 103030 153 128 100035 118 108 100040 106 102 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 31751 30789 24408002 21608 19739 10891003 13478 11657 4840004 8308 6876 2306005 5201 4155 1188010 733 539 151015 208 165 102020 119 109 100025 102 101 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 02f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 34426 33892 29954002 27970 26539 18259003 20989 19099 10219004 15189 13301 5880005 10878 9216 3457010 2302 1751 424015 662 485 142020 265 203 104025 151 128 100030 115 106 100035 104 101 100040 101 100 10005 100 100 100



shift 1198961 = 30025 1198962 = 26008 120582 = 03f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 35410 35054 32319002 30861 27975 22805003 25242 23571 14727004 19873 17957 9341005 15354 13461 5986010 4224 3327 899015 1380 1026 248020 551 405 129025 272 207 105030 167 138 100035 126 113 100040 109 104 10005 101 100 100














Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 42580 39336002 28338 22636003 17249 12112004 10370 6561005 6324 3670010 761 361015 191 123020 112 101025 101 100030 100 100035 100 100040 100 10005 100 100





Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 41209 37456002 25761 19883003 14808 9958004 8495 5120005 4982 2746010 550 257015 151 110020 105 100025 100 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 32223 17762002 13787 4157003 5859 1181004 2654 423005 1293 202010 939 100015 101 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 31751 27470002 21608 17455003 13478 9629004 8308 5385005 5201 3120010 733 374015 208 132020 119 103025 102 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 30789 18739002 19739 11429003 11657 7995004 6876 4263005 4155 2380010 539 275015 165 116020 109 101025 101 100030 100 100035 100 100040 100 10005 100 100




ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

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50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

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Y 143 200 168 181 148 178 162 215 161 141 175 187 187 186 172 182 177 204 178 196 160 183 179 194 181X 342 570 475 534 478 515 459 591 484 473 573 585 582 570 494 572 506 551 509 579 455 539 512 575 185 (32)



1198711198621198711 = 5004131198801198621198711 = 5155861198711198621198712 = 5022791198801198621198712 = 513721(33)


5 Conclusion




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 24408 13859002 10891 3505003 4840 1094004 2306 432005 1188 219010 151 100015 102 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

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ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

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ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

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20000

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RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

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ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

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ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

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ARL

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ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

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ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

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ARs at

ARLs at

ARLs at

roh=025

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ARs at

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roh=025

roh=050

roh=090

ARs at

ARLs at

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roh=025

roh=050

roh=090

40000

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ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

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RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

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ARL

s

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

60000

50000

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ARLs for BT


60000

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ARL

s

Shi (f)

ARLs for BT


40000

30000

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RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

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10

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40

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50

505

51

515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018










shift 1198961= 30958 119896

2= 2339 120582 = 03

f 120588 = 025 120588 = 050 120588 = 0900 50062 50062 50062001 47840 47324 43391002 41312 39769 29994003 33388 31058 18933004 25947 23321 11732005 19780 17226 7338010 5070 3932 952015 1521 1101 230020 553 393 121025 254 191 103030 153 128 100035 118 108 100040 106 102 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 01f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 31751 30789 24408002 21608 19739 10891003 13478 11657 4840004 8308 6876 2306005 5201 4155 1188010 733 539 151015 208 165 102020 119 109 100025 102 101 100030 100 100 100035 100 100 100040 100 100 10005 100 100 100


shift 1198961 = 30025 1198962 = 26008 120582 = 02f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 34426 33892 29954002 27970 26539 18259003 20989 19099 10219004 15189 13301 5880005 10878 9216 3457010 2302 1751 424015 662 485 142020 265 203 104025 151 128 100030 115 106 100035 104 101 100040 101 100 10005 100 100 100



shift 1198961 = 30025 1198962 = 26008 120582 = 03f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 35410 35054 32319002 30861 27975 22805003 25242 23571 14727004 19873 17957 9341005 15354 13461 5986010 4224 3327 899015 1380 1026 248020 551 405 129025 272 207 105030 167 138 100035 126 113 100040 109 104 10005 101 100 100














Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 42580 39336002 28338 22636003 17249 12112004 10370 6561005 6324 3670010 761 361015 191 123020 112 101025 101 100030 100 100035 100 100040 100 10005 100 100





Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 41209 37456002 25761 19883003 14808 9958004 8495 5120005 4982 2746010 550 257015 151 110020 105 100025 100 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 32223 17762002 13787 4157003 5859 1181004 2654 423005 1293 202010 939 100015 101 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 31751 27470002 21608 17455003 13478 9629004 8308 5385005 5201 3120010 733 374015 208 132020 119 103025 102 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 30789 18739002 19739 11429003 11657 7995004 6876 4263005 4155 2380010 539 275015 165 116020 109 101025 101 100030 100 100035 100 100040 100 10005 100 100




ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



Y 143 200 168 181 148 178 162 215 161 141 175 187 187 186 172 182 177 204 178 196 160 183 179 194 181X 342 570 475 534 478 515 459 591 484 473 573 585 582 570 494 572 506 551 509 579 455 539 512 575 185 (32)



1198711198621198711 = 5004131198801198621198711 = 5155861198711198621198712 = 5022791198801198621198712 = 513721(33)


5 Conclusion




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 24408 13859002 10891 3505003 4840 1094004 2306 432005 1188 219010 151 100015 102 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50




ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50






RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


50

505

51

515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018










shift 1198961 = 30025 1198962 = 26008 120582 = 03f 120588 = 025 120588 = 050 120588 = 0900 37098 37098 37098001 35410 35054 32319002 30861 27975 22805003 25242 23571 14727004 19873 17957 9341005 15354 13461 5986010 4224 3327 899015 1380 1026 248020 551 405 129025 272 207 105030 167 138 100035 126 113 100040 109 104 10005 101 100 100














Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 42580 39336002 28338 22636003 17249 12112004 10370 6561005 6324 3670010 761 361015 191 123020 112 101025 101 100030 100 100035 100 100040 100 10005 100 100





Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 41209 37456002 25761 19883003 14808 9958004 8495 5120005 4982 2746010 550 257015 151 110020 105 100025 100 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 32223 17762002 13787 4157003 5859 1181004 2654 423005 1293 202010 939 100015 101 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 31751 27470002 21608 17455003 13478 9629004 8308 5385005 5201 3120010 733 374015 208 132020 119 103025 102 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 30789 18739002 19739 11429003 11657 7995004 6876 4263005 4155 2380010 539 275015 165 116020 109 101025 101 100030 100 100035 100 100040 100 10005 100 100




ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



Y 143 200 168 181 148 178 162 215 161 141 175 187 187 186 172 182 177 204 178 196 160 183 179 194 181X 342 570 475 534 478 515 459 591 484 473 573 585 582 570 494 572 506 551 509 579 455 539 512 575 185 (32)



1198711198621198711 = 5004131198801198621198711 = 5155861198711198621198712 = 5022791198801198621198712 = 513721(33)


5 Conclusion




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 24408 13859002 10891 3505003 4840 1094004 2306 432005 1188 219010 151 100015 102 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50




ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50






RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


50

505

51

515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018



















Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 42580 39336002 28338 22636003 17249 12112004 10370 6561005 6324 3670010 761 361015 191 123020 112 101025 101 100030 100 100035 100 100040 100 10005 100 100





Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 41209 37456002 25761 19883003 14808 9958004 8495 5120005 4982 2746010 550 257015 151 110020 105 100025 100 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 32223 17762002 13787 4157003 5859 1181004 2654 423005 1293 202010 939 100015 101 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 31751 27470002 21608 17455003 13478 9629004 8308 5385005 5201 3120010 733 374015 208 132020 119 103025 102 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 30789 18739002 19739 11429003 11657 7995004 6876 4263005 4155 2380010 539 275015 165 116020 109 101025 101 100030 100 100035 100 100040 100 10005 100 100




ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



Y 143 200 168 181 148 178 162 215 161 141 175 187 187 186 172 182 177 204 178 196 160 183 179 194 181X 342 570 475 534 478 515 459 591 484 473 573 585 582 570 494 572 506 551 509 579 455 539 512 575 185 (32)



1198711198621198711 = 5004131198801198621198711 = 5155861198711198621198712 = 5022791198801198621198712 = 513721(33)


5 Conclusion




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 24408 13859002 10891 3505003 4840 1094004 2306 432005 1188 219010 151 100015 102 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50




ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50






RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


50

505

51

515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018










Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 41209 37456002 25761 19883003 14808 9958004 8495 5120005 4982 2746010 550 257015 151 110020 105 100025 100 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30958 1198962= 2339f BT HK0 50062 50062001 32223 17762002 13787 4157003 5859 1181004 2654 423005 1293 202010 939 100015 101 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 31751 27470002 21608 17455003 13478 9629004 8308 5385005 5201 3120010 733 374015 208 132020 119 103025 102 100030 100 100035 100 100040 100 10005 100 100


Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 30789 18739002 19739 11429003 11657 7995004 6876 4263005 4155 2380010 539 275015 165 116020 109 101025 101 100030 100 100035 100 100040 100 10005 100 100




ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



Y 143 200 168 181 148 178 162 215 161 141 175 187 187 186 172 182 177 204 178 196 160 183 179 194 181X 342 570 475 534 478 515 459 591 484 473 573 585 582 570 494 572 506 551 509 579 455 539 512 575 185 (32)



1198711198621198711 = 5004131198801198621198711 = 5155861198711198621198712 = 5022791198801198621198712 = 513721(33)


5 Conclusion




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 24408 13859002 10891 3505003 4840 1094004 2306 432005 1188 219010 151 100015 102 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50




ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50






RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


50

505

51

515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018









ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



Y 143 200 168 181 148 178 162 215 161 141 175 187 187 186 172 182 177 204 178 196 160 183 179 194 181X 342 570 475 534 478 515 459 591 484 473 573 585 582 570 494 572 506 551 509 579 455 539 512 575 185 (32)



1198711198621198711 = 5004131198801198621198711 = 5155861198711198621198712 = 5022791198801198621198712 = 513721(33)


5 Conclusion




Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 24408 13859002 10891 3505003 4840 1094004 2306 432005 1188 219010 151 100015 102 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50




ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50






RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


50

505

51

515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018










Shift 1198961 = 30025 1198962 = 26008f BT HK0 37098 37098001 24408 13859002 10891 3505003 4840 1094004 2306 432005 1188 219010 151 100015 102 100020 100 100025 100 100030 100 100035 100 100040 100 10005 100 100

ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50




ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARLs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50






RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


50

505

51

515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018









RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


50

505

51

515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018









RLs

Shi (f)

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

ARs at

ARLs at

ARLs at

roh=025

roh=050

roh=090

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


60000

50000

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


ARL

s

Shi (f)

ARLs for BT


40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50



RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


50

505

51

515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018









RLs

Shi (f)

ARLs for BT

ARLs for HK

ARLs for BT

ARLs for HK

40000

30000

20000

10000

00

00

01

02

03

04

05

10

15

20

25

30

35

40

50


50

505

51

515

52

1 2 3 4 5

EWM

A

Observations


UCL1UCL2



Appendix





Data Availability




References




















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








exponentially weighted moving average control charts for...

Documents