research article inferential statistics from black

Research ArticleInferential Statistics from Black HispanicBreast Cancer Survival Data

Hafiz M R Khan1 Anshul Saxena2 Elizabeth Ross3

Venkataraghavan Ramamoorthy4 and Diana Sheehan5

1 Department of Biostatistics Robert Stempel College of Public Health amp Social Work Florida International UniversityMiami FL 33199 USA

2Department of Health Promotion amp Disease Prevention Robert Stempel College of Public Health amp Social WorkFlorida International University Miami FL 33199 USA

3 Behavioral Science Research 2121 Ponce De Leon Coral Gables FL 33134 USA4Department of Dietetics and Nutrition Robert Stempel College of Public Health amp Social WorkFlorida International University Miami FL 33199 USA

5Department of Epidemiology Robert Stempel College of Public Health amp Social Work Florida International UniversityMiami FL 33199 USA

Correspondence should be addressed to Hafiz M R Khan hmkhanfiuedu

Received 28 November 2013 Accepted 16 December 2013 Published 11 February 2014

Academic Editors M Tsionas and J Yu

Copyright copy 2014 Hafiz M R Khan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In this paper we test the statistical probability models for breast cancer survival data for race and ethnicity Data was collectedfrom breast cancer patients diagnosed in United States during the years 1973ndash2009We selected a stratified random sample of BlackHispanic female patients from the Surveillance Epidemiology and End Results (SEER) database to derive the statistical probabilitymodels We used three common model building criteria which include Akaike Information Criteria (AIC) Bayesian InformationCriteria (BIC) and Deviance Information Criteria (DIC) to measure the goodness of fit tests and it was found that Black Hispanicfemale patients survival data better fit the exponentiated exponential probability model A novel Bayesian method was used toderive the posterior density function for the model parameters as well as to derive the predictive inference for future response Wespecifically focused on Black Hispanic race Markov Chain Monte Carlo (MCMC) method was used for obtaining the summaryresults of posterior parameters Additionally we reported predictive intervals for future survival times These findings would be ofgreat significance in treatment planning and healthcare resource allocation

1 Introduction

11 Problem Statement Breast Cancer in Black HispanicWomen Cancer is defined as a process that induces irre-versible mutations in cellular genetic processes resulting inuncontrolled growth and proliferations [1] Tumor is definedas any abnormal growth of cancer cells that form a lump ormass Human breast is primarily composed of fat connectivetissues lymphatic vessels and organized lobules of milksecreting glandsThese lobules are connected exteriorly to thenipple via secretory ducts Most breast cancers are carcinoma

in situ (CIS) because they are confined by epithelial bound-aries either to the duct (ductal carcinoma in situ (DCIS)) orto the lobule (lobular carcinoma in situ (LCIS))

Breast cancer is one of the most common life-threateningcancers in women of any age group According to the WorldHealth Organization (WHO) report 2004 breast cancercomprises approximately 16 of all cancer types and causes519000 deaths annually worldwide Surprisingly 69 ofthese deaths occurred in developing countries refuting themisconception that breast cancer is a disease of developedworld [2] According to the American Institute of Cancer

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 604581 13 pageshttpdxdoiorg1011552014604581

2 The Scientific World Journal

Research (AICR) over 226000 cases of breast cancer arediagnosed every year in USA and approximately 40000American women die of breast cancer every year [3]

12 Breast Cancer according to Race and Ethnicity In theUnited States breast cancer is one of the most frequentlydiagnosed cancers across different racial and ethnic groups[2] Raceethnicity specific incidence rates remained fairlyconstant for all racial and ethnic groups during the years2004ndash2008 [1] Previously it was believed that family historysocioeconomic status levels of education frequency ofmam-mograms and access to health care resources were some ofthemajor determinants affecting the prognosis of the diseaseHowever recent studies have shown that racial and ethnicfactors also contribute significant risk for the prognosis of thedisease

The American Cancer Society has found evidence thatthere are notable differences in breast cancer death ratesbetween different states across various socioeconomic strataand between different racialethnic groups [1] Although ageis the strongest predictor for breast cancer risk raceethnicitycould also be a major risk factor [1] Since the early 1990sbreast cancer mortality rates have decreased among all ethnicgroups except the American IndiansAlaska Natives therebyshowing another racial disparity associated with the diseaseIn the United StatesWhite women are more likely to developbreast cancer than African-American Hispanic Asian orAmerican IndianAlaska native women [1]

13 Hispanic Black Women Although Hispanics are thefastest growing minority population in the United Statesthere are notmuch breast cancer statistics onBlackHispanicsspecifically Breast cancer data for Hispanics are usuallytabulated under one ethnic group (Hispanic) therefore race-specific breast cancer data for the Hispanic population is notreadily available Overall the incidence andmortality rates ofbreast cancer among Hispanic (Black andWhite) women arelower than non-HispanicWhite women [3] Hispanic women(Black and White) show lower levels of awareness about therisk factors associated with the disease and have less access tohealth care facilities when compared to women of any otherethnicity

Unfortunately there are not many studies that elucidatebreast cancer disparities among different races within theHispanic ethnicity Usual research findings describe breastcancer incidence mortality and death rates and other vitalstatistics associated with the disease among Hispanic womenwithout any details about interracial differences within theethnicity Banegas and Li [4] have asserted that further studyof specific breast cancer outcomes among the different racesof Hispanic women could greatly enhance knowledge aboutthe distribution and determinants of the disease in this highrisk ethnic group Their study showed that non-HispanicBlacks had a 15ndash25-fold greater risk of having stage IV breastcancer types and 10ndash50 greater risk of breast cancer specificmortality compared to non-Hispanic whites [4] This findingagain shows the need for a study that tries to understand the

current state of affairs about breast cancer survival in thissubpopulation within United States

14 Statistical Probability Models Healthcare personnel hascollected vast amounts of phenomic and genomic datawhich should bemaximally utilized for research perspectivesThese large databases should be tested with newer statisticalmethods and statistical probability models This would bevery useful to predict future patterns of diseasemorbidity andmortality thereby enhancing our understanding the severityand outcomes

Data may follow several statistical probability modelslike exponential gamma Weibull normal half-normal log-normal Rayleigh inverse Gaussian exponentiated expo-nential (EE) exponentiated Weibull (EW) beta generalizedexponential (BGE) beta inverseWeibull (BIW) and so forthStatistical models are immensely useful to characterize thedata and derive reliable scientific inferences

The biomedical and engineering fields often use exponen-tiated exponentialmodel (EEM) for datamodelingTheEEMa generalization of exponential distribution was introducedby Gupta and Kundu [5] and received rapid and widespreadacceptance The EEM considers two parameters ldquoshaperdquo andldquoscalerdquo Moreover Gupta and Kundu [6] observed that theEEM is similar to the Weibull family and suggested thepossibility of using the EE distribution as a substitute forWeibull model

A random variable 119909 is said to follow the exponentiatedexponential distribution if its probability density function(pdf) is given by

119901 (119909) = 120572120582 exp minus (120582119909) (1 minus exp minus (120582119909))120572minus1

(1)

where 120572 gt 0 is the shape parameter and 120582 gt 0 isthe scale parameter Gupta and Kundu [5] introduced theabove mentioned density function for exponentiated expo-nential distribution The random variable can be expressedas 119909 sim EE(120572 120582) Some interesting applications of EEMinclude designing rainfall estimation in the Coast of Chiapas[7] analysis of Los Angeles rainfall data [8] and softwarereliability growth models for vital quality metrics [9] A curerate model based on the generalized exponential distributionthat incorporates the effects of risk factors or covariates forthe probability of an individual being a long-time survivorwas proposed by Kannan et al [10] Also Gompertrsquoz form ofexponentiated exponential model was used to predict squidaxon voltage clamp conductance [11]

For a beta generalized exponential model the probabilitydensity function is given by

119901 (119909) =

120572120582

119861 (119886 119887)

exp minus (120582119909)

times (1 minus exp minus (120582119909))119886120572minus1

(1 minus (1 minus exp minus (120582119909))120572

)

119887minus1

(2)

where 120572 gt 0 is the shape parameter and 120582 gt 0 is thescale parameter There are two additional parameters 119886 gt

0 and 119887 gt 0 The role of these parameters is to describe

The Scientific World Journal 3

skewness and tail weight [12] The BGE model generalizessome well-known models for example beta exponential andgeneralized exponential models as special cases

A Swedish physicist called Weibull [13] introduced theWeibull distribution primarily for examining the breakingstrength of materials The first EW model with bathtubshaped distribution and unimodal failure rates was intro-duced by Mudholkar and Srivastava [14] Since then appli-cation of the EW model to analyze lifetime data has beenrecommended by Nassar and Eissa [15] and Choudhury [16]

For the EWmodel the probability density function (pdf)is given by

119901 (119909) = 120572120573120582119909120573minus1 exp minus (120582119909

120573

) (1 minus exp minus (120582119909120573

))

120572minus1

(3)

where 120572 gt 0 and 120573 gt 0 are the shape parameters and 120582 gt 0 isthe scale parameter

The BIW model has several applications for problemsin engineering health and medical fields It shows best fitfor several data sets for instance the amount of time takenfor breakdown of insulating fluids subjected to tensions [17]For the BIW model the probability density function (pdf) isgiven by

119901 (119909) =

120573119909minus(120573+1)

119861 (119886 119887)

exp minus (119909minus120573

) (exp minus (119909minus120573

))

120572minus1

times (1 minus exp minus (119909minus120573

))

119887minus1

(4)

where 120573 represents the shape parameter and two parameters119886 gt 0 and 119887 gt 0 represent skewness and tail weight

A novel Bayesian method can be used to derive theposterior probability for the parameters to calculate posteriorinferenceModel parameters and data are considered randomvariables in a Bayesian estimation technique Their jointprobability distribution is stated by a probabilistic modelData are considered as ldquoobserved variablesrdquo and parametersas ldquounobserved variablesrdquo in a Bayesian method Multiplyinglikelihood and prior gives the joint distribution for theparameters The ldquopriorrdquo contains information about theparameter The likelihood depends on the model of underly-ing process andmeasured as a conditional distribution whichspecifies the probability of the observed data All the infor-mation available about the parameters is combined by priorand likelihood Bymanipulating the joint distribution of priorand likelihood inference about parameters of the probabilitymodel can be derived from the given data The Bayesianinference intends to develop the posterior distribution of theparameters for given sets of observed data

Readers can refer to Berger [18] Geisser [19] Bernardoand Smith [20] Ahsanullah and Ahmed [21] Gelman [22]and Baklizi [23 24] for further information on Bayesianmethods Khan et al [25] Thabane [26] Thabane and Haq[27] Ali-Mousa andAl-Sagheer [28] andRaqab [29 30] havediscussed several additional applications of Bayesian methodfor predictive inferences

Objectives of this paper include (i) studying some demo-graphic and socioeconomic variables (ii) reviewing right

skewed models EE EW BGE and BIW (iii) justifying thatthe given sample data follows a specific model by applyingmodel selection criteria through goodness of fit tests (iv)performing a Bayesian analysis of the posterior distributionof the parameters and (v) deriving Bayesian predictivemodelfor future response

The structural organization of this paper is as followsSection 2 includes a real example of breast cancer datadiscussed in detail Section 3 includes the measure of good-ness of fit tests log-likelihood functions and the posteriorinference for the model parameters for raceethnicity (BlackHispanic females only) Section 4 includes the Bayesianpredictive model which includes the likelihood functionposterior density function and the predictive density for afuture response given a set of observations from the bestmodel Section 5 includes the results and discussions andSection 6 includes the conclusion

2 Real Life Data Example

Breast cancer data (119873 = 657 712) from SurveillanceEpidemiology and End Results (SEER 1973ndash2009) websitehas beenused as a real life data exampleData on breast cancerpatients collected from twelve states have been stored in SEERdatabase Stratified random sampling scheme was used topick nine sates randomly from these twelve states Data fromthese nine states included state-wise raceethnicity categoriesfor breast cancer distribution

This data included 4269males and 653443 females Sincebreast cancers are rare in males data from females onlywere used in our analysis A simple random sampling (SRS)method was used to select 298 female subjects from BlackHispanic data

Figure 1 shows the pedigree chart for the selection ofBlack Hispanic breast cancer patients out of total femalebreast cancer patients 119873 = 608 032 In the total populationthere were 300 Black Hispanic female patients but data weremissing for 2 participants Figure 2 describes the nine states(dark blue regions) which were randomly selected and werefollowed by a random selection of Black Hispanic breastcancer patients

The descriptive statistics (frequency distribution andsummary statistics) are shown in Tables 1 and 2 respectivelyTable 1 contains the state wise frequency and its corre-sponding percentages for the selected patients Table 2 hasthe descriptive statistics (mean standard deviation medianquartiles and variance) for somedemographic characteristics(age at diagnosis survival times andmarital status at diagno-sis) of the selected random sample of Black Hispanic breastcancer patients

We selected 2000non-Hispanic Blacks out of 53531 Blacknon-Hispanics for comparing with 298 Black Hispanics inthis sample The mean survival for non-Hispanic Blacks was6676 (standard deviation 3020) and for Black Hispanics7138 (standard deviation 6133) Cox Proportional Regres-sion was used to calculate hazard ratios by ethnicity Hazardratios compare the probability of an event occurring in onegroup versus another and take into account the time elapsed


Total female cancer patients

Total Black female Hispanic cancer patients

Selected Black female Hispanic cancer patients

n = 298

N1 = 300

(1973ndash2009)N = 608032

Figure 1 Selection of stratified random sample of Black Hispanicbreast cancer patients from SEER (1973ndash2009) dataset representedas a pedigree chart

WA

CAUT

NM

IA MI

HI

GA

CT

Figure 2 The nine states from which the Black Hispanic breastcancer patients were selectedNoteThe Black Hispanic females (119899 =

298) patients were randomly selected from the dark blue colorednine states in Figure 2

Table 1 Frequency distribution of the selected Black Hispanicbreast cancer patients from the nine states

States Black HispanicCount ()

Georgia 94 315Hawaii 1 03Iowa 3 10Michigan 44 148New Mexico 5 17Utah 4 13Washington 9 30California 67 225Connecticut 71 238Total 298 100

until the event should occur In survival analysis the eventunder consideration is death A hazard ratio of 10 representsan equal risk of death between the groups being compareda hazard ratio above 10 means an increased risk of deathand a hazard ratio below 10 represents a decreased riskof death compared to the referent group In this analysis

Table 2 Given below is the marital status survival time and ageat diagnosis for female Black Hispanic breast cancer patients Ageat diagnosis and survival time are classified by mean standarddeviation (SD) median range and quartiles Marital status isclassified into six categories and their respective frequencies arereported

Characteristics Categories Black Hispanic

Age at diagnosis (years)

Mean 5411SD 1441Median 52Quartile

1198761

441198762

521198763

64Variance 20765

Survival time (months)


1198761

251198762

50501198763

10125Variance 376104

Marital status at diagnosis

Single 57Married 124Separated 7Divorced 51Widowed 50Unknown 9

we used non-Hispanic Blacks as a referent group Statisticalsignificance is established if the 95 confidence intervaldid not include 1 Non-Hispanic Blacks had a significantlyincreased risk of death compared to Black Hispanics (Hazardratio 1445 95 Wald Robust Confidence Limits 1210ndash1724 95Profile LikelihoodConfidence Limits 1265ndash1659)When compared to non-Hispanic Blacks Hispanic Blackshad a significantly decreased risk of death (hazard ratio0692 95Wald Robust Confidence Limits 0580ndash0826 95Profile Likelihood Confidence Limits 0603ndash0791) Theseresults are consistent with the mean survival times for eachgroup as well as the observed survival curve confirming thelonger survival among Hispanic Blacks and shorter survivalamong non-Hispanic Blacks

3 Methods of Goodness of Fit

Akaike Information Criterion (AIC) Deviance InformationCriterion (DIC) and Bayesian Information Criterion (BIC)are themost commonly usedmodels tomeasure the goodnessof fit DIC a Bayesian measure of fit is used for comparisonof different models for example the use of public data byCongdon [31 32] The values of DIC can be either positive ornegativeModels with lower values are considered better thanothers DIC is similar to AIC and provides the same results


as AIC when models with only fixed effects are fitted BIC isan asymptotic result which assumes that the data distributionis an exponential family and can only be used to compareestimated models when numerical values of the dependentvariable are identical for all estimates being compared TheBIC penalizes free parameters more than AIC As is the casewith AIC given any two estimated models the model withlower value of BIC is preferred

31 The Log-Likelihood Function and Reparameterization Areparameterization method from the Birnbaum-Saunderslifetime model was proposed by Ahmed et al [33] LaterAchcar et al [34] considered a reparameterization fromcertain skewedmodels A reparameterizationmethodmay beapplied in terms of the log-likelihood functions consideringdata x = (119909

1 1199092 119909

119899) from the models described earlier

which are given in the followingThe log-likelihood function from the EE model is given

by

ℓ (120572 120582 | x) = 119899 log (120572) + 119899 log (120582) + (120572 minus 1)

times

119899

sum

119894=1

log (1 minus exp minus (120582119909119894)) minus 120582

119899

sum

119894=1

119909119894

(5)

Assume 1205881= log(120572) and 120588

2= log(120582) It is assumed that 120588

1and

1205882are independently distributed To obtain noninformative

prior for 1205881and 1205882 let a uniform prior distribution for 120588

119894be

119880(minus119888119894 119888119894) for all 119894 = 1 2 Then the joint posterior density is

given by

119901 (1205881 1205882| x)

= 119901 (1205881 1205882) times exp119899120588

1+ 1198991205882minus 119899119909 exp (120588

2)

+ (exp (1205881) minus 1)

times

119899

sum

119894=1

log (1 minus exp minus119909119894exp (120588

2))

(6)

The log-likelihood function from the beta generalized expo-nentiated model is given by

ℓ (120572 120582 119886 119887 | x) = 119899 log(120572

120582

119861 (119886 119887)

) minus 120582

119899

sum

119894=1

119909119894+ (119886120572 minus 1)

times

119899

sum

119894=1

log (1 minus exp minus (120582119909119894)) + (119887 minus 1)

times

119899

sum

119894=1

log (1 minus [1 minus exp minus (120582119909119894)]120572

)

(7)

Assume 1205881

= log(119886) 1205882

= log(119887) 1205883

= log(120572) and1205884

= log(120582) We further assume that 1205881 1205882 1205883 and 120588

4are

independently distributed To obtain noninformative prior

for 1205881 1205882 1205883 and 120588

4 let a uniform prior distribution for 120588

119895

be 119880(minus119889119895 119889119895) for all 119895 = 1 2 3 4 Then the joint posterior

density is given by

119901 (1205881 1205882 1205883 1205884| x)

= 119901 (1205881 1205882 1205883 1205884)

times [119899 log(

1198901205883+1205884

119861 (1198901205881 1198901205882)

) minus 1198901205884

119899

sum

119894=1

119909119894+ (1198901205881+1205883

minus 1)

times

119899

sum

119894=1

log (1 minus exp minus1198901205884119909119894) (1198901205882

minus 1)

times(

119899

sum

119894=1

log (1 minus (1 minus exp minus1198901205884119909119894)1198901205883

))]

(8)

The log-likelihood function from the EWmodel is derived by

ℓ (120572 120573 120582 | x)

= 119899 log (120572) + 119899 log (120573) + 119899 log (120582) + (120572 minus 1)

times

119899

sum

119894=1

log (1 minus exp minus (120582119909120573

119894))

minus 120582

119899

sum

119894=1

119909120573

119894+ (120573 minus 1)

119899

sum

119894=1

log (119909119894)

(9)

Assume 1205881= log(120572) 120588

2= log(120573) and 120588

3= log(120582) It is further

assumed that 1205881 1205882 and 120588

3are independently distributed To

obtain non-informative prior for 1205881 1205882 and 120588

3 let a uniform

prior distribution for 120588119896be 119880(minus119890

119896 119890119896) for all 119896 = 1 2 3

Then the joint posterior density is derived by

119901 (1205881 1205882 1205883| x)

= 119901 (1205881 1205882 1205883) times exp119899 (120588

1+ 1205882+ 1205883)

minus 1198901205883

119899

sum

119894=1

1199091198901205882

119894+ (1198901205881

minus 1)

times

119899

sum

119894=1

log (1 minus exp minus11989012058831199091198901205882

119894)

+ (1198901205882

minus 1)

119899

sum

119894=1

log (119909119894)

(10)

The log-likelihood function from the BIWmodel is given by

ℓ (120573 119886 119887 | x)

= 119899 log(

120573

119861 (119886 119887)

) minus 119886

119899

sum

119894=1

119909minus120573

119894+ (120573 minus 1)

times

119899

sum

119894=1

log (119909119894) + (119887 minus 1)

119899

sum

119894=1

log (1 minus exp minus119909minus120573

119894)

(11)


Table 3 Selection of the best model for Black Hispanic females onthe basis of AIC BIC and DIC criterions

Model criterions AIC BIC DICExponentiated exponential 313672 314411 3136732Exponentiated Weibull 318425 319164 3182328Beta generalized exponential 326220 327698 3256208Beta inverse Weibull 332365 333474 3317650

Assume 1205881= log(120573) 120588

2= log(119886) and 120588

3= log(119887) We further

assume that 1205881 1205882 and 120588



3 let a uniform

prior distribution for 120588ℎbe 119880(minus119891

ℎ 119891ℎ) for all ℎ = 1 2 3

Then the joint posterior density is given by

119901 (1205881 1205882 1205883| x)

= 119901 (1205881 1205882 1205883) [119899 log(

1198901205881

119861 (1198901205882 1198901205883)

)

minus 1198901205882

119899

sum

119894=1

119909minus1198901205881

119894+ (1198901205881

minus 1)

times

119899

sum

119894=1

log (119909119894) + (1198901205883

minus 1)

times

119899

sum

119894=1

log(1 minus exp minus119890minus119909119894

minus1198901205881

)]

(12)

A better performance of the posterior distributions for theparameters can be achieved with the reparameterizationmethod Table 3 gives the results of the measures of goodnessof fit for Black Hispanic females Tables 4ndash7 summarize theresults of the posterior parameters Figures 3ndash6 show theposterior kernel densities for the parameters

32The Results of Goodness of Fit Tests and Posterior Inferencefor the Parameters from the Black Hispanic Survival DataTable 3 includes the AIC BIC and DIC values for the EEEW BGE and BIWmodels Better model fit is inferred if thevalues of AIC BIC and DIC are the least The data fits EEmodel better than the other models The estimated value ofAIC is the lowest (313672) while the DIC value is very closeto AIC Comparing the estimated values of all AIC BIC andDIC for the models the EEM fits better for the survival daysbecause it produces smaller values for all three criteria AICBIC and DIC

Table 4 summarizes the results of the posterior distri-bution of the parameters from the EE for the Black His-panic breast cancer patientsrsquo survival data In the Bayesianapproach the knowledge of the distribution of the parametersis updated through the use of observed data resulting in whatis known as the posterior distribution of the parameters Inthe case of breast cancer data we are interested in estimatingthe posterior distribution of the parameters assuming thatobserved random sample form an appropriate statisticalprobability distribution

The values of the 1205881and 120588

2are generated from the data

and the results of the posterior distribution parameters 120572

and 120582 are estimated using the MCMCmethod Samples froma probability distribution can be generated using MarkovChain Monte Carlo which is a class of algorithms usedin statistics [35] The EE model is used to derive the log-likelihood function and the parameter values are assignedto the appropriate theoretical probability distributions Thesummary results (mean SD MC error median and con-fidence intervals) of the parameters are obtained by usingthe WinBUGS software In this process the early iterationsup to 1000 are ignored in order to remove any biases ofestimated values of the parameters resulting from the valueof x utilized to initialize the chain This process is known asburn-in The remaining samples are treated as if the samplesare from the original distribution (after elimination of theburn-in samples) Fifty thousand (50000) Monte Carlo rep-etitions were used to produce the inference for the posteriorparameters as shown in Table 4The graphical representationof the parametersrsquo behavior is displayed in Figure 3 After50000 Monte Carlo repetitions the kernel densities for bothshape and scale parameters follow approximately symmetricdistributions

Table 5 shows the summary results of the posterior dis-tribution of the parameters from the exponentiated WeibullThe Black Hispanic female breast cancer patientsrsquo survivaldata has been used for these results The values 120588

1 1205882 and

1205883have been generated from the data Using the MCMC

method the results of the posterior distribution parameters120572 120573 and 120582 are estimated by setting the generated valuesThe log-likelihood function is derived from the EW modelSubsequently the parameter values which are assigned toappropriate probability distributions are derived The sum-mary results (mean SD MC Error median and confidenceintervals) of the parameters are derived using the WinBugssoftware The graphical representation of distributions of theparameter behaviors has been summarized in Figure 4 Theshape parameters120573 and 120588

2show a normal distribution while

other model parameters show skewed distributionsTable 6 shows the summary results of posterior distribu-

tion of the parameters from the BGE model Black Hispanicfemale breast cancer patientsrsquo data has been used for theseresults We used the WinBugs software to obtain the sum-mary results (mean SD MC error median and confidenceintervals) of the parameters The graphical representationof the parameters for female in the case of BGE has beendisplayed in Figure 5 A symmetrical pattern of distributionis shown by the parameters 120582 and 120588

4 while a nonsymmetrical

distribution is shown by other parametersTable 7 shows the summary results of the posterior

distribution of the parameters from the BIW model TheBlack Hispanic female breast cancer patientsrsquo survival datahas been used The summary results which include (meanSD MC error median and confidence intervals) of theparameters have been derived by using theWinBugs softwareThe graphical representations of the parameters for femalesin the case of BIW model have been displayed in Figure 6It is noted that the skewed distribution pattern is shown by


Table 4 Summary of the results of the posterior parameters from exponentiated exponential for BlackHispanic females breast cancer patients(119899 = 298)

Node Mean SD MC error Median 95 CI Sample120572 1234 009684 894119864 minus 04 1231 (1053 1434) 50000120582 001595 0001142 106119864 minus 05 001593 (001377 001824) 500001205881

02073 007844 726119864 minus 04 02076 (005193 03608) 500001205882

minus4141 007182 669119864 minus 04 minus4139 (minus4285 minus4004) 50000Dbar Dhat pD

Zeros 3134730 3132720 2006Total 3134730 3132720 2006Dbar postmean of minus2 log119871 Dhat minus2 log119871 at postmean of stochastic nodes1205881simdunif (minus10 10) 120588

2simdunif (minus10 10)

Table 5 Summary results of the posterior parameters for exponentiatedWeibull for Black Hispanic females breast cancer patients (119899 = 298)

Node Mean SD MC error Median 95 CI Sample120572 2743 002486 243119864 minus 04 2736 (2719 2809) 50000120573 08414 001005 904119864 minus 05 0841 (08227 08625) 50000120582 004858 0001167 131119864 minus 05 004892 (004547 004976) 500001205881

1009 0008985 877119864 minus 05 1006 (1000 1033) 500001205882

minus01728 001193 107119864 minus 04 minus01731 (minus01952 minus01479) 500001205883


Zeros 3181290 3180250 1039Total 3181290 3180250 1039Dbar postmean of minus2 log119871 Dhat minus2 log119871 at postmean of stochastic nodes1205881simdunif (1 2) 120588

2simdunif (minus10 1) 120588

3simdunif (minus4 minus3)

parameters 120573 and 1205881from the BIW while other parameters

show approximately uniform distributions

4 The Bayesian Predictive Survival Model

Due to the current economic crisis health care costs areincreasing tremendously It is important for health careresearchers and providers to promptly identify the highrisk population variables for several diseases The goal isto identify and provide preventive interventions withoutsignificantly increasing the cost of management Currentlypredictive modeling is a popular technique used for high-risk assessment at very low costs Health care providers andresearcherswill greatly benefit frompredictivemodeling bothto improve present health care services and reduce futurehealth care costs

Predictive modeling is a process that can be applied toavailable healthcare data for instance identification of peoplewho have high medical need and who are ldquoat riskrdquo for above-average future medical service utilization We are deriving anovel Bayesian method which can predict the breast cancersurvival days based on past data collected from patients

The Bayesian predictive method is growing extremelypopular finding newer applications in the fields of healthsciences engineering environmental sciences business andeconomics and social sciences among others The Bayesianpredictive approach is used for the design and analysis ofsurvival research studies in the health sciences It is widely

used to reduce healthcare costs and to economically allocatehealthcare resources

In this section a predictive survival model for breastcancer patients is developed by using a novel Bayesianmethod It is found that the Black Hispanic female breastcancer patientsrsquo data follow the EE model

Let us assume that the data x = (1199091 119909

119899) represents 119899

female breast cancer patients survival days that follow the EEmodel and let 119911 be a future response (or future survival days)The predictive density of 119911 for the observed data x is

119901 (119911 | x) = ∬119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572 (13)

where 119901(120572 120582 | x) is the posterior density function and119901(119911 | 120572 120582) represents the probability density function of afuture response (119911) that may be defined from model (1) Theposterior density is given by

119901 (120572 120582 | x) = Ψ (x) 119871 (120572 120582 | x) 119901 (120572 120582) (14)

where 119871(120572 120582 | x) is the likelihood function 119901(120572 120582) is theprior density for the parameters and the reciprocal of thenormalizing constant is

Ψ(x)minus1 = ∬119871 (120572 120582 | x) 119901 (120572 120582) 119889120582 119889120572 (15)

To derive the likelihood function let 1199091 119909

119899be a ran-

dom sample of size 119899 from model (1) Thus x = (1199091 119909

119899)1015840


Table 6 Summary results of the posterior parameters in the case of BGE for Black Hispanic females breast cancer patients (119899 = 298)

Node Mean SD MC error Median 95 CI Samplea 2735 001647 164119864 minus 04 273 (2719 2779) 50000120572 1006 0005935 538119864 minus 05 1004 (1000 1022) 50000b 1029 002016 155119864 minus 04 1025 (1001 1069) 50000120582 002494 0001025 724119864 minus 06 002492 (002298 002698) 500001205881

1006 0005986 594119864 minus 05 1004 (100 1022) 500001205882

002805 001952 151119864 minus 04 002482 (0001068 006674) 500001205883

0005888 0005865 532119864 minus 05 0004063 (148119864 minus 04 002165) 500001205884



2simdunif (0 007) 120588

3simdunif (0 1) and 120588


Table 7 Summary of the posterior parameters in the case of BIW for Black Hispanic females breast cancer patients (119899 = 298)

Node Mean SD MC error Median 95 CI Samplea 1031 001789 838119864 minus 05 103 (1001 106) 50000b 1047 002736 121119864 minus 04 1047 (1002 1092) 50000120573 4021 1344 001014 4025 (3984 4034) 500001205881

5997 0003353 253119864 minus 05 5998 (5988 6000) 500001205882

003002 001736 814119864 minus 05 002997 (0001484 005849) 500001205883

00454 002614 116119864 minus 04 004559 (000228 008786) 50000Dbar Dhat pD

Zeros 3317650 3317650 minus0000Total 3317650 3317650 minus0000Dbar postmean of minus2 log119871 Dhat minus2 log119871 at postmean of stochastic nodes1205881simdunif (0 6) 120588

2simdunif (0 006) and 120588

3simdunif (0 009)

075 10 125 15

00

20

40

60

001 0015 002

00

1000

2000

3000

4000

00

20

40

60

00 02 04

00

20

40

60

minus02 minus46 minus44 minus42 minus40

120572 sample 50000 120582 sample 50000

1205881 sample 50000 1205882 sample 50000

Figure 3 Kernel density of the posterior parameters in the case of exponentiated exponential for BlackHispanic females breast cancer patients(119899 = 298)


00

100

200

300

400

00

100

200

300

400

0975 10 1025 1075

00

250

500

750

1000

0035 004 0045

00

2000

4000

6000

8000

08 0825 085 0875

00

200

400

600

27 28 29

00

100

200

300

400

120572 sample 50000 120573 sample 50000

120582 sample 50000 1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus025 minus02 minus015 minus33 minus32 minus31 minus30

Figure 4 Kernel density of the posterior parameters for the exponentiated Weibull for Black Hispanic females breast cancer patients(119899 = 298)

forms an observed sample Then given a set of data x =

(1199091 119909

119899) from (1) the likelihood function is given by

119871 (120572 120582 | x) prop (120572120582)119899 expminus

119899

sum

119894=1

(120582119909119894)

times [

119899

prod

119894=1

(1 minus exp minus (120582119909119894))120572minus1

]

(16)

An estimation theory under uncertain prior informationwas discussed in detail by Ahmed [36] Further detailson Bayes and empirical Bayes estimates of survival andhazard functions of a class of distribution were discussed byAhsanullah and Ahmed [21] The estimation of lognormalmean by making use of uncertain prior information wasalso discussed by Ahmed and Tomkins [37] The Bayesianpredictive model from the Weibull life model by means ofa conjugate prior for the scale parameter and a uniform priorfor the shape parameter has been discussed at length byKhanet al [25] The prior density for the scale parameter (120582) canbe given by

119901 (120582) prop 120582 exp minus120582 120582 gt 0 (17)

With reference from Khan et al [25] the shape parameter 120572has a uniform prior over the interval (0 120572) which is given asfollows

119901 (120572) prop

1

120572

120572 gt 0 (18)

Thus the joint prior density is

119901 (120572 120582) prop

120582 exp minus120582

120572

120572 120582 gt 0 (19)

Considering the prior density in (19) the posterior density of120572 and 120582 is given by

119901 (120572 120582 | x) = Ψ0(x) (120572)

119899minus1

120582119899+1 expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times [

119899

prod

119894=1


]

(20)

where Ψ0(x) is a normalizing constant


500

1000

1500

27 275 28 285

00

200

400

600

098 10 102 104

00

500

1000

1500

002 00225 0025 00275

00

1000

2000

3000

4000

00 002 004 006

00

100

200

300

00

25

50

75

100

098 10 102 104 106

00

100

200

300

098 10 102 104 106

00

500

1000

1500

00

00

002 004

120572 sample 50000

120582 sample 50000

1205881 sample 50000 1205882 sample 50000

1205883 sample 50000 1205884 sample 50000

minus39 minus38 minus37 minus36

minus002

minus002

a sample 50000

b sample 50000

Figure 5 Kernel density of the posterior parameters in the case of beta generalized exponential for Black Hispanic females breast cancerpatients (119899 = 298)

41 Predictive Density for a Single Future Response Let 119911 be asingle future response from the model specified by (1) where119911 is independent of the observed data Then the predictivedensity for a single future response (119911) given x = (119909

1 119909

119899)

is

119901 (119911 | x) = int

+infin

120572=0

int

+infin

120582=0

119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572

(21)

where 119901(119911 | 120572 120582) may be defined from model (1) Thus thepredictive density for a single future response is given by

119901 (119911 | x)

=

Ψ1(x) int+infin

120572=0

int

+infin

120582=0

(120572)119899

120582(119899+2)

times exp minus (120582119911) (1 minus exp minus (120582119911))120572minus1

times expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times[

119899

prod

119894=1

(1minusexp minus (120582119909119894))120572minus1

]119889120582119889120572 for 119911gt0 120572 120582gt0

0 elsewhere(22)



098 10 102 106

00

50

100

150

200

095 10 105

00

50

100

150

3850 3900 3950 4000

00

02

04

06

08

596 598 60

00

1000

2000

3000

002 006

00

50

100

150

200

0025 0075

00

50

100

150

1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus002 minus0025

120573 sample 50000

a sample 50000 b sample 50000

Figure 6 Kernel density of the posterior parameters in the case of BIW for Black Hispanic females breast cancer patients (119899 = 298)

0014

0012

0010

0008

0006

0004

0002

Pdf

Hispanic

100 200 300 400

z

Figure 7 The predictive density for a single future response forBlack Hispanic survival data

Figure 7 shows the graphical representation of the predic-tive density based on the Black Hispanic female breast cancerpatientsrsquo survival days It is noted that the predictive densityformed right skewed model

The summary results of Black Hispanic female predictivemeans standard errors and predictive intervals for futuresurvival days are given in Table 8 The predictive shape char-acteristics raw moments corrected moments and measuresof skewness and kurtosis are also presented in Table 8 Thesefindings are very important for health care researchers to

characterize future disease patterns and to make an effectivefuture plans for prevention strategies for the diseases

5 Results and Discussion

The mean plusmn SD of age at diagnosis is 5411 plusmn 1441 yearsfor Black Hispanic group The minimum age at diagnosis forBlack Hispanic was 24 years The mean plusmn SD of survival time(months) for Black Hispanic females was 7138 plusmn 6133 Themajority of these patients were married

The EE model is shown to be a better fit compared toother models for Black Hispanic survival data The lowestDIC value for Black Hispanics is 3136732 In the case of theEE model mean plusmn SD for 120572 and 120582 values is 1234 plusmn 009684

and 001595 plusmn 0001142 respectively Rho (120588) values are asfollows 120588

1= (minus10 10) and 120588

2= (minus10 10)

We used the Bayesian method to determine the inferencefor posterior parameters given the breast cancer survivalmodel Tables 4ndash7 summarize the inferences for the poste-rior parameters using less Markov Chain errors for BlackHispanic females Figures 3ndash6 report the dynamic kerneldensities for each of the parameters for Black Hispanicfemales This helps us to observe the shapes of the kerneldensities

The graphical representation of Black Hispanic femalesrsquobased on future survival times is shown in Figure 7 It shouldbe noted that future survival times forHispanic Black females


Table 8 Predictive inference for Black Hispanic female breastcancer patients survival data

Summary Black HispanicMean 897881SE 12802Raw moments

1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108

Corrected moments1205831

8978811205832

3277841205833

3965821205834

73029 times 107

Skewness and Kurtosis1205731

4465821205732

6797031205741

2113251205742

379703Predictive intervals

90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)

show positively skewed distribution Table 8 summarizes thepredictive raw and corrected moments predictive skewnessand kurtosis and predictive intervals for future response forBlack Hispanic female future survival times

6 Conclusions

There were four types of statistical probability models usedto the Black Hispanic females cancer survival data Theexponentiated exponential model was found to be the bestfitted model to the Black Hispanic females cancer survivaldata compared to the other widely used models

The results of the predictive inference under the fittedmodel were obtained and it was noticed that the shape of thefuture survival model for BlackHispanic is positively skewedGiven the patientrsquos current and past history of reportedconditions these models help the healthcare providers andresearchers to predict a patientrsquos future survival outcomesThus a combination of current knowledge and future predic-tions can be used to enhance and improve the rationales forbetter utilization of current facilities and planned allocationof future resources

Descriptive statistics were obtained by using the SPSSsoftware version 190 The geographic maps of the randomlyselected nine states out of the twelve states were derived usingthe ldquoGoogle fusion tablerdquo [38] We used the SPSS version190 software [39] to obtain basic summary statistics for thebreast cancer survival times for Black Hispanic subset Toshow the graphical representations of the predictive density

for a single future response for Black Hispanic womenwe used advanced computational software package calledldquoMathematica version 80rdquo [40]Weused the same software toderive additional predictive inferences for the ethnicity abouttheir survival times We used the WinBugs software to checkthe goodness of fit tests to derive the summary results of theposterior parameters to determine the kernel densities of theparameters and to carry out all related calculations

Disclosure

All authors have completed the CITI course in the protectionof human research subjects that was required in order torequest the SEERrsquos data from the National Cancer Institutein the United States

Conflict of Interests

Theauthors have declared that there is no conflict of interests

Acknowledgments

The authors would like to thank the editor and the refereesfor their valuable comments and suggestions

References

[1] National Cancer Institute ldquoWhat You Need to Know aboutCancer Understanding Cancerrdquo 2012 httpwwwcancergov

[2] WHO World Health Report 2004 Global Burden of DiseaseWorld Health Organization Geneva Switzerland 2004

[3] ACS Breast Cancer Facts amp Figures 2011-2012 American CancerSociety Atlanta Ga USA 2012

[4] M P Banegas and C I Li ldquoBreast cancer characteristics andoutcomes among Hispanic Black and Hispanic White womenrdquoBreast Cancer Research and Treatment vol 134 no 3 pp 1297ndash1304 2012

[5] R D Gupta and D Kundu ldquoGeneralized exponential distribu-tionsrdquo Australian and New Zealand Journal of Statistics vol 41no 2 pp 173ndash188 1999

[6] R D Gupta and D Kundu ldquoExponentiated exponential familyan alternative to gamma andWeibull distributionsrdquo BiometricalJournal vol 43 no 1 pp 117ndash130 2001

[7] C Escalante-Sandoval ldquoApplication of bivariate extreme valuedistribution to flood frequency analysis a case study of North-westernMexicordquoNaturalHazards vol 42 no 1 pp 37ndash46 2007

[8] M T Madi and M Z Raqab ldquoBayesian prediction of rainfallrecords using the generalized exponential distributionrdquo Envi-ronmetrics vol 18 no 5 pp 541ndash549 2007

[9] R Subburaj G Gopal and P K Kapur ldquoA software reliabilitygrowth model for vital quality metricsrdquo The South AfricanJournal of Industrial Engineering vol 18 pp 93ndash108 2007

[10] N Kannan D Kundu P Nair and R C Tripathi ldquoThe gener-alized exponential cure rate model with covariatesrdquo Journal ofApplied Statistics vol 37 no 10 pp 1625ndash1636 2010

[11] D M Easton and C E Swenberg ldquoTemperature and impulsevelocity in giant axon of squid Loligo pealeirdquo The AmericanJournal of Physiology vol 229 no 5 pp 1249ndash1253 1975


[12] W Barreto-Souza A H S Santos and G M Cordeiro ldquoThebeta generalized exponential distributionrdquo Journal of StatisticalComputation and Simulation vol 80 no 2 pp 159ndash172 2010

[13] W Weibull A Statistical Theory of the Strength of Materialsvol 151 of Ingeniorsvetenskapsakademiens handlingar General-stabens litografiska anstalts forlag Stockholm Sweden 1939

[14] G S Mudholkar and D K Srivastava ldquoExponentiated Weibullfamily for analyzing bathtub failure-rate datardquo IEEE Transac-tions on Reliability vol 42 no 2 pp 299ndash302 1993

[15] M M Nassar and F H Eissa ldquoOn the exponentiated Weibulldistributionrdquo Communications in Statistics vol 32 no 7 pp1317ndash1336 2003

[16] A Choudhury ldquoA simple derivation of moments of the expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

[17] W Nelson Applied Life Data Analysis JohnWiley amp Sons NewYork NY USA 1982

[18] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer New York NY USA 1985

[19] S Geisser Predictive Inference An Introduction Chapman ampHall New York NY USA 1993

[20] J M Bernardo and A F M Smith BayesianTheory JohnWileyamp Sons New York NY USA 1994

[21] M Ahsanullah and S E Ahmed ldquoBayes and empirical Bayesestimates of survival and hazard functions of a class of distribu-tionsrdquo Lecture Notes in Statistics vol 148 pp 81ndash87 2001

[22] A Gelman ldquoParameterization and Bayesian modelingrdquo Journalof the American Statistical Association vol 99 no 466 pp 537ndash545 2004

[23] A Baklizi ldquoPreliminary test estimation in theRayleigh distribu-tion usingminimax regret significance levelsrdquoPertanika Journalof Science amp Technology vol 12 no 1 pp 45ndash50 2004

[24] A Baklizi ldquoLikelihood and Bayesian estimation of Pr (X lt

Y) using lower record values from the generalized exponentialdistributionrdquo Computational Statistics and Data Analysis vol52 no 7 pp 3468ndash3473 2008

[25] H M R Khan M S Haq and S B Provost ldquoPredictivedistributions for responses from theWeibull life testing modelrdquoJournal of Statistical Theory and Applications vol 3 no 1 pp53ndash73 2004

[26] LThabaneContributions to Bayesian statistical inference [PhDthesis] The University of Western Ontario Ontario Canada1998

[27] L Thabane and M S Haq ldquoBayesian prediction from acompound statistical model an actual applicationrdquo PakistanJournal of Statistics vol 16 no 2 pp 110ndash125 2000

[28] M A M Ali Mousa and S A Al-Sagheer ldquoStatistical inferencefor the Rayleigh model based on progressively type-II censoreddatardquo Statistics vol 40 no 2 pp 149ndash157 2006

[29] M Z Raqab ldquoOn the maximum likelihood prediction of theexponential distribution based on double type II censoredsamplesrdquo Pakistan Journal of Statistics vol 11 no 1 pp 1ndash101995

[30] M Z Raqab ldquoDistribution-free prediction intervals for thefuture current record statisticsrdquo Statistical Papers vol 50 no2 pp 429ndash439 2009

[31] P Congdon ldquoBayesian predictivemodel comparison via parallelsamplingrdquo Computational Statistics and Data Analysis vol 48no 4 pp 735ndash753 2005

[32] P Congdon ldquoMixtures of spatial and unstructured effectsfor spatially discontinuous health outcomesrdquo ComputationalStatistics and Data Analysis vol 51 no 6 pp 3197ndash3212 2007

[33] S E Ahmed K Budsaba S Lisawadi and A I VolodinldquoParametric estimation for the Birnbaum-Saunders lifetimedistribution based on a new parameterizationrdquoThailand Statis-tician vol 6 pp 213ndash240 2008

[34] J A Achcar E A Coelho-Barros and G M CordeiroldquoBeta distributions and related exponential models a Bayesianapproachrdquo 2009

[35] WRGilks S Richardson andD J SpiegelhalterMarkovChainMonte Carlo in Practice Chapman amp Hall Suffolk UK 1996

[36] S E Ahmed ldquoImproved shrinkage estimation of relativepotencyrdquo Biometrical Journal vol 37 pp 627ndash638 1995

[37] S E Ahmed and R J Tomkins ldquoEstimating lognormal meanusing uncertain prior informationrdquo Pakistan Journal of Statis-tics vol 11 pp 67ndash92 1995

[38] H Gonzalez A Y Halevy C S Jensen et al ldquoGoogle fusiontables web-centered data management and collaborationrdquo inProceedings of the International Conference on Management ofData (SIGMOD rsquo10) pp 1061ndash1066 June 2010

[39] IBM Corp IBM SPSS Statistics for Windows Version 190 IBMCorp Armonk NY USA 2010

[40] Wolfram ResearchTheMathematica Archive Mathematica 80Wolfram Research Inc Illinois Ill USA 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Research (AICR) over 226000 cases of breast cancer arediagnosed every year in USA and approximately 40000American women die of breast cancer every year [3]

12 Breast Cancer according to Race and Ethnicity In theUnited States breast cancer is one of the most frequentlydiagnosed cancers across different racial and ethnic groups[2] Raceethnicity specific incidence rates remained fairlyconstant for all racial and ethnic groups during the years2004ndash2008 [1] Previously it was believed that family historysocioeconomic status levels of education frequency ofmam-mograms and access to health care resources were some ofthemajor determinants affecting the prognosis of the diseaseHowever recent studies have shown that racial and ethnicfactors also contribute significant risk for the prognosis of thedisease

The American Cancer Society has found evidence thatthere are notable differences in breast cancer death ratesbetween different states across various socioeconomic strataand between different racialethnic groups [1] Although ageis the strongest predictor for breast cancer risk raceethnicitycould also be a major risk factor [1] Since the early 1990sbreast cancer mortality rates have decreased among all ethnicgroups except the American IndiansAlaska Natives therebyshowing another racial disparity associated with the diseaseIn the United StatesWhite women are more likely to developbreast cancer than African-American Hispanic Asian orAmerican IndianAlaska native women [1]

13 Hispanic Black Women Although Hispanics are thefastest growing minority population in the United Statesthere are notmuch breast cancer statistics onBlackHispanicsspecifically Breast cancer data for Hispanics are usuallytabulated under one ethnic group (Hispanic) therefore race-specific breast cancer data for the Hispanic population is notreadily available Overall the incidence andmortality rates ofbreast cancer among Hispanic (Black andWhite) women arelower than non-HispanicWhite women [3] Hispanic women(Black and White) show lower levels of awareness about therisk factors associated with the disease and have less access tohealth care facilities when compared to women of any otherethnicity

Unfortunately there are not many studies that elucidatebreast cancer disparities among different races within theHispanic ethnicity Usual research findings describe breastcancer incidence mortality and death rates and other vitalstatistics associated with the disease among Hispanic womenwithout any details about interracial differences within theethnicity Banegas and Li [4] have asserted that further studyof specific breast cancer outcomes among the different racesof Hispanic women could greatly enhance knowledge aboutthe distribution and determinants of the disease in this highrisk ethnic group Their study showed that non-HispanicBlacks had a 15ndash25-fold greater risk of having stage IV breastcancer types and 10ndash50 greater risk of breast cancer specificmortality compared to non-Hispanic whites [4] This findingagain shows the need for a study that tries to understand the

current state of affairs about breast cancer survival in thissubpopulation within United States

14 Statistical Probability Models Healthcare personnel hascollected vast amounts of phenomic and genomic datawhich should bemaximally utilized for research perspectivesThese large databases should be tested with newer statisticalmethods and statistical probability models This would bevery useful to predict future patterns of diseasemorbidity andmortality thereby enhancing our understanding the severityand outcomes

Data may follow several statistical probability modelslike exponential gamma Weibull normal half-normal log-normal Rayleigh inverse Gaussian exponentiated expo-nential (EE) exponentiated Weibull (EW) beta generalizedexponential (BGE) beta inverseWeibull (BIW) and so forthStatistical models are immensely useful to characterize thedata and derive reliable scientific inferences

The biomedical and engineering fields often use exponen-tiated exponentialmodel (EEM) for datamodelingTheEEMa generalization of exponential distribution was introducedby Gupta and Kundu [5] and received rapid and widespreadacceptance The EEM considers two parameters ldquoshaperdquo andldquoscalerdquo Moreover Gupta and Kundu [6] observed that theEEM is similar to the Weibull family and suggested thepossibility of using the EE distribution as a substitute forWeibull model

A random variable 119909 is said to follow the exponentiatedexponential distribution if its probability density function(pdf) is given by

119901 (119909) = 120572120582 exp minus (120582119909) (1 minus exp minus (120582119909))120572minus1

(1)

where 120572 gt 0 is the shape parameter and 120582 gt 0 isthe scale parameter Gupta and Kundu [5] introduced theabove mentioned density function for exponentiated expo-nential distribution The random variable can be expressedas 119909 sim EE(120572 120582) Some interesting applications of EEMinclude designing rainfall estimation in the Coast of Chiapas[7] analysis of Los Angeles rainfall data [8] and softwarereliability growth models for vital quality metrics [9] A curerate model based on the generalized exponential distributionthat incorporates the effects of risk factors or covariates forthe probability of an individual being a long-time survivorwas proposed by Kannan et al [10] Also Gompertrsquoz form ofexponentiated exponential model was used to predict squidaxon voltage clamp conductance [11]

For a beta generalized exponential model the probabilitydensity function is given by

119901 (119909) =

120572120582

119861 (119886 119887)

exp minus (120582119909)

times (1 minus exp minus (120582119909))119886120572minus1

(1 minus (1 minus exp minus (120582119909))120572

)

119887minus1

(2)

where 120572 gt 0 is the shape parameter and 120582 gt 0 is thescale parameter There are two additional parameters 119886 gt

0 and 119887 gt 0 The role of these parameters is to describe





119901 (119909) = 120572120573120582119909120573minus1 exp minus (120582119909

120573

) (1 minus exp minus (120582119909120573

))

120572minus1

(3)



119901 (119909) =

120573119909minus(120573+1)

119861 (119886 119887)



))

120572minus1


))

119887minus1

(4)

















n = 298

N1 = 300

(1973ndash2009)N = 608032


WA

CAUT

NM

IA MI

HI

GA

CT











1198761

441198762

521198763

64Variance 20765



1198761

251198762

50501198763










1 1199092 119909



by

ℓ (120572 120582 | x) = 119899 log (120572) + 119899 log (120582) + (120572 minus 1)

times

119899

sum

119894=1


119899

sum

119894=1

119909119894

(5)

Assume 1205881= log(120572) and 120588


1and



119894be


given by

119901 (1205881 1205882| x)

= 119901 (1205881 1205882) times exp119899120588

1+ 1198991205882minus 119899119909 exp (120588

2)

+ (exp (1205881) minus 1)

times

119899

sum

119894=1


2))

(6)


ℓ (120572 120582 119886 119887 | x) = 119899 log(120572

120582

119861 (119886 119887)

) minus 120582

119899

sum

119894=1

119909119894+ (119886120572 minus 1)

times

119899

sum

119894=1


times

119899

sum

119894=1


)

(7)

Assume 1205881

= log(119886) 1205882

= log(119887) 1205883

= log(120572) and1205884


4are


for 1205881 1205882 1205883 and 120588


119895


density is given by

119901 (1205881 1205882 1205883 1205884| x)

= 119901 (1205881 1205882 1205883 1205884)

times [119899 log(

1198901205883+1205884

119861 (1198901205881 1198901205882)

) minus 1198901205884

119899

sum

119894=1

119909119894+ (1198901205881+1205883

minus 1)

times

119899

sum

119894=1


minus 1)

times(

119899

sum

119894=1


))]

(8)


ℓ (120572 120573 120582 | x)

= 119899 log (120572) + 119899 log (120573) + 119899 log (120582) + (120572 minus 1)

times

119899

sum

119894=1


119894))

minus 120582

119899

sum

119894=1

119909120573

119894+ (120573 minus 1)

119899

sum

119894=1

log (119909119894)

(9)

Assume 1205881= log(120572) 120588

2= log(120573) and 120588


assumed that 1205881 1205882 and 120588



3 let a uniform


119896 119890119896) for all 119896 = 1 2 3


119901 (1205881 1205882 1205883| x)

= 119901 (1205881 1205882 1205883) times exp119899 (120588

1+ 1205882+ 1205883)

minus 1198901205883

119899

sum

119894=1

1199091198901205882

119894+ (1198901205881

minus 1)

times

119899

sum

119894=1


119894)

+ (1198901205882

minus 1)

119899

sum

119894=1

log (119909119894)

(10)


ℓ (120573 119886 119887 | x)

= 119899 log(

120573

119861 (119886 119887)

) minus 119886

119899

sum

119894=1

119909minus120573

119894+ (120573 minus 1)

times

119899

sum

119894=1

log (119909119894) + (119887 minus 1)

119899

sum

119894=1


119894)

(11)




Assume 1205881= log(120573) 120588

2= log(119886) and 120588


assume that 1205881 1205882 and 120588



3 let a uniform


ℎ 119891ℎ) for all ℎ = 1 2 3


119901 (1205881 1205882 1205883| x)

= 119901 (1205881 1205882 1205883) [119899 log(

1198901205881

119861 (1198901205882 1198901205883)

)

minus 1198901205882

119899

sum

119894=1

119909minus1198901205881

119894+ (1198901205881

minus 1)

times

119899

sum

119894=1

log (119909119894) + (1198901205883

minus 1)

times

119899

sum

119894=1


minus1198901205881

)]

(12)









1 1205882 and












02073 007844 726119864 minus 04 02076 (005193 03608) 500001205882






1009 0008985 877119864 minus 05 1006 (1000 1033) 500001205882

















119901 (119911 | x) = ∬119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572 (13)


119901 (120572 120582 | x) = Ψ (x) 119871 (120572 120582 | x) 119901 (120572 120582) (14)


Ψ(x)minus1 = ∬119871 (120572 120582 | x) 119901 (120572 120582) 119889120582 119889120572 (15)


119899be a ran-


119899)1015840




1006 0005986 594119864 minus 05 1004 (100 1022) 500001205882

002805 001952 151119864 minus 04 002482 (0001068 006674) 500001205883

0005888 0005865 532119864 minus 05 0004063 (148119864 minus 04 002165) 500001205884



2simdunif (0 007) 120588





5997 0003353 253119864 minus 05 5998 (5988 6000) 500001205882

003002 001736 814119864 minus 05 002997 (0001484 005849) 500001205883



2simdunif (0 006) and 120588

3simdunif (0 009)

075 10 125 15

00

20

40

60

001 0015 002

00

1000

2000

3000

4000

00

20

40

60

00 02 04

00

20

40

60


120572 sample 50000 120582 sample 50000

1205881 sample 50000 1205882 sample 50000



00

100

200

300

400

00

100

200

300

400

0975 10 1025 1075

00

250

500

750

1000

0035 004 0045

00

2000

4000

6000

8000

08 0825 085 0875

00

200

400

600

27 28 29

00

100

200

300

400

120572 sample 50000 120573 sample 50000

120582 sample 50000 1205881 sample 50000

1205882 sample 50000 1205883 sample 50000




(1199091 119909


119871 (120572 120582 | x) prop (120572120582)119899 expminus

119899

sum

119894=1

(120582119909119894)

times [

119899

prod

119894=1


]

(16)


119901 (120582) prop 120582 exp minus120582 120582 gt 0 (17)


119901 (120572) prop

1

120572

120572 gt 0 (18)


119901 (120572 120582) prop

120582 exp minus120582

120572

120572 120582 gt 0 (19)


119901 (120572 120582 | x) = Ψ0(x) (120572)

119899minus1

120582119899+1 expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times [

119899

prod

119894=1


]

(20)



500

1000

1500

27 275 28 285

00

200

400

600

098 10 102 104

00

500

1000

1500

002 00225 0025 00275

00

1000

2000

3000

4000

00 002 004 006

00

100

200

300

00

25

50

75

100

098 10 102 104 106

00

100

200

300

098 10 102 104 106

00

500

1000

1500

00

00

002 004

120572 sample 50000

120582 sample 50000

1205881 sample 50000 1205882 sample 50000

1205883 sample 50000 1205884 sample 50000


minus002

minus002

a sample 50000

b sample 50000



1 119909

119899)

is

119901 (119911 | x) = int

+infin

120572=0

int

+infin

120582=0

119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572

(21)


119901 (119911 | x)

=

Ψ1(x) int+infin

120572=0

int

+infin

120582=0

(120572)119899

120582(119899+2)


times expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times[

119899

prod

119894=1


]119889120582119889120572 for 119911gt0 120572 120582gt0

0 elsewhere(22)



098 10 102 106

00

50

100

150

200

095 10 105

00

50

100

150

3850 3900 3950 4000

00

02

04

06

08

596 598 60

00

1000

2000

3000

002 006

00

50

100

150

200

0025 0075

00

50

100

150

1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus002 minus0025

120573 sample 50000



0014

0012

0010

0008

0006

0004

0002

Pdf

Hispanic

100 200 300 400

z









1= (minus10 10) and 120588

2= (minus10 10)






1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108


8978811205832

3277841205833

3965821205834

73029 times 107


4465821205732

6797031205741

2113251205742


90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)


6 Conclusions





Disclosure




Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119901 (119909) = 120572120573120582119909120573minus1 exp minus (120582119909

120573

) (1 minus exp minus (120582119909120573

))

120572minus1

(3)



119901 (119909) =

120573119909minus(120573+1)

119861 (119886 119887)



))

120572minus1


))

119887minus1

(4)

















n = 298

N1 = 300

(1973ndash2009)N = 608032


WA

CAUT

NM

IA MI

HI

GA

CT











1198761

441198762

521198763

64Variance 20765



1198761

251198762

50501198763










1 1199092 119909



by

ℓ (120572 120582 | x) = 119899 log (120572) + 119899 log (120582) + (120572 minus 1)

times

119899

sum

119894=1


119899

sum

119894=1

119909119894

(5)

Assume 1205881= log(120572) and 120588


1and



119894be


given by

119901 (1205881 1205882| x)

= 119901 (1205881 1205882) times exp119899120588

1+ 1198991205882minus 119899119909 exp (120588

2)

+ (exp (1205881) minus 1)

times

119899

sum

119894=1


2))

(6)


ℓ (120572 120582 119886 119887 | x) = 119899 log(120572

120582

119861 (119886 119887)

) minus 120582

119899

sum

119894=1

119909119894+ (119886120572 minus 1)

times

119899

sum

119894=1


times

119899

sum

119894=1


)

(7)

Assume 1205881

= log(119886) 1205882

= log(119887) 1205883

= log(120572) and1205884


4are


for 1205881 1205882 1205883 and 120588


119895


density is given by

119901 (1205881 1205882 1205883 1205884| x)

= 119901 (1205881 1205882 1205883 1205884)

times [119899 log(

1198901205883+1205884

119861 (1198901205881 1198901205882)

) minus 1198901205884

119899

sum

119894=1

119909119894+ (1198901205881+1205883

minus 1)

times

119899

sum

119894=1


minus 1)

times(

119899

sum

119894=1


))]

(8)


ℓ (120572 120573 120582 | x)

= 119899 log (120572) + 119899 log (120573) + 119899 log (120582) + (120572 minus 1)

times

119899

sum

119894=1


119894))

minus 120582

119899

sum

119894=1

119909120573

119894+ (120573 minus 1)

119899

sum

119894=1

log (119909119894)

(9)

Assume 1205881= log(120572) 120588

2= log(120573) and 120588


assumed that 1205881 1205882 and 120588



3 let a uniform


119896 119890119896) for all 119896 = 1 2 3


119901 (1205881 1205882 1205883| x)

= 119901 (1205881 1205882 1205883) times exp119899 (120588

1+ 1205882+ 1205883)

minus 1198901205883

119899

sum

119894=1

1199091198901205882

119894+ (1198901205881

minus 1)

times

119899

sum

119894=1


119894)

+ (1198901205882

minus 1)

119899

sum

119894=1

log (119909119894)

(10)


ℓ (120573 119886 119887 | x)

= 119899 log(

120573

119861 (119886 119887)

) minus 119886

119899

sum

119894=1

119909minus120573

119894+ (120573 minus 1)

times

119899

sum

119894=1

log (119909119894) + (119887 minus 1)

119899

sum

119894=1


119894)

(11)




Assume 1205881= log(120573) 120588

2= log(119886) and 120588


assume that 1205881 1205882 and 120588



3 let a uniform


ℎ 119891ℎ) for all ℎ = 1 2 3


119901 (1205881 1205882 1205883| x)

= 119901 (1205881 1205882 1205883) [119899 log(

1198901205881

119861 (1198901205882 1198901205883)

)

minus 1198901205882

119899

sum

119894=1

119909minus1198901205881

119894+ (1198901205881

minus 1)

times

119899

sum

119894=1

log (119909119894) + (1198901205883

minus 1)

times

119899

sum

119894=1


minus1198901205881

)]

(12)









1 1205882 and












02073 007844 726119864 minus 04 02076 (005193 03608) 500001205882






1009 0008985 877119864 minus 05 1006 (1000 1033) 500001205882

















119901 (119911 | x) = ∬119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572 (13)


119901 (120572 120582 | x) = Ψ (x) 119871 (120572 120582 | x) 119901 (120572 120582) (14)


Ψ(x)minus1 = ∬119871 (120572 120582 | x) 119901 (120572 120582) 119889120582 119889120572 (15)


119899be a ran-


119899)1015840




1006 0005986 594119864 minus 05 1004 (100 1022) 500001205882

002805 001952 151119864 minus 04 002482 (0001068 006674) 500001205883

0005888 0005865 532119864 minus 05 0004063 (148119864 minus 04 002165) 500001205884



2simdunif (0 007) 120588





5997 0003353 253119864 minus 05 5998 (5988 6000) 500001205882

003002 001736 814119864 minus 05 002997 (0001484 005849) 500001205883



2simdunif (0 006) and 120588

3simdunif (0 009)

075 10 125 15

00

20

40

60

001 0015 002

00

1000

2000

3000

4000

00

20

40

60

00 02 04

00

20

40

60


120572 sample 50000 120582 sample 50000

1205881 sample 50000 1205882 sample 50000



00

100

200

300

400

00

100

200

300

400

0975 10 1025 1075

00

250

500

750

1000

0035 004 0045

00

2000

4000

6000

8000

08 0825 085 0875

00

200

400

600

27 28 29

00

100

200

300

400

120572 sample 50000 120573 sample 50000

120582 sample 50000 1205881 sample 50000

1205882 sample 50000 1205883 sample 50000




(1199091 119909


119871 (120572 120582 | x) prop (120572120582)119899 expminus

119899

sum

119894=1

(120582119909119894)

times [

119899

prod

119894=1


]

(16)


119901 (120582) prop 120582 exp minus120582 120582 gt 0 (17)


119901 (120572) prop

1

120572

120572 gt 0 (18)


119901 (120572 120582) prop

120582 exp minus120582

120572

120572 120582 gt 0 (19)


119901 (120572 120582 | x) = Ψ0(x) (120572)

119899minus1

120582119899+1 expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times [

119899

prod

119894=1


]

(20)



500

1000

1500

27 275 28 285

00

200

400

600

098 10 102 104

00

500

1000

1500

002 00225 0025 00275

00

1000

2000

3000

4000

00 002 004 006

00

100

200

300

00

25

50

75

100

098 10 102 104 106

00

100

200

300

098 10 102 104 106

00

500

1000

1500

00

00

002 004

120572 sample 50000

120582 sample 50000

1205881 sample 50000 1205882 sample 50000

1205883 sample 50000 1205884 sample 50000


minus002

minus002

a sample 50000

b sample 50000



1 119909

119899)

is

119901 (119911 | x) = int

+infin

120572=0

int

+infin

120582=0

119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572

(21)


119901 (119911 | x)

=

Ψ1(x) int+infin

120572=0

int

+infin

120582=0

(120572)119899

120582(119899+2)


times expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times[

119899

prod

119894=1


]119889120582119889120572 for 119911gt0 120572 120582gt0

0 elsewhere(22)



098 10 102 106

00

50

100

150

200

095 10 105

00

50

100

150

3850 3900 3950 4000

00

02

04

06

08

596 598 60

00

1000

2000

3000

002 006

00

50

100

150

200

0025 0075

00

50

100

150

1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus002 minus0025

120573 sample 50000



0014

0012

0010

0008

0006

0004

0002

Pdf

Hispanic

100 200 300 400

z









1= (minus10 10) and 120588

2= (minus10 10)






1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108


8978811205832

3277841205833

3965821205834

73029 times 107


4465821205732

6797031205741

2113251205742


90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)


6 Conclusions





Disclosure




Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













n = 298

N1 = 300

(1973ndash2009)N = 608032


WA

CAUT

NM

IA MI

HI

GA

CT











1198761

441198762

521198763

64Variance 20765



1198761

251198762

50501198763










1 1199092 119909



by

ℓ (120572 120582 | x) = 119899 log (120572) + 119899 log (120582) + (120572 minus 1)

times

119899

sum

119894=1


119899

sum

119894=1

119909119894

(5)

Assume 1205881= log(120572) and 120588


1and



119894be


given by

119901 (1205881 1205882| x)

= 119901 (1205881 1205882) times exp119899120588

1+ 1198991205882minus 119899119909 exp (120588

2)

+ (exp (1205881) minus 1)

times

119899

sum

119894=1


2))

(6)


ℓ (120572 120582 119886 119887 | x) = 119899 log(120572

120582

119861 (119886 119887)

) minus 120582

119899

sum

119894=1

119909119894+ (119886120572 minus 1)

times

119899

sum

119894=1


times

119899

sum

119894=1


)

(7)

Assume 1205881

= log(119886) 1205882

= log(119887) 1205883

= log(120572) and1205884


4are


for 1205881 1205882 1205883 and 120588


119895


density is given by

119901 (1205881 1205882 1205883 1205884| x)

= 119901 (1205881 1205882 1205883 1205884)

times [119899 log(

1198901205883+1205884

119861 (1198901205881 1198901205882)

) minus 1198901205884

119899

sum

119894=1

119909119894+ (1198901205881+1205883

minus 1)

times

119899

sum

119894=1


minus 1)

times(

119899

sum

119894=1


))]

(8)


ℓ (120572 120573 120582 | x)

= 119899 log (120572) + 119899 log (120573) + 119899 log (120582) + (120572 minus 1)

times

119899

sum

119894=1


119894))

minus 120582

119899

sum

119894=1

119909120573

119894+ (120573 minus 1)

119899

sum

119894=1

log (119909119894)

(9)

Assume 1205881= log(120572) 120588

2= log(120573) and 120588


assumed that 1205881 1205882 and 120588



3 let a uniform


119896 119890119896) for all 119896 = 1 2 3


119901 (1205881 1205882 1205883| x)

= 119901 (1205881 1205882 1205883) times exp119899 (120588

1+ 1205882+ 1205883)

minus 1198901205883

119899

sum

119894=1

1199091198901205882

119894+ (1198901205881

minus 1)

times

119899

sum

119894=1


119894)

+ (1198901205882

minus 1)

119899

sum

119894=1

log (119909119894)

(10)


ℓ (120573 119886 119887 | x)

= 119899 log(

120573

119861 (119886 119887)

) minus 119886

119899

sum

119894=1

119909minus120573

119894+ (120573 minus 1)

times

119899

sum

119894=1

log (119909119894) + (119887 minus 1)

119899

sum

119894=1


119894)

(11)




Assume 1205881= log(120573) 120588

2= log(119886) and 120588


assume that 1205881 1205882 and 120588



3 let a uniform


ℎ 119891ℎ) for all ℎ = 1 2 3


119901 (1205881 1205882 1205883| x)

= 119901 (1205881 1205882 1205883) [119899 log(

1198901205881

119861 (1198901205882 1198901205883)

)

minus 1198901205882

119899

sum

119894=1

119909minus1198901205881

119894+ (1198901205881

minus 1)

times

119899

sum

119894=1

log (119909119894) + (1198901205883

minus 1)

times

119899

sum

119894=1


minus1198901205881

)]

(12)









1 1205882 and












02073 007844 726119864 minus 04 02076 (005193 03608) 500001205882






1009 0008985 877119864 minus 05 1006 (1000 1033) 500001205882

















119901 (119911 | x) = ∬119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572 (13)


119901 (120572 120582 | x) = Ψ (x) 119871 (120572 120582 | x) 119901 (120572 120582) (14)


Ψ(x)minus1 = ∬119871 (120572 120582 | x) 119901 (120572 120582) 119889120582 119889120572 (15)


119899be a ran-


119899)1015840




1006 0005986 594119864 minus 05 1004 (100 1022) 500001205882

002805 001952 151119864 minus 04 002482 (0001068 006674) 500001205883

0005888 0005865 532119864 minus 05 0004063 (148119864 minus 04 002165) 500001205884



2simdunif (0 007) 120588





5997 0003353 253119864 minus 05 5998 (5988 6000) 500001205882

003002 001736 814119864 minus 05 002997 (0001484 005849) 500001205883



2simdunif (0 006) and 120588

3simdunif (0 009)

075 10 125 15

00

20

40

60

001 0015 002

00

1000

2000

3000

4000

00

20

40

60

00 02 04

00

20

40

60


120572 sample 50000 120582 sample 50000

1205881 sample 50000 1205882 sample 50000



00

100

200

300

400

00

100

200

300

400

0975 10 1025 1075

00

250

500

750

1000

0035 004 0045

00

2000

4000

6000

8000

08 0825 085 0875

00

200

400

600

27 28 29

00

100

200

300

400

120572 sample 50000 120573 sample 50000

120582 sample 50000 1205881 sample 50000

1205882 sample 50000 1205883 sample 50000




(1199091 119909


119871 (120572 120582 | x) prop (120572120582)119899 expminus

119899

sum

119894=1

(120582119909119894)

times [

119899

prod

119894=1


]

(16)


119901 (120582) prop 120582 exp minus120582 120582 gt 0 (17)


119901 (120572) prop

1

120572

120572 gt 0 (18)


119901 (120572 120582) prop

120582 exp minus120582

120572

120572 120582 gt 0 (19)


119901 (120572 120582 | x) = Ψ0(x) (120572)

119899minus1

120582119899+1 expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times [

119899

prod

119894=1


]

(20)



500

1000

1500

27 275 28 285

00

200

400

600

098 10 102 104

00

500

1000

1500

002 00225 0025 00275

00

1000

2000

3000

4000

00 002 004 006

00

100

200

300

00

25

50

75

100

098 10 102 104 106

00

100

200

300

098 10 102 104 106

00

500

1000

1500

00

00

002 004

120572 sample 50000

120582 sample 50000

1205881 sample 50000 1205882 sample 50000

1205883 sample 50000 1205884 sample 50000


minus002

minus002

a sample 50000

b sample 50000



1 119909

119899)

is

119901 (119911 | x) = int

+infin

120572=0

int

+infin

120582=0

119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572

(21)


119901 (119911 | x)

=

Ψ1(x) int+infin

120572=0

int

+infin

120582=0

(120572)119899

120582(119899+2)


times expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times[

119899

prod

119894=1


]119889120582119889120572 for 119911gt0 120572 120582gt0

0 elsewhere(22)



098 10 102 106

00

50

100

150

200

095 10 105

00

50

100

150

3850 3900 3950 4000

00

02

04

06

08

596 598 60

00

1000

2000

3000

002 006

00

50

100

150

200

0025 0075

00

50

100

150

1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus002 minus0025

120573 sample 50000



0014

0012

0010

0008

0006

0004

0002

Pdf

Hispanic

100 200 300 400

z









1= (minus10 10) and 120588

2= (minus10 10)






1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108


8978811205832

3277841205833

3965821205834

73029 times 107


4465821205732

6797031205741

2113251205742


90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)


6 Conclusions





Disclosure




Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 1199092 119909



by

ℓ (120572 120582 | x) = 119899 log (120572) + 119899 log (120582) + (120572 minus 1)

times

119899

sum

119894=1


119899

sum

119894=1

119909119894

(5)

Assume 1205881= log(120572) and 120588


1and



119894be


given by

119901 (1205881 1205882| x)

= 119901 (1205881 1205882) times exp119899120588

1+ 1198991205882minus 119899119909 exp (120588

2)

+ (exp (1205881) minus 1)

times

119899

sum

119894=1


2))

(6)


ℓ (120572 120582 119886 119887 | x) = 119899 log(120572

120582

119861 (119886 119887)

) minus 120582

119899

sum

119894=1

119909119894+ (119886120572 minus 1)

times

119899

sum

119894=1


times

119899

sum

119894=1


)

(7)

Assume 1205881

= log(119886) 1205882

= log(119887) 1205883

= log(120572) and1205884


4are


for 1205881 1205882 1205883 and 120588


119895


density is given by

119901 (1205881 1205882 1205883 1205884| x)

= 119901 (1205881 1205882 1205883 1205884)

times [119899 log(

1198901205883+1205884

119861 (1198901205881 1198901205882)

) minus 1198901205884

119899

sum

119894=1

119909119894+ (1198901205881+1205883

minus 1)

times

119899

sum

119894=1


minus 1)

times(

119899

sum

119894=1


))]

(8)


ℓ (120572 120573 120582 | x)

= 119899 log (120572) + 119899 log (120573) + 119899 log (120582) + (120572 minus 1)

times

119899

sum

119894=1


119894))

minus 120582

119899

sum

119894=1

119909120573

119894+ (120573 minus 1)

119899

sum

119894=1

log (119909119894)

(9)

Assume 1205881= log(120572) 120588

2= log(120573) and 120588


assumed that 1205881 1205882 and 120588



3 let a uniform


119896 119890119896) for all 119896 = 1 2 3


119901 (1205881 1205882 1205883| x)

= 119901 (1205881 1205882 1205883) times exp119899 (120588

1+ 1205882+ 1205883)

minus 1198901205883

119899

sum

119894=1

1199091198901205882

119894+ (1198901205881

minus 1)

times

119899

sum

119894=1


119894)

+ (1198901205882

minus 1)

119899

sum

119894=1

log (119909119894)

(10)


ℓ (120573 119886 119887 | x)

= 119899 log(

120573

119861 (119886 119887)

) minus 119886

119899

sum

119894=1

119909minus120573

119894+ (120573 minus 1)

times

119899

sum

119894=1

log (119909119894) + (119887 minus 1)

119899

sum

119894=1


119894)

(11)




Assume 1205881= log(120573) 120588

2= log(119886) and 120588


assume that 1205881 1205882 and 120588



3 let a uniform


ℎ 119891ℎ) for all ℎ = 1 2 3


119901 (1205881 1205882 1205883| x)

= 119901 (1205881 1205882 1205883) [119899 log(

1198901205881

119861 (1198901205882 1198901205883)

)

minus 1198901205882

119899

sum

119894=1

119909minus1198901205881

119894+ (1198901205881

minus 1)

times

119899

sum

119894=1

log (119909119894) + (1198901205883

minus 1)

times

119899

sum

119894=1


minus1198901205881

)]

(12)









1 1205882 and












02073 007844 726119864 minus 04 02076 (005193 03608) 500001205882






1009 0008985 877119864 minus 05 1006 (1000 1033) 500001205882

















119901 (119911 | x) = ∬119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572 (13)


119901 (120572 120582 | x) = Ψ (x) 119871 (120572 120582 | x) 119901 (120572 120582) (14)


Ψ(x)minus1 = ∬119871 (120572 120582 | x) 119901 (120572 120582) 119889120582 119889120572 (15)


119899be a ran-


119899)1015840




1006 0005986 594119864 minus 05 1004 (100 1022) 500001205882

002805 001952 151119864 minus 04 002482 (0001068 006674) 500001205883

0005888 0005865 532119864 minus 05 0004063 (148119864 minus 04 002165) 500001205884



2simdunif (0 007) 120588





5997 0003353 253119864 minus 05 5998 (5988 6000) 500001205882

003002 001736 814119864 minus 05 002997 (0001484 005849) 500001205883



2simdunif (0 006) and 120588

3simdunif (0 009)

075 10 125 15

00

20

40

60

001 0015 002

00

1000

2000

3000

4000

00

20

40

60

00 02 04

00

20

40

60


120572 sample 50000 120582 sample 50000

1205881 sample 50000 1205882 sample 50000



00

100

200

300

400

00

100

200

300

400

0975 10 1025 1075

00

250

500

750

1000

0035 004 0045

00

2000

4000

6000

8000

08 0825 085 0875

00

200

400

600

27 28 29

00

100

200

300

400

120572 sample 50000 120573 sample 50000

120582 sample 50000 1205881 sample 50000

1205882 sample 50000 1205883 sample 50000




(1199091 119909


119871 (120572 120582 | x) prop (120572120582)119899 expminus

119899

sum

119894=1

(120582119909119894)

times [

119899

prod

119894=1


]

(16)


119901 (120582) prop 120582 exp minus120582 120582 gt 0 (17)


119901 (120572) prop

1

120572

120572 gt 0 (18)


119901 (120572 120582) prop

120582 exp minus120582

120572

120572 120582 gt 0 (19)


119901 (120572 120582 | x) = Ψ0(x) (120572)

119899minus1

120582119899+1 expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times [

119899

prod

119894=1


]

(20)



500

1000

1500

27 275 28 285

00

200

400

600

098 10 102 104

00

500

1000

1500

002 00225 0025 00275

00

1000

2000

3000

4000

00 002 004 006

00

100

200

300

00

25

50

75

100

098 10 102 104 106

00

100

200

300

098 10 102 104 106

00

500

1000

1500

00

00

002 004

120572 sample 50000

120582 sample 50000

1205881 sample 50000 1205882 sample 50000

1205883 sample 50000 1205884 sample 50000


minus002

minus002

a sample 50000

b sample 50000



1 119909

119899)

is

119901 (119911 | x) = int

+infin

120572=0

int

+infin

120582=0

119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572

(21)


119901 (119911 | x)

=

Ψ1(x) int+infin

120572=0

int

+infin

120582=0

(120572)119899

120582(119899+2)


times expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times[

119899

prod

119894=1


]119889120582119889120572 for 119911gt0 120572 120582gt0

0 elsewhere(22)



098 10 102 106

00

50

100

150

200

095 10 105

00

50

100

150

3850 3900 3950 4000

00

02

04

06

08

596 598 60

00

1000

2000

3000

002 006

00

50

100

150

200

0025 0075

00

50

100

150

1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus002 minus0025

120573 sample 50000



0014

0012

0010

0008

0006

0004

0002

Pdf

Hispanic

100 200 300 400

z









1= (minus10 10) and 120588

2= (minus10 10)






1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108


8978811205832

3277841205833

3965821205834

73029 times 107


4465821205732

6797031205741

2113251205742


90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)


6 Conclusions





Disclosure




Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Assume 1205881= log(120573) 120588

2= log(119886) and 120588


assume that 1205881 1205882 and 120588



3 let a uniform


ℎ 119891ℎ) for all ℎ = 1 2 3


119901 (1205881 1205882 1205883| x)

= 119901 (1205881 1205882 1205883) [119899 log(

1198901205881

119861 (1198901205882 1198901205883)

)

minus 1198901205882

119899

sum

119894=1

119909minus1198901205881

119894+ (1198901205881

minus 1)

times

119899

sum

119894=1

log (119909119894) + (1198901205883

minus 1)

times

119899

sum

119894=1


minus1198901205881

)]

(12)









1 1205882 and












02073 007844 726119864 minus 04 02076 (005193 03608) 500001205882






1009 0008985 877119864 minus 05 1006 (1000 1033) 500001205882

















119901 (119911 | x) = ∬119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572 (13)


119901 (120572 120582 | x) = Ψ (x) 119871 (120572 120582 | x) 119901 (120572 120582) (14)


Ψ(x)minus1 = ∬119871 (120572 120582 | x) 119901 (120572 120582) 119889120582 119889120572 (15)


119899be a ran-


119899)1015840




1006 0005986 594119864 minus 05 1004 (100 1022) 500001205882

002805 001952 151119864 minus 04 002482 (0001068 006674) 500001205883

0005888 0005865 532119864 minus 05 0004063 (148119864 minus 04 002165) 500001205884



2simdunif (0 007) 120588





5997 0003353 253119864 minus 05 5998 (5988 6000) 500001205882

003002 001736 814119864 minus 05 002997 (0001484 005849) 500001205883



2simdunif (0 006) and 120588

3simdunif (0 009)

075 10 125 15

00

20

40

60

001 0015 002

00

1000

2000

3000

4000

00

20

40

60

00 02 04

00

20

40

60


120572 sample 50000 120582 sample 50000

1205881 sample 50000 1205882 sample 50000



00

100

200

300

400

00

100

200

300

400

0975 10 1025 1075

00

250

500

750

1000

0035 004 0045

00

2000

4000

6000

8000

08 0825 085 0875

00

200

400

600

27 28 29

00

100

200

300

400

120572 sample 50000 120573 sample 50000

120582 sample 50000 1205881 sample 50000

1205882 sample 50000 1205883 sample 50000




(1199091 119909


119871 (120572 120582 | x) prop (120572120582)119899 expminus

119899

sum

119894=1

(120582119909119894)

times [

119899

prod

119894=1


]

(16)


119901 (120582) prop 120582 exp minus120582 120582 gt 0 (17)


119901 (120572) prop

1

120572

120572 gt 0 (18)


119901 (120572 120582) prop

120582 exp minus120582

120572

120572 120582 gt 0 (19)


119901 (120572 120582 | x) = Ψ0(x) (120572)

119899minus1

120582119899+1 expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times [

119899

prod

119894=1


]

(20)



500

1000

1500

27 275 28 285

00

200

400

600

098 10 102 104

00

500

1000

1500

002 00225 0025 00275

00

1000

2000

3000

4000

00 002 004 006

00

100

200

300

00

25

50

75

100

098 10 102 104 106

00

100

200

300

098 10 102 104 106

00

500

1000

1500

00

00

002 004

120572 sample 50000

120582 sample 50000

1205881 sample 50000 1205882 sample 50000

1205883 sample 50000 1205884 sample 50000


minus002

minus002

a sample 50000

b sample 50000



1 119909

119899)

is

119901 (119911 | x) = int

+infin

120572=0

int

+infin

120582=0

119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572

(21)


119901 (119911 | x)

=

Ψ1(x) int+infin

120572=0

int

+infin

120582=0

(120572)119899

120582(119899+2)


times expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times[

119899

prod

119894=1


]119889120582119889120572 for 119911gt0 120572 120582gt0

0 elsewhere(22)



098 10 102 106

00

50

100

150

200

095 10 105

00

50

100

150

3850 3900 3950 4000

00

02

04

06

08

596 598 60

00

1000

2000

3000

002 006

00

50

100

150

200

0025 0075

00

50

100

150

1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus002 minus0025

120573 sample 50000



0014

0012

0010

0008

0006

0004

0002

Pdf

Hispanic

100 200 300 400

z









1= (minus10 10) and 120588

2= (minus10 10)






1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108


8978811205832

3277841205833

3965821205834

73029 times 107


4465821205732

6797031205741

2113251205742


90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)


6 Conclusions





Disclosure




Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












02073 007844 726119864 minus 04 02076 (005193 03608) 500001205882






1009 0008985 877119864 minus 05 1006 (1000 1033) 500001205882

















119901 (119911 | x) = ∬119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572 (13)


119901 (120572 120582 | x) = Ψ (x) 119871 (120572 120582 | x) 119901 (120572 120582) (14)


Ψ(x)minus1 = ∬119871 (120572 120582 | x) 119901 (120572 120582) 119889120582 119889120572 (15)


119899be a ran-


119899)1015840




1006 0005986 594119864 minus 05 1004 (100 1022) 500001205882

002805 001952 151119864 minus 04 002482 (0001068 006674) 500001205883

0005888 0005865 532119864 minus 05 0004063 (148119864 minus 04 002165) 500001205884



2simdunif (0 007) 120588





5997 0003353 253119864 minus 05 5998 (5988 6000) 500001205882

003002 001736 814119864 minus 05 002997 (0001484 005849) 500001205883



2simdunif (0 006) and 120588

3simdunif (0 009)

075 10 125 15

00

20

40

60

001 0015 002

00

1000

2000

3000

4000

00

20

40

60

00 02 04

00

20

40

60


120572 sample 50000 120582 sample 50000

1205881 sample 50000 1205882 sample 50000



00

100

200

300

400

00

100

200

300

400

0975 10 1025 1075

00

250

500

750

1000

0035 004 0045

00

2000

4000

6000

8000

08 0825 085 0875

00

200

400

600

27 28 29

00

100

200

300

400

120572 sample 50000 120573 sample 50000

120582 sample 50000 1205881 sample 50000

1205882 sample 50000 1205883 sample 50000




(1199091 119909


119871 (120572 120582 | x) prop (120572120582)119899 expminus

119899

sum

119894=1

(120582119909119894)

times [

119899

prod

119894=1


]

(16)


119901 (120582) prop 120582 exp minus120582 120582 gt 0 (17)


119901 (120572) prop

1

120572

120572 gt 0 (18)


119901 (120572 120582) prop

120582 exp minus120582

120572

120572 120582 gt 0 (19)


119901 (120572 120582 | x) = Ψ0(x) (120572)

119899minus1

120582119899+1 expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times [

119899

prod

119894=1


]

(20)



500

1000

1500

27 275 28 285

00

200

400

600

098 10 102 104

00

500

1000

1500

002 00225 0025 00275

00

1000

2000

3000

4000

00 002 004 006

00

100

200

300

00

25

50

75

100

098 10 102 104 106

00

100

200

300

098 10 102 104 106

00

500

1000

1500

00

00

002 004

120572 sample 50000

120582 sample 50000

1205881 sample 50000 1205882 sample 50000

1205883 sample 50000 1205884 sample 50000


minus002

minus002

a sample 50000

b sample 50000



1 119909

119899)

is

119901 (119911 | x) = int

+infin

120572=0

int

+infin

120582=0

119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572

(21)


119901 (119911 | x)

=

Ψ1(x) int+infin

120572=0

int

+infin

120582=0

(120572)119899

120582(119899+2)


times expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times[

119899

prod

119894=1


]119889120582119889120572 for 119911gt0 120572 120582gt0

0 elsewhere(22)



098 10 102 106

00

50

100

150

200

095 10 105

00

50

100

150

3850 3900 3950 4000

00

02

04

06

08

596 598 60

00

1000

2000

3000

002 006

00

50

100

150

200

0025 0075

00

50

100

150

1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus002 minus0025

120573 sample 50000



0014

0012

0010

0008

0006

0004

0002

Pdf

Hispanic

100 200 300 400

z









1= (minus10 10) and 120588

2= (minus10 10)






1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108


8978811205832

3277841205833

3965821205834

73029 times 107


4465821205732

6797031205741

2113251205742


90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)


6 Conclusions





Disclosure




Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1006 0005986 594119864 minus 05 1004 (100 1022) 500001205882

002805 001952 151119864 minus 04 002482 (0001068 006674) 500001205883

0005888 0005865 532119864 minus 05 0004063 (148119864 minus 04 002165) 500001205884



2simdunif (0 007) 120588





5997 0003353 253119864 minus 05 5998 (5988 6000) 500001205882

003002 001736 814119864 minus 05 002997 (0001484 005849) 500001205883



2simdunif (0 006) and 120588

3simdunif (0 009)

075 10 125 15

00

20

40

60

001 0015 002

00

1000

2000

3000

4000

00

20

40

60

00 02 04

00

20

40

60


120572 sample 50000 120582 sample 50000

1205881 sample 50000 1205882 sample 50000



00

100

200

300

400

00

100

200

300

400

0975 10 1025 1075

00

250

500

750

1000

0035 004 0045

00

2000

4000

6000

8000

08 0825 085 0875

00

200

400

600

27 28 29

00

100

200

300

400

120572 sample 50000 120573 sample 50000

120582 sample 50000 1205881 sample 50000

1205882 sample 50000 1205883 sample 50000




(1199091 119909


119871 (120572 120582 | x) prop (120572120582)119899 expminus

119899

sum

119894=1

(120582119909119894)

times [

119899

prod

119894=1


]

(16)


119901 (120582) prop 120582 exp minus120582 120582 gt 0 (17)


119901 (120572) prop

1

120572

120572 gt 0 (18)


119901 (120572 120582) prop

120582 exp minus120582

120572

120572 120582 gt 0 (19)


119901 (120572 120582 | x) = Ψ0(x) (120572)

119899minus1

120582119899+1 expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times [

119899

prod

119894=1


]

(20)



500

1000

1500

27 275 28 285

00

200

400

600

098 10 102 104

00

500

1000

1500

002 00225 0025 00275

00

1000

2000

3000

4000

00 002 004 006

00

100

200

300

00

25

50

75

100

098 10 102 104 106

00

100

200

300

098 10 102 104 106

00

500

1000

1500

00

00

002 004

120572 sample 50000

120582 sample 50000

1205881 sample 50000 1205882 sample 50000

1205883 sample 50000 1205884 sample 50000


minus002

minus002

a sample 50000

b sample 50000



1 119909

119899)

is

119901 (119911 | x) = int

+infin

120572=0

int

+infin

120582=0

119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572

(21)


119901 (119911 | x)

=

Ψ1(x) int+infin

120572=0

int

+infin

120582=0

(120572)119899

120582(119899+2)


times expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times[

119899

prod

119894=1


]119889120582119889120572 for 119911gt0 120572 120582gt0

0 elsewhere(22)



098 10 102 106

00

50

100

150

200

095 10 105

00

50

100

150

3850 3900 3950 4000

00

02

04

06

08

596 598 60

00

1000

2000

3000

002 006

00

50

100

150

200

0025 0075

00

50

100

150

1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus002 minus0025

120573 sample 50000



0014

0012

0010

0008

0006

0004

0002

Pdf

Hispanic

100 200 300 400

z









1= (minus10 10) and 120588

2= (minus10 10)






1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108


8978811205832

3277841205833

3965821205834

73029 times 107


4465821205732

6797031205741

2113251205742


90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)


6 Conclusions





Disclosure




Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

100

200

300

400

00

100

200

300

400

0975 10 1025 1075

00

250

500

750

1000

0035 004 0045

00

2000

4000

6000

8000

08 0825 085 0875

00

200

400

600

27 28 29

00

100

200

300

400

120572 sample 50000 120573 sample 50000

120582 sample 50000 1205881 sample 50000

1205882 sample 50000 1205883 sample 50000




(1199091 119909


119871 (120572 120582 | x) prop (120572120582)119899 expminus

119899

sum

119894=1

(120582119909119894)

times [

119899

prod

119894=1


]

(16)


119901 (120582) prop 120582 exp minus120582 120582 gt 0 (17)


119901 (120572) prop

1

120572

120572 gt 0 (18)


119901 (120572 120582) prop

120582 exp minus120582

120572

120572 120582 gt 0 (19)


119901 (120572 120582 | x) = Ψ0(x) (120572)

119899minus1

120582119899+1 expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times [

119899

prod

119894=1


]

(20)



500

1000

1500

27 275 28 285

00

200

400

600

098 10 102 104

00

500

1000

1500

002 00225 0025 00275

00

1000

2000

3000

4000

00 002 004 006

00

100

200

300

00

25

50

75

100

098 10 102 104 106

00

100

200

300

098 10 102 104 106

00

500

1000

1500

00

00

002 004

120572 sample 50000

120582 sample 50000

1205881 sample 50000 1205882 sample 50000

1205883 sample 50000 1205884 sample 50000


minus002

minus002

a sample 50000

b sample 50000



1 119909

119899)

is

119901 (119911 | x) = int

+infin

120572=0

int

+infin

120582=0

119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572

(21)


119901 (119911 | x)

=

Ψ1(x) int+infin

120572=0

int

+infin

120582=0

(120572)119899

120582(119899+2)


times expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times[

119899

prod

119894=1


]119889120582119889120572 for 119911gt0 120572 120582gt0

0 elsewhere(22)



098 10 102 106

00

50

100

150

200

095 10 105

00

50

100

150

3850 3900 3950 4000

00

02

04

06

08

596 598 60

00

1000

2000

3000

002 006

00

50

100

150

200

0025 0075

00

50

100

150

1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus002 minus0025

120573 sample 50000



0014

0012

0010

0008

0006

0004

0002

Pdf

Hispanic

100 200 300 400

z









1= (minus10 10) and 120588

2= (minus10 10)






1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108


8978811205832

3277841205833

3965821205834

73029 times 107


4465821205732

6797031205741

2113251205742


90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)


6 Conclusions





Disclosure




Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










500

1000

1500

27 275 28 285

00

200

400

600

098 10 102 104

00

500

1000

1500

002 00225 0025 00275

00

1000

2000

3000

4000

00 002 004 006

00

100

200

300

00

25

50

75

100

098 10 102 104 106

00

100

200

300

098 10 102 104 106

00

500

1000

1500

00

00

002 004

120572 sample 50000

120582 sample 50000

1205881 sample 50000 1205882 sample 50000

1205883 sample 50000 1205884 sample 50000


minus002

minus002

a sample 50000

b sample 50000



1 119909

119899)

is

119901 (119911 | x) = int

+infin

120572=0

int

+infin

120582=0

119901 (119911 | 120572 120582) 119901 (120572 120582 | x) 119889120582 119889120572

(21)


119901 (119911 | x)

=

Ψ1(x) int+infin

120572=0

int

+infin

120582=0

(120572)119899

120582(119899+2)


times expminus

119899

sum

119894=1

(120582119909119894) minus 120582

times[

119899

prod

119894=1


]119889120582119889120572 for 119911gt0 120572 120582gt0

0 elsewhere(22)



098 10 102 106

00

50

100

150

200

095 10 105

00

50

100

150

3850 3900 3950 4000

00

02

04

06

08

596 598 60

00

1000

2000

3000

002 006

00

50

100

150

200

0025 0075

00

50

100

150

1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus002 minus0025

120573 sample 50000



0014

0012

0010

0008

0006

0004

0002

Pdf

Hispanic

100 200 300 400

z









1= (minus10 10) and 120588

2= (minus10 10)






1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108


8978811205832

3277841205833

3965821205834

73029 times 107


4465821205732

6797031205741

2113251205742


90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)


6 Conclusions





Disclosure




Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










098 10 102 106

00

50

100

150

200

095 10 105

00

50

100

150

3850 3900 3950 4000

00

02

04

06

08

596 598 60

00

1000

2000

3000

002 006

00

50

100

150

200

0025 0075

00

50

100

150

1205881 sample 50000

1205882 sample 50000 1205883 sample 50000

minus002 minus0025

120573 sample 50000



0014

0012

0010

0008

0006

0004

0002

Pdf

Hispanic

100 200 300 400

z









1= (minus10 10) and 120588

2= (minus10 10)






1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108


8978811205832

3277841205833

3965821205834

73029 times 107


4465821205732

6797031205741

2113251205742


90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)


6 Conclusions





Disclosure




Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198981

8978811198982

11339901198983

20034 times 106

1198984

439016 times 108


8978811205832

3277841205833

3965821205834

73029 times 107


4465821205732

6797031205741

2113251205742


90 (28532 325350)95 (24120 332450)98 (22013 346092)99 (21051 358301)


6 Conclusions





Disclosure




Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article inferential statistics from black

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