research article statistical inferences and applications...

Hindawi Publishing CorporationJournal of Quality and Reliability EngineeringVolume 2013 Article ID 219473 9 pageshttpdxdoiorg1011552013219473

Research ArticleStatistical Inferences and Applications of the Half ExponentialPower Distribution

Wenhao Gui

Department of Mathematics and Statistics University of Minnesota Duluth Duluth MN 55812 USA

Correspondence should be addressed to Wenhao Gui guiwenhaogmailcom

Received 23 October 2012 Revised 13 December 2012 Accepted 4 February 2013

Academic Editor Kai Yuan Cai

Copyright copy 2013 Wenhao Gui This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate the statistical inferences and applications of the half exponential power distribution for the first time The proposedmodel defined on the nonnegative reals extends the half normal distribution and is more flexible The characterizations andproperties involving moments and some measures based on moments of this distribution are derived The inference aspects usingmethods of moment and maximum likelihood are presented We also study the performance of the estimators using the MonteCarlo simulation Finally we illustrate it with two real applications

1 Introduction

The well-known exponential power (EP) distribution or thegeneralized normal distribution has the following densityfunction

119891 (119909) =1199011minus1119901

2Γ (1119901)119890minus|119909|

119901119901 minusinfin lt 119909 lt infin (1)

where 119901 gt 0 is the shape parameter This family consists of awide range of symmetric distributions and allows continuousvariation from normality to nonnormality It includes thenormal distribution 119885 sim 119873(0 1) as the special case when119901 = 2 and the Laplace distribution when 119901 = 1 Nadarajah[1] provided a comprehensive treatment of its mathematicalproperties

Its tails can be more platykurtic (119901 gt 2) or more lep-tokurtic (119901 lt 2) than the normal distribution (119901 = 2) Thedistribution has been widely used in the Bayes analysis androbustness studies (see Box and Tiao [2] Genc [3] Goodmanand Kotz [4] and Tiao and Lund [5])

On the other hand since the most popular models usedto describe the lifetime process are defined on nonnegativemeasurements which motivate us to take a positive trunca-tion in the model (1) and develop a half exponential power(HEP) distribution As far as we know this model has notbeen previously studied although we believe it plays animportant role in data analysisThe resulting nonnegative half

exponential power distribution generalizes the half normal(HN) distribution and it ismore flexible In our work we aimto investigate the statistical features of the nonnegativemodeland apply them to fit the lifetime data

The rest of this paper is organized as follows in Section 2we present the new distribution and study its propertiesSection 3 discusses the inference moments and maximumlikelihood estimation for the parameters In Section 4 wediscuss a useful technique a half normal plot with a simulatedenvelope to assess the model adequacy Simulation studiesare performed in Section 5 Section 6 gives two illustrativeexamples and reports the results Section 7 concludes ourwork

2 The Half Exponential Power Distribution

21 The Density and Hazard Function

Definition 1 A random variable 119883 has a half exponentialpower slash distribution if its density function with scaleparameter 120590 gt 0 takes

119891 (119909) =1199011minus1119901

120590Γ (1119901)119890minus119909119901119901120590119901

119909 ge 0 (2)

where 120590 gt 0 and 119901 gt 0 We denote it as119883 sim HEP(120590 119901)

2 Journal of Quality and Reliability Engineering

10

08

06

04

02

00

0 1 2 3 4 5 6

Den

sity119891(119909)

119901 = 05119901 = 1

119901 = 2119901 = 8

119909

(a) Density function

10

8

6

4

2

0

0 1 2 3 4 5 6

ℎ(119909)

119901 = 05119901 = 1

119901 = 2119901 = 8

119909

(b) Hazard function

Figure 1 The density and hazard rate functions of HEP(120590 119901) for 120590 = 1

Figure 1(a) displays some plots of the density functionof the half exponential power distribution with variousparameters

The cumulative distribution function of the half exponen-tial power distribution119883 sim HEP(120590 119901) is given as follows For119909 ge 0

119865 (119909) = int119909

0

119891119883(119906) 119889119906 = int

119909

0

1199011minus1119901

120590Γ (1119901)119890minus119906119901119901120590119901

119889119906

=120574 (1119901 119909119901119901120590119901)

Γ (1119901)

(3)

where 120574( ) is the lower incomplete gamma function definedas 120574(119904 119909) = int

119909

0119905119904minus1119890minus119905119889119905

The hazard rate function (also known as the failure ratefunction) of the half exponential power distribution is givenby for 119909 ge 0

ℎ (119909) =119891 (119909)

1 minus 119865 (119909)=

1199011minus1119901119890minus119909119901119901120590119901

120590 [Γ (1119901) minus 120574 (1119901 119909119901119901120590119901)] (4)

Since Γ(119904) minus 120574(119904 119909) sim 119909119904minus1119890minus119909 as 119909 rarr infin weobtain ℎ(119909) sim 119909119901minus1120590119901 Therefore the hazard rate functionis increasing for 119901 ge 1 and decreasing for 0 lt 119901 lt 1Figure 1(b) displays some plots of the hazard rate functionof the half exponential power distribution with variousparameters

22 Moments and Measures Based on Moments

Proposition 2 Let119883 sim HEP(120590 119901) for 119896 = 1 2 3 the 119896thnoncentral moments are given by

120583119896= E119883119896 =

119901119896119901120590119896

Γ (1119901)Γ (

119896 + 1

119901) (5)

The following results are immediate consequences of (5)

Corollary 3 Let 119883 sim HEP(120590 119901) The mean and variance of119883 are given by

E119883 =1199011119901120590

Γ (1119901)Γ (

2

119901)

Var (119883) =11990121199011205902 [Γ (1119901) Γ (3119901) minus [Γ (2119901)]

2

]

[Γ (1119901)]2

(6)

Corollary 4 Let 119883 sim HEP(120590 119901) The skewness and kurtosiscoefficients of 119883 are given by

radic1205731=

2[Γ (2119901)]3

minus 3Γ (1119901) Γ (2119901) Γ (3119901)

(Γ (1119901) Γ (3119901) minus [Γ(2119901)]2

)32

+[Γ (1119901)]

2

Γ (4119901)

(Γ (1119901) Γ (3119901) minus [Γ(2119901)]2

)32

Journal of Quality and Reliability Engineering 3

1086420

400

300

200

100

0

Skew

nessradic1205731

119901

(a) Skewness coefficient

1086420

6

4

2

0

minus2

log(radic1205731)

119901

(b) Skewness coefficient in log scale

1086420

2000

1500

1000

500

0

Kurt

osis1205732

119901

(c) Kurtosis coefficient

1086420

14

12

10

8

6

4

2

log(radic1205732)

119901

(d) Kurtosis coefficient in log scale

Figure 2 The plot for the skewness and kurtosis coefficients with various parameters

1205732=

minus3[Γ (2119901)]4

+ 6Γ (1119901) [Γ (2119901)]2

Γ (3119901)

(Γ (1119901) Γ (3119901) minus [Γ(2119901)]2

)2

minus4[Γ (1119901)]

2

Γ (2119901) Γ (4119901) + [Γ (1119901)]3

Γ (5119901)

(Γ (1119901) Γ (3119901) minus [Γ(2119901)]2

)2

(7)

Figure 2 shows the skewness and kurtosis coefficientswith various parameters for the HEP model

3 Inference

31 Moment Estimation Let 1198831 1198832 119883

119899be a

random sample from the distribution HEP(120590 119901)From (5) we have E119883 = (1199011119901120590Γ(1119901))Γ(2119901) and


E1198832 = (11990121199011205902Γ(1119901))Γ(3119901) Replacing E119883 and E1198832 withthe corresponding sample estimators we obtain the momentequations

119883 =1

119899

119899

sum119894=1

119883119894=

1199011119901120590

Γ (1119901)Γ (

2

119901)

1198832

=1

119899

119899

sum119894=1

1198832119894=

11990121199011205902

Γ (1119901)Γ (

3

119901)

(8)

The estimate is the solution to

Γ (1119901) Γ (3119901)

[Γ (2119901)]2

=1198832

1198832 (9)

which can be solved numerically And the estimate is givenby

=119883Γ (1)

1

Γ (2 ) (10)

It is clear that for the special case when 119901 is knownestimator is unbiased and its mean squared error (MSE) isgiven by

MSE () =1205902 [Γ (1119901) Γ (3119901) minus [Γ (2119901)]

2

]

119899[Γ (2119901)]2

(11)

In the following proposition we present the asymtoticproperty of the moment estimators

Proposition 5 Let1198831 1198832 119883

119899be a random sample of size

119899 from the distribution HEP(120590 119901) and let 120579 = (120590 119901) then if1205836= E1198836 lt infin and is the moment estimator of 120579 one has

radic119899 ( minus 120579)119889

997888rarr 1198732(0Hminus1Σ[Hminus1]

119879

) (12)

as 119899 rarr infin where Σ = (120583119894+119895

minus 120583119894120583119895119894119895) andH is given by

H = H (120579) = (

1205971205831

120597120590

1205971205831

1205971199011205971205832

120597120590

1205971205832

120597119901

) (13)

whose entries are given by

1205971205831

120597120590=

1199011119901Γ (2119901)

Γ (1119901)

1205971205831

120597119901= minus

119901minus2+1119901120590Γ (2119901) [minus1 + log119901 minus 120595 (1119901) + 2120595 (2119901)]

Γ (1119901)

1205971205832

120597120590=

21199012119901120590Γ (3119901)

Γ (1119901)

1205971205832

120597119901=minus

119901minus2+21199011199042Γ (3119901) [minus2+2 log119901minus120595 (1119901)+3120595 (3119901)]

Γ (1119901)

(14)

where120595() is the digamma function defined as the logarithmicderivative of the gamma function 120595(119909) = (119889119889119909) log Γ(119909) =Γ1015840(119909)Γ(119909)

Remark 6 A consistent estimator for the asymptotic covari-ance matrix Hminus1Σ[Hminus1]119879 can be obtained by replacingparameters with their corresponding moment estimators

32 Maximum Likelihood Estimation In this section weconsider the maximum likelihood estimation about theparameter 120579 = (120590 119901) of the HEP model defined in (2) Thelog likelihood for a random sample 119909

1 1199092 119909

119899is

119897 (120579) = log119899

prod119894=1

119891 (119909119894) = 119899 (1 minus

1

119901) log119901 minus 119899 log120590

minus 119899 log Γ(1

119901) minus

1

119901120590119901

119899

sum119894=1

119909119901

119894

(15)

By taking the partial derivatives of the log-likelihoodfunction with respect to 120590 and119901 respectively and equalizingthe obtained expressions to zero the following maximumlikelihood estimating equations are obtained

119897120590= minus

119899

120590+

1

120590119901+1

119899

sum119894=1

119909119901

119894= 0

119897119901=

119899 (log119901 + 119901 minus 1)

1199012+

119899120595 (1119901)

1199012

+1 + 119901 log120590

1205901199011199012

119899

sum119894=1

119909119901

119894minus

1

119901120590119901

119899

sum119894=1

119909119901

119894log119909119894= 0

(16)

In general there are no explicit solutions for the abovemaximum likelihood estimating equations The estimatescan be obtained by means of numerical procedures suchas the Newton-Raphson method The program 119877 providesthe nonlinear optimization routine optim for solving suchproblems

For asymptotic inference of 120579 = (120590 119901) we need theFisher information matrix I(120579) It is known that its inverse isthe asymptotic variance matrix of the maximum likelihoodestimators For the case of a single observation (119899 = 1)we take the second-order derivatives of the log-likelihoodfunction in (15)


Table 1 Empirical means and SD for the moment estimators of 120590 and 119901

120590 119901119899 = 100 119899 = 150 119899 = 200

(SD) (SD) (SD) (SD) (SD) (SD)1 1 10116 (01274) 10643 (01949) 10099 (01077) 10450 (01675) 10084 (00935) 10380 (01426)1 2 10046 (01014) 20544 (03443) 09989 (00816) 20369 (03167) 10034 (00745) 20484 (02869)1 3 09972 (00844) 30454 (04233) 09998 (00714) 30375 (04089) 10044 (00640) 30547 (03970)2 1 20365 (02499) 10660 (01959) 20390 (02099) 10559 (01635) 20233 (01872) 10443 (01505)2 2 20090 (01983) 20726 (03453) 20111 (01710) 20541 (03117) 20014 (01424) 20372 (02814)2 3 20033 (01660) 30516 (04338) 20013 (01392) 30344 (04054) 20116 (01275) 30607 (03974)

Table 2 Empirical means and SD for the MLE estimators of 120590 and 119901

120590 119901119899 = 100 119899 = 150 119899 = 200

(SD) (SD) (SD) (SD) (SD) (SD)1 1 10119 (01272) 10515 (02055) 10134 (01079) 10397 (01695) 10026 (00890) 10270 (01401)1 2 10153 (01106) 22028 (06168) 10048 (00883) 20995 (04420) 10063 (00770) 20876 (03644)1 3 10193 (01102) 34735 (13164) 10099 (00816) 32477 (07742) 10068 (00736) 31542 (06405)2 1 20202 (02631) 10566 (02107) 20309 (02178) 10409 (01697) 20153 (01766) 10242 (01372)2 2 20250 (02266) 21944 (06224) 20136 (01798) 21194 (04469) 20031 (01531) 20695 (03449)2 3 20332 (02235) 34523 (14561) 20241 (01682) 32700 (08226) 20218 (01432) 32229 (07221)

Consider

119897120590120590

=1

1205902minus

119901 + 1

120590119901+2119909119901

119897120590119901

=1

120590119901+1119909119901 (log119909 minus log120590)

119897119901119901

= minus1

1199014120590119901[ minus 3119901120590119901 + 1199012120590119901 + 2119901119909119901 + 2119901120590119901 log119901

+ 21199012119909119901 log120590 + 1199013119909119901[log120590]2

minus 21199012119909119901 log119909 minus 21199013119909119901 log120590 log119909

+ 1199013119909119901[log119909]2 + 2119901120590119901120595(1

119901)

+ 1205901199011205951015840 (1

119901)]

(17)

Using the facts

E119909119901 = 120590119901

E (119909119901 log119909) =120590119901 [119901 log120590 + log119901 + 120595 (1 + 1119901)]

119901

E (119909119901[log119909]2)

=120590119901 [(119901 log120590 + log119901 + 120595 (1 + 1119901))

2

+ 1205951015840 (1 + 1119901)]

1199012

(18)

Table 3 Summary of the plasma ferritin concentration measure-ments

Sample size Mean Standard deviation radic1198871

1198872

202 7688 4750 128 442

we can obtain the elements of the Fisher information matrix

11986811

= minusE119897120590120590

=119901

1205902

11986812

= minusE119897120590119901

=log119901 + 120595 (1 + 1119901)

120590119901

11986821

= minusE119897119901120590

=log119901 + 120595 (1 + 1119901)

120590119901

11986822

= minusE119897119901119901

=minus119901 minus 1199012 + 119901[log119901 + 120595 (1 + 1119901)]

2

1199014

+1199011205951015840 (1 + 1119901) + 1205951015840 (1119901)

1199014

(19)

Proposition 7 Let1198831 1198832 119883


119899 from the distribution HEP(120590 119901) let 120579 = (120590 119901) and is themaximum likelihood estimator of 120579 one has

radic119899 ( minus 120579)119889

997888rarr 1198732(0 I(120579)minus1) (20)

4 Assessment of Model Adequacy

In this section we introduce a useful tool a half normal plotwith a simulated envelope which will be used to evaluate


Table 4 Maximum likelihood parameter estimates (with (SD)) of the HN and HEP models for the plasma ferritin concentration data

Model Log lik AIC BICHN 769436 (30588) mdash minus1062037 2126074 2129382HEP 971311 (61496) 25109 (03318) minus1054739 2113478 2120095

the HEP model in Section 6 The advantage of this techniqueis its ease of interpretation without knowing the distributionof the residuals

Atkinson [6] proposed this diagnostic plot to detectpotential outliers and influential observations in linearregression models A simulated envelope is added to the plotto aid overall assessment whereby the observed residuals areexpected to lie within the boundary of the envelope if thepresumed model has been correctly specified

The method of simulated envelope and its correspondingtransformations have been widely applied in many appli-cations (see Flack and Flores [7] Ferrari and Cribari-Neto[8] da Silva Ferreira et al [9] and so forth) The simulatedenvelope technique compares the observed statistics withthose of the data generated from the proposed modelAny sizeble departure of the observed residuals from thesimulated quantities may be thought as evidence against theadequacy of the proposed model Here is the procedure toproduce the half normal plot with simulated envelopes

(1) Fit the model to the observed data (sample size = 119899)(2) Generate a sample of 119899 observations based on the

fitted model(3) Fit the model to the above generated sample and

compute the ordered absolute values of the standardresiduals

(4) Repeat the above steps 119896 times(5) Consider the 119899 sets of the 119896-ordered statistics cal-

culate the average minimum and maximum valuesacross each set

(6) Plot these values together with the ordered residualsfrom the original data against the half normal scoresΦminus1((119894+119899minus18)(2119899+12))

The minimum and maximum values of the 119896-orderedstatistics constitute a simulated envelope to guide assessmentof the model adequacy Atkinson [6] suggested using 119896 = 19since there is a 5 chance to detect the largest residual beingoutside the boundary of the simulated envelope Moreoverother types of residuals such as deviance or score residualmaybe used in the procedure For example da Silva Ferreira et al[9] used theMahalanobis distance to assess their modelsThehorizontal axis can also show other variables such as index

5 Simulation Study

In this section we conduct some simulations and study theproperties of the estimators numerically

We perform a simulation to illustrate the behaviors ofthe moment and MLE estimators for parameters 120579 = (120590 119901)

Table 5 Summaryof the life of fatigue fracture

sample size Mean Standard deviation radic1198871

1198872

101 1025 1119 3001 16709

respectively The simulation is conducted by the software 119877We generate 1000 samples of size 119899 = 100 119899 = 150 and119899 = 200 from the HEP(120590 119901) distribution for fixed parameters120590 and 119901

The random numbers can be generated as follows Wefirst generate random numbers 119884 from an exponential powerdistributionwith120583 = 0120590 and119901 the procedures can be foundin Chiodi [10] then we take the absolute value of the randomnumbers119883 = |119884| It follows that119883 sim HEP(120590 119901)

The estimators are computed using the results inSection 3 The empirical means and standard deviations ofthe estimators are presented in Tables 1 and 2 respectivelyThe simulation studies show that the parameters are wellestimated and the estimates are asymptotically unbiasedTheempiricalMSEs decrease as sample size increases as expectedFurther MLEs are more efficient than moment estimators

6 Real Data Illustration

In this section we analyze two real datasets to fit with theproposed model The applications demonstrate that the HEPmodel fits the data better than the HN model

61 Application 1 The data are the plasma ferritin concentra-tion measurements of 202 athletes collected at the AustralianInstitute of Sport This dataset has been studied by severalauthors (see Azzalini andDalla Valle [11] Cook andWeisberc[12] and Elal-Olivero et al [13])

The descriptive statistics for the dataset are shown inTable 3 whereradic119887

1and 1198872are the sample skewness and kurto-

sis coefficients Notice that the dataset presents nonnegativemeasurements

We fit the dataset with the half normal and the halfexponential power distribution respectively usingmaximumlikelihood method The MLE estimators are computed using119877 and the results are reported in Table 4 The usual Akaikeinformation criterion (AIC) and Bayesian information cri-terion (BIC) to measure of the goodness of fit are alsocomputed AIC = 2119896 minus 2 log 119871 and BIC = 119896 log 119899 minus 2 log 119871where 119896 is the number of parameters in the distribution and119871is the maximized value of the likelihood functionThe resultsindicate that HEPmodel has the lower values for the AIC andBIC statistics and thus it is a better model Figures 3(a) and3(b) display the fitted models using the MLE estimates


Table 6 Maximum likelihood parameter estimates (with (SD)) of the HN and HEP models for the life of fatigue fracture data


0008

0006

0004

0002

0000

0 50 100 150 200 250

Den

sity

Plasma ferritin concentration

HEPHN

(a) Histogram and fitted curves

10

08

06

04

02

00

50 100 150 200 250

Dist

ribut

ion

func

tion


ECDFHEPHN

0

(b) Empirical and fitted CDF

Figure 3 Models fitted for the plasma ferritin concentration dataset

3

2

1

0

00 05 01 15 20 25 30

Scores

Stan

dard

resid

uals

(a) Half normal

006

005

004

003

002

001

000

00 05 10 15 20 25 30

Scores

Stan

dard

resid

uals

(b) Half exponential power

Figure 4 Simulated envelopes for on HN and HEP models


06

05

04

03

02

01

00

00 2 4 6 8

Den

sity

Life of fatigue fracture

HEPHN


10

08

06

04

02

00

0 2 4 6 8

Dist

ribut

ion

func

tion


ECDFHEPHN


Figure 5 Models fitted for the life of fatigue fracture dataset

The diagnostic procedure introduced in Section 4 isimplemented for both models The simulated envelope plotsare shown in Figures 4(a) and 4(b) Most of the observedresiduals are either near or outside the boundary of theenvelope indicating inadequacy of the fitted HN model Onthe other hand the observed residuals corresponding to theHEP model in Figure 4(b) are well within the simulated

6

4

2

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores

(a) Half normal

6

5

4

3

2

1

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores



envelope indicating that the HEP model provides a better fitto the data

62 Application 2 We consider the stress-rupture datasetand the life of fatigue fracture of Kevlar 49epoxy that aresubject to the pressure at the 90 level The dataset has beenpreviously studied by Andrews and Herzberg [14] Barlowet al [15] and Olmos et al [16]

Table 5 summarizes the dataset This dataset also showsnonnegative asymmetry Same as before we fit the datasetwith the half normal and the half exponential power dis-tribution respectively using maximum likelihood method


The results are reported in Table 6 The AIC and BIC arepresented as well and the results show that HEP model fitsbetter Figures 5(a) and 5(b) display the fitted models usingthe MLE estimates

The diagnostic procedure introduced in Section 4 isimplemented for both models The simulated envelope plotsare shown in Figures 6(a) and 6(b) The observed residualscorresponding to the HEP model in Figure 6(b) are wellwithin the simulated envelope indicating that theHEPmodelprovides a better fit to the data

7 Concluding Remarks

In this paper we have studied the half exponential powerdistribution HEP(120590 119901) in detail This nonnegative distribu-tion contains the half normal distribution as its special caseProbabilistic and inferential properties are studied A simula-tion is conducted and demonstrates the good performance ofthe moment and maximum likelihood estimators We applythe model to two real datasets illustrating that the proposedmodel is appropriate and flexible in real applications Thereare a number of possible extensions of the current workMixture modeling using the proposed distributions is themost natural extension Other extensions of the current workinclude a generalization of the distribution to multivariatesettings

Appendix

Proofs of Propositions

Proof of Proposition 2 Consider

E119883119896 = intinfin

0

1199091198961199011minus1119901

120590Γ (1119901)119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)intinfin

0

119909119896119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)120590119896+1119901(119896+1)119901minus1Γ(

119896 + 1

119901)

=119901119896119901120590119896

Γ (1119901)Γ (

119896 + 1

119901)

(A1)

Proof of Proposition 5 This result follows directly by usingstandard large sample theory for moment estimators asdiscussed in Sen and Singer [17]

Proof of Proposition 7 It follows directly by using the largesample theory for maximum likelihood estimators and theFisher information matrix given above

References

[1] S Nadarajah ldquoA generalized normal distributionrdquo Journal ofApplied Statistics vol 32 no 7 pp 685ndash694 2005

[2] G Box and G Tiao ldquoA further look at robustness via bayesrsquostheoremrdquo Biometrika vol 49 no 3-4 pp 419ndash432 1962

[3] A I Genc ldquoA generalization of the univariate slash by a scale-mixtured exponential power distributionrdquo Communications inStatistics vol 36 no 5 pp 937ndash947 2007

[4] I R Goodman and S Kotz ldquoMultivariate 120579-generalized normaldistributionsrdquo Journal of Multivariate Analysis vol 3 no 2 pp204ndash219 1973

[5] G Tiao and D Lund ldquoThe use of olumv estimators in inferencerobustness studies of the location parameter of a class ofsymmetric distributionsrdquo Journal of the American StatisticalAssociation vol 65 pp 370ndash386 1970

[6] A Atkinson Plots Transformations and Regression An Intro-duction to Graphical Methods of Diagnostic Regression AnalysisClarendon Press Oxford 1985

[7] V F Flack and R A Flores ldquoUsing simulated envelopes in theevaluation of normal probability plots of regression residualsrdquoTechnometrics vol 31 no 2 pp 219ndash225 1989

[8] S L P Ferrari and F Cribari-Neto ldquoBeta regression formodelling rates and proportionsrdquo Journal of Applied Statisticsvol 31 no 7 pp 799ndash815 2004

[9] C da Silva Ferreira H Bolfarine and V H Lachos ldquoSkew scalemixtures of normal distributions properties and estimationrdquoStatistical Methodology vol 8 no 2 pp 154ndash171 2011

[10] M Chiodi ldquoProcedures for generating pseudo-random num-bers from a normal distribution of order p (119875 gt 1)rdquo StatisticaApplicata vol 1 pp 7ndash26 1986

[11] A Azzalini and A Dalla Valle ldquoThe multivariate skew-normaldistributionrdquo Biometrika vol 83 no 4 pp 715ndash726 1996

[12] R Cook and S Weisberc ldquoAn introduction to regressiongraphicrdquoMethods vol 17 article 640 1994

[13] D Elal-Olivero J F Olivares-Pacheco H W Gomez and HBolfarine ldquoA new class of non negative distributions generatedby symmetric distributionsrdquo Communications in StatisticsmdashTheory and Methods vol 38 no 7 pp 993ndash1008 2009

[14] D Andrews and A Herzberg Data A Collection of Problemsfrom Many Fields for the Student and Research Worker vol 18Springer New York NY USA 1985

[15] R Barlow R Toland andT Freeman ldquoA bayesian analysis of thestress-rupture life of kevlarepoxy spherical pressure vesselsrdquo inAccelerated Life Testing and Experts Opinions in Reliability C AClarotti and D V Lindley Eds 1988

[16] N M Olmos H Varela H W Gomez and H Bolfarine ldquoAnextension of the half-normal distributionrdquo Statistical Papers pp1ndash12 2011

[17] P Sen and J M Singer Large Sample Methods in Statistics AnIntroduction with Applications Chapman and HallCRC 1993

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



10

08

06

04

02

00

0 1 2 3 4 5 6

Den

sity119891(119909)

119901 = 05119901 = 1

119901 = 2119901 = 8

119909

(a) Density function

10

8

6

4

2

0

0 1 2 3 4 5 6

ℎ(119909)

119901 = 05119901 = 1

119901 = 2119901 = 8

119909

(b) Hazard function

Figure 1 The density and hazard rate functions of HEP(120590 119901) for 120590 = 1

Figure 1(a) displays some plots of the density functionof the half exponential power distribution with variousparameters

The cumulative distribution function of the half exponen-tial power distribution119883 sim HEP(120590 119901) is given as follows For119909 ge 0

119865 (119909) = int119909

0

119891119883(119906) 119889119906 = int

119909

0

1199011minus1119901

120590Γ (1119901)119890minus119906119901119901120590119901

119889119906

=120574 (1119901 119909119901119901120590119901)

Γ (1119901)

(3)

where 120574( ) is the lower incomplete gamma function definedas 120574(119904 119909) = int

119909

0119905119904minus1119890minus119905119889119905

The hazard rate function (also known as the failure ratefunction) of the half exponential power distribution is givenby for 119909 ge 0

ℎ (119909) =119891 (119909)

1 minus 119865 (119909)=

1199011minus1119901119890minus119909119901119901120590119901

120590 [Γ (1119901) minus 120574 (1119901 119909119901119901120590119901)] (4)

Since Γ(119904) minus 120574(119904 119909) sim 119909119904minus1119890minus119909 as 119909 rarr infin weobtain ℎ(119909) sim 119909119901minus1120590119901 Therefore the hazard rate functionis increasing for 119901 ge 1 and decreasing for 0 lt 119901 lt 1Figure 1(b) displays some plots of the hazard rate functionof the half exponential power distribution with variousparameters

22 Moments and Measures Based on Moments

Proposition 2 Let119883 sim HEP(120590 119901) for 119896 = 1 2 3 the 119896thnoncentral moments are given by

120583119896= E119883119896 =

119901119896119901120590119896

Γ (1119901)Γ (

119896 + 1

119901) (5)

The following results are immediate consequences of (5)

Corollary 3 Let 119883 sim HEP(120590 119901) The mean and variance of119883 are given by

E119883 =1199011119901120590

Γ (1119901)Γ (

2

119901)

Var (119883) =11990121199011205902 [Γ (1119901) Γ (3119901) minus [Γ (2119901)]

2

]

[Γ (1119901)]2

(6)

Corollary 4 Let 119883 sim HEP(120590 119901) The skewness and kurtosiscoefficients of 119883 are given by

radic1205731=

2[Γ (2119901)]3

minus 3Γ (1119901) Γ (2119901) Γ (3119901)

(Γ (1119901) Γ (3119901) minus [Γ(2119901)]2

)32

+[Γ (1119901)]

2

Γ (4119901)

(Γ (1119901) Γ (3119901) minus [Γ(2119901)]2

)32


1086420

400

300

200

100

0

Skew

nessradic1205731

119901


1086420

6

4

2

0

minus2

log(radic1205731)

119901


1086420

2000

1500

1000

500

0

Kurt

osis1205732

119901


1086420

14

12

10

8

6

4

2

log(radic1205732)

119901



1205732=

minus3[Γ (2119901)]4

+ 6Γ (1119901) [Γ (2119901)]2

Γ (3119901)

(Γ (1119901) Γ (3119901) minus [Γ(2119901)]2

)2

minus4[Γ (1119901)]

2

Γ (2119901) Γ (4119901) + [Γ (1119901)]3

Γ (5119901)

(Γ (1119901) Γ (3119901) minus [Γ(2119901)]2

)2

(7)


3 Inference


119899be a




119883 =1

119899

119899

sum119894=1

119883119894=

1199011119901120590

Γ (1119901)Γ (

2

119901)

1198832

=1

119899

119899

sum119894=1

1198832119894=

11990121199011205902

Γ (1119901)Γ (

3

119901)

(8)


Γ (1119901) Γ (3119901)

[Γ (2119901)]2

=1198832

1198832 (9)


=119883Γ (1)

1

Γ (2 ) (10)


MSE () =1205902 [Γ (1119901) Γ (3119901) minus [Γ (2119901)]

2

]

119899[Γ (2119901)]2

(11)


Proposition 5 Let1198831 1198832 119883



radic119899 ( minus 120579)119889


119879

) (12)



H = H (120579) = (

1205971205831

120597120590

1205971205831

1205971199011205971205832

120597120590

1205971205832

120597119901

) (13)


1205971205831

120597120590=

1199011119901Γ (2119901)

Γ (1119901)

1205971205831

120597119901= minus


Γ (1119901)

1205971205832

120597120590=

21199012119901120590Γ (3119901)

Γ (1119901)

1205971205832

120597119901=minus


Γ (1119901)

(14)




1 1199092 119909

119899is

119897 (120579) = log119899

prod119894=1

119891 (119909119894) = 119899 (1 minus

1

119901) log119901 minus 119899 log120590


119901) minus

1

119901120590119901

119899

sum119894=1

119909119901

119894

(15)


119897120590= minus

119899

120590+

1

120590119901+1

119899

sum119894=1

119909119901

119894= 0

119897119901=

119899 (log119901 + 119901 minus 1)

1199012+

119899120595 (1119901)

1199012

+1 + 119901 log120590

1205901199011199012

119899

sum119894=1

119909119901

119894minus

1

119901120590119901

119899

sum119894=1

119909119901

119894log119909119894= 0

(16)





120590 119901119899 = 100 119899 = 150 119899 = 200

(SD) (SD) (SD) (SD) (SD) (SD)1 1 10116 (01274) 10643 (01949) 10099 (01077) 10450 (01675) 10084 (00935) 10380 (01426)1 2 10046 (01014) 20544 (03443) 09989 (00816) 20369 (03167) 10034 (00745) 20484 (02869)1 3 09972 (00844) 30454 (04233) 09998 (00714) 30375 (04089) 10044 (00640) 30547 (03970)2 1 20365 (02499) 10660 (01959) 20390 (02099) 10559 (01635) 20233 (01872) 10443 (01505)2 2 20090 (01983) 20726 (03453) 20111 (01710) 20541 (03117) 20014 (01424) 20372 (02814)2 3 20033 (01660) 30516 (04338) 20013 (01392) 30344 (04054) 20116 (01275) 30607 (03974)


120590 119901119899 = 100 119899 = 150 119899 = 200

(SD) (SD) (SD) (SD) (SD) (SD)1 1 10119 (01272) 10515 (02055) 10134 (01079) 10397 (01695) 10026 (00890) 10270 (01401)1 2 10153 (01106) 22028 (06168) 10048 (00883) 20995 (04420) 10063 (00770) 20876 (03644)1 3 10193 (01102) 34735 (13164) 10099 (00816) 32477 (07742) 10068 (00736) 31542 (06405)2 1 20202 (02631) 10566 (02107) 20309 (02178) 10409 (01697) 20153 (01766) 10242 (01372)2 2 20250 (02266) 21944 (06224) 20136 (01798) 21194 (04469) 20031 (01531) 20695 (03449)2 3 20332 (02235) 34523 (14561) 20241 (01682) 32700 (08226) 20218 (01432) 32229 (07221)

Consider

119897120590120590

=1

1205902minus

119901 + 1

120590119901+2119909119901

119897120590119901

=1

120590119901+1119909119901 (log119909 minus log120590)

119897119901119901

= minus1

1199014120590119901[ minus 3119901120590119901 + 1199012120590119901 + 2119901119909119901 + 2119901120590119901 log119901

+ 21199012119909119901 log120590 + 1199013119909119901[log120590]2


+ 1199013119909119901[log119909]2 + 2119901120590119901120595(1

119901)

+ 1205901199011205951015840 (1

119901)]

(17)

Using the facts

E119909119901 = 120590119901

E (119909119901 log119909) =120590119901 [119901 log120590 + log119901 + 120595 (1 + 1119901)]

119901

E (119909119901[log119909]2)

=120590119901 [(119901 log120590 + log119901 + 120595 (1 + 1119901))

2

+ 1205951015840 (1 + 1119901)]

1199012

(18)



1198872

202 7688 4750 128 442


11986811

= minusE119897120590120590

=119901

1205902

11986812

= minusE119897120590119901

=log119901 + 120595 (1 + 1119901)

120590119901

11986821

= minusE119897119901120590

=log119901 + 120595 (1 + 1119901)

120590119901

11986822

= minusE119897119901119901

=minus119901 minus 1199012 + 119901[log119901 + 120595 (1 + 1119901)]

2

1199014

+1199011205951015840 (1 + 1119901) + 1205951015840 (1119901)

1199014

(19)

Proposition 7 Let1198831 1198832 119883



radic119899 ( minus 120579)119889

997888rarr 1198732(0 I(120579)minus1) (20)
















5 Simulation Study





1198872

101 1025 1119 3001 16709














0008

0006

0004

0002

0000

0 50 100 150 200 250

Den

sity


HEPHN


10

08

06

04

02

00

50 100 150 200 250

Dist

ribut

ion

func

tion


ECDFHEPHN

0



3

2

1

0

00 05 01 15 20 25 30

Scores

Stan

dard

resid

uals

(a) Half normal

006

005

004

003

002

001

000

00 05 10 15 20 25 30

Scores

Stan

dard

resid

uals




06

05

04

03

02

01

00

00 2 4 6 8

Den

sity


HEPHN


10

08

06

04

02

00

0 2 4 6 8

Dist

ribut

ion

func

tion


ECDFHEPHN




6

4

2

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores

(a) Half normal

6

5

4

3

2

1

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores











Appendix



E119883119896 = intinfin

0

1199091198961199011minus1119901

120590Γ (1119901)119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)intinfin

0

119909119896119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)120590119896+1119901(119896+1)119901minus1Γ(

119896 + 1

119901)

=119901119896119901120590119896

Γ (1119901)Γ (

119896 + 1

119901)

(A1)



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1086420

400

300

200

100

0

Skew

nessradic1205731

119901


1086420

6

4

2

0

minus2

log(radic1205731)

119901


1086420

2000

1500

1000

500

0

Kurt

osis1205732

119901


1086420

14

12

10

8

6

4

2

log(radic1205732)

119901



1205732=

minus3[Γ (2119901)]4

+ 6Γ (1119901) [Γ (2119901)]2

Γ (3119901)

(Γ (1119901) Γ (3119901) minus [Γ(2119901)]2

)2

minus4[Γ (1119901)]

2

Γ (2119901) Γ (4119901) + [Γ (1119901)]3

Γ (5119901)

(Γ (1119901) Γ (3119901) minus [Γ(2119901)]2

)2

(7)


3 Inference


119899be a




119883 =1

119899

119899

sum119894=1

119883119894=

1199011119901120590

Γ (1119901)Γ (

2

119901)

1198832

=1

119899

119899

sum119894=1

1198832119894=

11990121199011205902

Γ (1119901)Γ (

3

119901)

(8)


Γ (1119901) Γ (3119901)

[Γ (2119901)]2

=1198832

1198832 (9)


=119883Γ (1)

1

Γ (2 ) (10)


MSE () =1205902 [Γ (1119901) Γ (3119901) minus [Γ (2119901)]

2

]

119899[Γ (2119901)]2

(11)


Proposition 5 Let1198831 1198832 119883



radic119899 ( minus 120579)119889


119879

) (12)



H = H (120579) = (

1205971205831

120597120590

1205971205831

1205971199011205971205832

120597120590

1205971205832

120597119901

) (13)


1205971205831

120597120590=

1199011119901Γ (2119901)

Γ (1119901)

1205971205831

120597119901= minus


Γ (1119901)

1205971205832

120597120590=

21199012119901120590Γ (3119901)

Γ (1119901)

1205971205832

120597119901=minus


Γ (1119901)

(14)




1 1199092 119909

119899is

119897 (120579) = log119899

prod119894=1

119891 (119909119894) = 119899 (1 minus

1

119901) log119901 minus 119899 log120590


119901) minus

1

119901120590119901

119899

sum119894=1

119909119901

119894

(15)


119897120590= minus

119899

120590+

1

120590119901+1

119899

sum119894=1

119909119901

119894= 0

119897119901=

119899 (log119901 + 119901 minus 1)

1199012+

119899120595 (1119901)

1199012

+1 + 119901 log120590

1205901199011199012

119899

sum119894=1

119909119901

119894minus

1

119901120590119901

119899

sum119894=1

119909119901

119894log119909119894= 0

(16)





120590 119901119899 = 100 119899 = 150 119899 = 200

(SD) (SD) (SD) (SD) (SD) (SD)1 1 10116 (01274) 10643 (01949) 10099 (01077) 10450 (01675) 10084 (00935) 10380 (01426)1 2 10046 (01014) 20544 (03443) 09989 (00816) 20369 (03167) 10034 (00745) 20484 (02869)1 3 09972 (00844) 30454 (04233) 09998 (00714) 30375 (04089) 10044 (00640) 30547 (03970)2 1 20365 (02499) 10660 (01959) 20390 (02099) 10559 (01635) 20233 (01872) 10443 (01505)2 2 20090 (01983) 20726 (03453) 20111 (01710) 20541 (03117) 20014 (01424) 20372 (02814)2 3 20033 (01660) 30516 (04338) 20013 (01392) 30344 (04054) 20116 (01275) 30607 (03974)


120590 119901119899 = 100 119899 = 150 119899 = 200

(SD) (SD) (SD) (SD) (SD) (SD)1 1 10119 (01272) 10515 (02055) 10134 (01079) 10397 (01695) 10026 (00890) 10270 (01401)1 2 10153 (01106) 22028 (06168) 10048 (00883) 20995 (04420) 10063 (00770) 20876 (03644)1 3 10193 (01102) 34735 (13164) 10099 (00816) 32477 (07742) 10068 (00736) 31542 (06405)2 1 20202 (02631) 10566 (02107) 20309 (02178) 10409 (01697) 20153 (01766) 10242 (01372)2 2 20250 (02266) 21944 (06224) 20136 (01798) 21194 (04469) 20031 (01531) 20695 (03449)2 3 20332 (02235) 34523 (14561) 20241 (01682) 32700 (08226) 20218 (01432) 32229 (07221)

Consider

119897120590120590

=1

1205902minus

119901 + 1

120590119901+2119909119901

119897120590119901

=1

120590119901+1119909119901 (log119909 minus log120590)

119897119901119901

= minus1

1199014120590119901[ minus 3119901120590119901 + 1199012120590119901 + 2119901119909119901 + 2119901120590119901 log119901

+ 21199012119909119901 log120590 + 1199013119909119901[log120590]2


+ 1199013119909119901[log119909]2 + 2119901120590119901120595(1

119901)

+ 1205901199011205951015840 (1

119901)]

(17)

Using the facts

E119909119901 = 120590119901

E (119909119901 log119909) =120590119901 [119901 log120590 + log119901 + 120595 (1 + 1119901)]

119901

E (119909119901[log119909]2)

=120590119901 [(119901 log120590 + log119901 + 120595 (1 + 1119901))

2

+ 1205951015840 (1 + 1119901)]

1199012

(18)



1198872

202 7688 4750 128 442


11986811

= minusE119897120590120590

=119901

1205902

11986812

= minusE119897120590119901

=log119901 + 120595 (1 + 1119901)

120590119901

11986821

= minusE119897119901120590

=log119901 + 120595 (1 + 1119901)

120590119901

11986822

= minusE119897119901119901

=minus119901 minus 1199012 + 119901[log119901 + 120595 (1 + 1119901)]

2

1199014

+1199011205951015840 (1 + 1119901) + 1205951015840 (1119901)

1199014

(19)

Proposition 7 Let1198831 1198832 119883



radic119899 ( minus 120579)119889

997888rarr 1198732(0 I(120579)minus1) (20)
















5 Simulation Study





1198872

101 1025 1119 3001 16709














0008

0006

0004

0002

0000

0 50 100 150 200 250

Den

sity


HEPHN


10

08

06

04

02

00

50 100 150 200 250

Dist

ribut

ion

func

tion


ECDFHEPHN

0



3

2

1

0

00 05 01 15 20 25 30

Scores

Stan

dard

resid

uals

(a) Half normal

006

005

004

003

002

001

000

00 05 10 15 20 25 30

Scores

Stan

dard

resid

uals




06

05

04

03

02

01

00

00 2 4 6 8

Den

sity


HEPHN


10

08

06

04

02

00

0 2 4 6 8

Dist

ribut

ion

func

tion


ECDFHEPHN




6

4

2

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores

(a) Half normal

6

5

4

3

2

1

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores











Appendix



E119883119896 = intinfin

0

1199091198961199011minus1119901

120590Γ (1119901)119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)intinfin

0

119909119896119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)120590119896+1119901(119896+1)119901minus1Γ(

119896 + 1

119901)

=119901119896119901120590119896

Γ (1119901)Γ (

119896 + 1

119901)

(A1)



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119883 =1

119899

119899

sum119894=1

119883119894=

1199011119901120590

Γ (1119901)Γ (

2

119901)

1198832

=1

119899

119899

sum119894=1

1198832119894=

11990121199011205902

Γ (1119901)Γ (

3

119901)

(8)


Γ (1119901) Γ (3119901)

[Γ (2119901)]2

=1198832

1198832 (9)


=119883Γ (1)

1

Γ (2 ) (10)


MSE () =1205902 [Γ (1119901) Γ (3119901) minus [Γ (2119901)]

2

]

119899[Γ (2119901)]2

(11)


Proposition 5 Let1198831 1198832 119883



radic119899 ( minus 120579)119889


119879

) (12)



H = H (120579) = (

1205971205831

120597120590

1205971205831

1205971199011205971205832

120597120590

1205971205832

120597119901

) (13)


1205971205831

120597120590=

1199011119901Γ (2119901)

Γ (1119901)

1205971205831

120597119901= minus


Γ (1119901)

1205971205832

120597120590=

21199012119901120590Γ (3119901)

Γ (1119901)

1205971205832

120597119901=minus


Γ (1119901)

(14)




1 1199092 119909

119899is

119897 (120579) = log119899

prod119894=1

119891 (119909119894) = 119899 (1 minus

1

119901) log119901 minus 119899 log120590


119901) minus

1

119901120590119901

119899

sum119894=1

119909119901

119894

(15)


119897120590= minus

119899

120590+

1

120590119901+1

119899

sum119894=1

119909119901

119894= 0

119897119901=

119899 (log119901 + 119901 minus 1)

1199012+

119899120595 (1119901)

1199012

+1 + 119901 log120590

1205901199011199012

119899

sum119894=1

119909119901

119894minus

1

119901120590119901

119899

sum119894=1

119909119901

119894log119909119894= 0

(16)





120590 119901119899 = 100 119899 = 150 119899 = 200

(SD) (SD) (SD) (SD) (SD) (SD)1 1 10116 (01274) 10643 (01949) 10099 (01077) 10450 (01675) 10084 (00935) 10380 (01426)1 2 10046 (01014) 20544 (03443) 09989 (00816) 20369 (03167) 10034 (00745) 20484 (02869)1 3 09972 (00844) 30454 (04233) 09998 (00714) 30375 (04089) 10044 (00640) 30547 (03970)2 1 20365 (02499) 10660 (01959) 20390 (02099) 10559 (01635) 20233 (01872) 10443 (01505)2 2 20090 (01983) 20726 (03453) 20111 (01710) 20541 (03117) 20014 (01424) 20372 (02814)2 3 20033 (01660) 30516 (04338) 20013 (01392) 30344 (04054) 20116 (01275) 30607 (03974)


120590 119901119899 = 100 119899 = 150 119899 = 200

(SD) (SD) (SD) (SD) (SD) (SD)1 1 10119 (01272) 10515 (02055) 10134 (01079) 10397 (01695) 10026 (00890) 10270 (01401)1 2 10153 (01106) 22028 (06168) 10048 (00883) 20995 (04420) 10063 (00770) 20876 (03644)1 3 10193 (01102) 34735 (13164) 10099 (00816) 32477 (07742) 10068 (00736) 31542 (06405)2 1 20202 (02631) 10566 (02107) 20309 (02178) 10409 (01697) 20153 (01766) 10242 (01372)2 2 20250 (02266) 21944 (06224) 20136 (01798) 21194 (04469) 20031 (01531) 20695 (03449)2 3 20332 (02235) 34523 (14561) 20241 (01682) 32700 (08226) 20218 (01432) 32229 (07221)

Consider

119897120590120590

=1

1205902minus

119901 + 1

120590119901+2119909119901

119897120590119901

=1

120590119901+1119909119901 (log119909 minus log120590)

119897119901119901

= minus1

1199014120590119901[ minus 3119901120590119901 + 1199012120590119901 + 2119901119909119901 + 2119901120590119901 log119901

+ 21199012119909119901 log120590 + 1199013119909119901[log120590]2


+ 1199013119909119901[log119909]2 + 2119901120590119901120595(1

119901)

+ 1205901199011205951015840 (1

119901)]

(17)

Using the facts

E119909119901 = 120590119901

E (119909119901 log119909) =120590119901 [119901 log120590 + log119901 + 120595 (1 + 1119901)]

119901

E (119909119901[log119909]2)

=120590119901 [(119901 log120590 + log119901 + 120595 (1 + 1119901))

2

+ 1205951015840 (1 + 1119901)]

1199012

(18)



1198872

202 7688 4750 128 442


11986811

= minusE119897120590120590

=119901

1205902

11986812

= minusE119897120590119901

=log119901 + 120595 (1 + 1119901)

120590119901

11986821

= minusE119897119901120590

=log119901 + 120595 (1 + 1119901)

120590119901

11986822

= minusE119897119901119901

=minus119901 minus 1199012 + 119901[log119901 + 120595 (1 + 1119901)]

2

1199014

+1199011205951015840 (1 + 1119901) + 1205951015840 (1119901)

1199014

(19)

Proposition 7 Let1198831 1198832 119883



radic119899 ( minus 120579)119889

997888rarr 1198732(0 I(120579)minus1) (20)
















5 Simulation Study





1198872

101 1025 1119 3001 16709














0008

0006

0004

0002

0000

0 50 100 150 200 250

Den

sity


HEPHN


10

08

06

04

02

00

50 100 150 200 250

Dist

ribut

ion

func

tion


ECDFHEPHN

0



3

2

1

0

00 05 01 15 20 25 30

Scores

Stan

dard

resid

uals

(a) Half normal

006

005

004

003

002

001

000

00 05 10 15 20 25 30

Scores

Stan

dard

resid

uals




06

05

04

03

02

01

00

00 2 4 6 8

Den

sity


HEPHN


10

08

06

04

02

00

0 2 4 6 8

Dist

ribut

ion

func

tion


ECDFHEPHN




6

4

2

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores

(a) Half normal

6

5

4

3

2

1

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores











Appendix



E119883119896 = intinfin

0

1199091198961199011minus1119901

120590Γ (1119901)119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)intinfin

0

119909119896119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)120590119896+1119901(119896+1)119901minus1Γ(

119896 + 1

119901)

=119901119896119901120590119896

Γ (1119901)Γ (

119896 + 1

119901)

(A1)



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120590 119901119899 = 100 119899 = 150 119899 = 200

(SD) (SD) (SD) (SD) (SD) (SD)1 1 10116 (01274) 10643 (01949) 10099 (01077) 10450 (01675) 10084 (00935) 10380 (01426)1 2 10046 (01014) 20544 (03443) 09989 (00816) 20369 (03167) 10034 (00745) 20484 (02869)1 3 09972 (00844) 30454 (04233) 09998 (00714) 30375 (04089) 10044 (00640) 30547 (03970)2 1 20365 (02499) 10660 (01959) 20390 (02099) 10559 (01635) 20233 (01872) 10443 (01505)2 2 20090 (01983) 20726 (03453) 20111 (01710) 20541 (03117) 20014 (01424) 20372 (02814)2 3 20033 (01660) 30516 (04338) 20013 (01392) 30344 (04054) 20116 (01275) 30607 (03974)


120590 119901119899 = 100 119899 = 150 119899 = 200

(SD) (SD) (SD) (SD) (SD) (SD)1 1 10119 (01272) 10515 (02055) 10134 (01079) 10397 (01695) 10026 (00890) 10270 (01401)1 2 10153 (01106) 22028 (06168) 10048 (00883) 20995 (04420) 10063 (00770) 20876 (03644)1 3 10193 (01102) 34735 (13164) 10099 (00816) 32477 (07742) 10068 (00736) 31542 (06405)2 1 20202 (02631) 10566 (02107) 20309 (02178) 10409 (01697) 20153 (01766) 10242 (01372)2 2 20250 (02266) 21944 (06224) 20136 (01798) 21194 (04469) 20031 (01531) 20695 (03449)2 3 20332 (02235) 34523 (14561) 20241 (01682) 32700 (08226) 20218 (01432) 32229 (07221)

Consider

119897120590120590

=1

1205902minus

119901 + 1

120590119901+2119909119901

119897120590119901

=1

120590119901+1119909119901 (log119909 minus log120590)

119897119901119901

= minus1

1199014120590119901[ minus 3119901120590119901 + 1199012120590119901 + 2119901119909119901 + 2119901120590119901 log119901

+ 21199012119909119901 log120590 + 1199013119909119901[log120590]2


+ 1199013119909119901[log119909]2 + 2119901120590119901120595(1

119901)

+ 1205901199011205951015840 (1

119901)]

(17)

Using the facts

E119909119901 = 120590119901

E (119909119901 log119909) =120590119901 [119901 log120590 + log119901 + 120595 (1 + 1119901)]

119901

E (119909119901[log119909]2)

=120590119901 [(119901 log120590 + log119901 + 120595 (1 + 1119901))

2

+ 1205951015840 (1 + 1119901)]

1199012

(18)



1198872

202 7688 4750 128 442


11986811

= minusE119897120590120590

=119901

1205902

11986812

= minusE119897120590119901

=log119901 + 120595 (1 + 1119901)

120590119901

11986821

= minusE119897119901120590

=log119901 + 120595 (1 + 1119901)

120590119901

11986822

= minusE119897119901119901

=minus119901 minus 1199012 + 119901[log119901 + 120595 (1 + 1119901)]

2

1199014

+1199011205951015840 (1 + 1119901) + 1205951015840 (1119901)

1199014

(19)

Proposition 7 Let1198831 1198832 119883



radic119899 ( minus 120579)119889

997888rarr 1198732(0 I(120579)minus1) (20)
















5 Simulation Study





1198872

101 1025 1119 3001 16709














0008

0006

0004

0002

0000

0 50 100 150 200 250

Den

sity


HEPHN


10

08

06

04

02

00

50 100 150 200 250

Dist

ribut

ion

func

tion


ECDFHEPHN

0



3

2

1

0

00 05 01 15 20 25 30

Scores

Stan

dard

resid

uals

(a) Half normal

006

005

004

003

002

001

000

00 05 10 15 20 25 30

Scores

Stan

dard

resid

uals




06

05

04

03

02

01

00

00 2 4 6 8

Den

sity


HEPHN


10

08

06

04

02

00

0 2 4 6 8

Dist

ribut

ion

func

tion


ECDFHEPHN




6

4

2

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores

(a) Half normal

6

5

4

3

2

1

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores











Appendix



E119883119896 = intinfin

0

1199091198961199011minus1119901

120590Γ (1119901)119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)intinfin

0

119909119896119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)120590119896+1119901(119896+1)119901minus1Γ(

119896 + 1

119901)

=119901119896119901120590119896

Γ (1119901)Γ (

119896 + 1

119901)

(A1)



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















5 Simulation Study





1198872

101 1025 1119 3001 16709














0008

0006

0004

0002

0000

0 50 100 150 200 250

Den

sity


HEPHN


10

08

06

04

02

00

50 100 150 200 250

Dist

ribut

ion

func

tion


ECDFHEPHN

0



3

2

1

0

00 05 01 15 20 25 30

Scores

Stan

dard

resid

uals

(a) Half normal

006

005

004

003

002

001

000

00 05 10 15 20 25 30

Scores

Stan

dard

resid

uals




06

05

04

03

02

01

00

00 2 4 6 8

Den

sity


HEPHN


10

08

06

04

02

00

0 2 4 6 8

Dist

ribut

ion

func

tion


ECDFHEPHN




6

4

2

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores

(a) Half normal

6

5

4

3

2

1

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores











Appendix



E119883119896 = intinfin

0

1199091198961199011minus1119901

120590Γ (1119901)119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)intinfin

0

119909119896119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)120590119896+1119901(119896+1)119901minus1Γ(

119896 + 1

119901)

=119901119896119901120590119896

Γ (1119901)Γ (

119896 + 1

119901)

(A1)



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












0008

0006

0004

0002

0000

0 50 100 150 200 250

Den

sity


HEPHN


10

08

06

04

02

00

50 100 150 200 250

Dist

ribut

ion

func

tion


ECDFHEPHN

0



3

2

1

0

00 05 01 15 20 25 30

Scores

Stan

dard

resid

uals

(a) Half normal

006

005

004

003

002

001

000

00 05 10 15 20 25 30

Scores

Stan

dard

resid

uals




06

05

04

03

02

01

00

00 2 4 6 8

Den

sity


HEPHN


10

08

06

04

02

00

0 2 4 6 8

Dist

ribut

ion

func

tion


ECDFHEPHN




6

4

2

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores

(a) Half normal

6

5

4

3

2

1

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores











Appendix



E119883119896 = intinfin

0

1199091198961199011minus1119901

120590Γ (1119901)119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)intinfin

0

119909119896119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)120590119896+1119901(119896+1)119901minus1Γ(

119896 + 1

119901)

=119901119896119901120590119896

Γ (1119901)Γ (

119896 + 1

119901)

(A1)



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










06

05

04

03

02

01

00

00 2 4 6 8

Den

sity


HEPHN


10

08

06

04

02

00

0 2 4 6 8

Dist

ribut

ion

func

tion


ECDFHEPHN




6

4

2

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores

(a) Half normal

6

5

4

3

2

1

0

00 05 10 15 20 25

Stan

dard

resid

uals

Scores











Appendix



E119883119896 = intinfin

0

1199091198961199011minus1119901

120590Γ (1119901)119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)intinfin

0

119909119896119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)120590119896+1119901(119896+1)119901minus1Γ(

119896 + 1

119901)

=119901119896119901120590119896

Γ (1119901)Γ (

119896 + 1

119901)

(A1)



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Appendix



E119883119896 = intinfin

0

1199091198961199011minus1119901

120590Γ (1119901)119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)intinfin

0

119909119896119890minus119909119901119901120590119901

119889119909

=1199011minus1119901

120590Γ (1119901)120590119896+1119901(119896+1)119901minus1Γ(

119896 + 1

119901)

=119901119896119901120590119896

Γ (1119901)Γ (

119896 + 1

119901)

(A1)



References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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