Saurashtra University Re Accredited Grade B by NAAC (CGPA 2.93)

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1 Saurashtra Unversty Re Accredted Grade B by NAAC (CGPA 2.93) Arora, Sumeet, 2009, Some advances n lfe testng and relablty, thess PhD, Saurashtra Unversty Copyrght and moral rghts for ths thess are retaned by the author A copy can be downloaded for personal non-commercal research or study, wthout pror permsson or charge. Ths thess cannot be reproduced or quoted extensvely from wthout frst obtanng permsson n wrtng from the Author. The content must not be changed n any way or sold commercally n any format or medum wthout the formal permsson of the Author When referrng to ths work, full bblographc detals ncludng the author, ttle, awardng nsttuton and date of the thess must be gven. Saurashtra Unversty Theses Servce repostory@sauun.ernet.n The Author

2 SOME ADVANCES IN LIFE TESTING AND RELIABILITY THESIS SUBMITTED TO SAURASHTRA UNIVERSITY FOR THE AWARD OF THE DEGREE OF DOCTOR OF PHILOSOPHY IN STATISITICS BY SUMEET ARORA INSITUTE OF MANAGEMENT NIRMA UNIVERSITY AHMEDABAD UNDER THE GUIDANCE OF DR. G.C.BHIMANI ASSOCIATE PROFESSOR DEPARTMENT OF STATISTICS SAURASHTRA UNIVERSITY RAJKOT

3 CERTIFICATE Ths s to certfy that Ms. Sumeet Arora, a canddate for the award of Doctor of Phlosophy n Statstcs, under the faculty of Scence, Saurashtra Unversty, Rajkot, has worked under my gudance for the perod requred by the unversty. The work emboded n thess s not submtted prevously to ths or any other unversty for Ph.D. or any other unversty for Ph.D. or any other degree or dploma. Ths s her orgnal research work. Dr. G.C.Bhman Research Gude Assocate Professor Department of Statstcs Saurashtra Unversty Rajkot INDIA

4 DECLARATION I, hereby, declare that the thess enttled SOME ADVANCES IN LIFE TESTING AND RELIABILITY s my own work conducted under the supervson of Dr. G.C. Bhman. I further declare that to the best of my knowledge the thess does not contan any part of any work whch has been submtted for the award of any degree or dploma ether n ths Unversty or any other Unversty or examnng body n Inda or any other country. Wherever the references have been made to prevous works of other, t has been clearly ndcated as such and ncluded n the Bblography Sgnature of Canddate October 2009 Sumeet Arora

5 EXPRESSION OF GRATITUDE I dedcate ths page to all those who have actvely or slently left an ndelble mark on my doctoral research endeavor so that they may be brought nto lmelght and gven the credt whch they rchly deserve. Wth profound sense of grattude and gratefulness, I express my sncere thanks to my mentor and research supervsor, Dr. G.C. Bhman, Assocate Professor, School of Scence, Rajkot. Hs much needed deas, gudance and suggestons were of mmense help. It was hs nsprng mentorshp and a contnuous zeal for perfecton whch brought out possbly the best of my efforts n completng ths work. It was ndeed a prvlege to be under hs tutelage and learn every moment from hs dlgence and creatve ntellect. I am grateful to Dr. Upnder Dhar, Vce Chancellor, J.K. Laxmpath Unversty, Japur for hs contnuous encouragement and support to complete my work. I take great pleasure to express my profuse and sncere grattude to Dr. C. Gopalakrshnan, Drector-In-Charge of Insttute of Management, Nrma Unversty. Rarely does one come across persons of hs stature. He nspred me to complete my work wth more enthusasm.

6 I would lke to extend my grattude to the entre staff of department of Statstcs, Gujarat Unversty, Ahmedabad. Specal thanks to Prof. (Dr.) R.G. Bhatt, Head of the Statstcs department, for extendng her co-operaton and provdng varous facltes n the department of Statstcs, Gujarat Unversty, for gvng constructve suggestons on the subject matter. My specal words of thanks to Dr.(Mrs.) Santosh Dhar, Professor, Japura Insttute of Managemnet,Japur, Dr. (Ms.) Sapna Parashar, Assstant Professor, Nrma Unversty, Ahmedabad, Mr. Sameer Pngle, Sr. Lecturer, Nrma Unversty, Ahmedabad for provdng nspraton and support. I sncerely acknowledge the co-operaton and support extended to me by faculty members and supportng staff of Chmanbha Patel Insttute of Management, Ahmedabad and Insttute of Management, Nrma Unversty durng varous stages of my project. I am also thankful to the lbrary staff of the Insttute of Management, Nrma Unversty. I would also lke to express my oblgaton to the authors of pervous relevant studes that have nspred me to take up ths study and have provded valuable nsght n my research area. I express my sncere thanks to

7 Mr. Gaurav Garg who has prepared computer programme for dfferent models, wthout hs help ths work would not have been possble. My profound thanks to my grandparents, Mrs. Mahendarkaur Mekan and Mr. Jogndersngh Mekan and my parents Mrs. Manjeetkaur Arora and Mr. Harpalsngh Arora, for ther nspraton, blessngs and untold sacrfces. Words are not enough to thank my brother, Mr. Santpreetsngh Arora, my sster Mrs. Sona Chabbra and my brother-n-law Mr. Sandeepsngh Chabbra who have nspred, encouraged and provded the moral support. I would also lke to thank my nece, Kawleen for cooperatng n her own way. Last but not the least, my thanks are also due to all those whom I am not namng here but have drectly or ndrectly facltated me at dfferent stages of ths project. Above all, I thank Almghty God for hs blessngs and gvng me an opportunty to undertake ths nterestng project for my study. Sumeet Arora

8 CONTENTS 1. Introducton Hstory of Lfe-testng and Relablty Pre-Work Present Work 7 2. Estmaton under Progressve Type-II censorng for Length Based Exponental Dstrbuton 2.1 Introducton Length Based Exponental Dstrbuton Maxmum Lkelhood Estmaton Bayes Estmate Smulaton Study Concluson and Suggeston Estmaton under Progressve Interval Type-I Censorng for Recprocal Exponental Dstrbuton 3.1 Introducton Recprocal Exponental Dstrbuton Maxmum Lkelhood Estmaton Comparng Censorng Patterns va Smulaton Confdence Interval Estmaton Expected Duraton of Lfe test Estmaton under Progressve Type-II censorng for Recprocal Exponental Dstrbuton 4.1 Introducton Maxmum Lkelhood Estmaton 57

9 4.3 Comparng Censorng Patterns va Smulaton Confdence Interval Estmaton Estmaton under Progressve Type-II Censorng for Type-II Generalzed Half Logstc Dstrbuton 5.1 Introducton Maxmum Lkelhood Dstrbuton Bayes Estmate Smulaton Study Confdence Interval Estmaton Some Results on Maxmum Lkelhood Estmators of Parameters of Generalzed Half Logstc Dstrbuton under Type-I Progressve Censorng wth Changng Falure Rate 6.1 Introducton Generalzed Half Logstc Dstrbuton Maxmum Lkelhood Dstrbuton Estmaton based on Interval-Censorng wth 100 dfferent parameters at each stage 6.5 Expected Duraton of Lfe Test 105 REFERENCES 108

10 CHAPTER 1 INTRODUCTION 1.1 Hstory of Lfe-testng and Relablty Relablty theory n nneteenth century was prmarly used as a tool to help martme and lfe nsurance companes fgure proftable rates to charge ther customers apart from the manstream of probablty and statstcs. In today s technologcal world nearly everyone depends upon the contnued functonng of a wde array of complex machnery and equpment for ther everyday health, safety, moblty and economc welfare. We expect our cars, computers, electrcal applances, lghts, televsons etc. to functon whenever we need them day after day, year after year. When they fal the results can be dsastrous whch lead to njury, loss of lfe and/or costly lawsuts. Moreover, repeated falure leads to annoyance, nconvenence and a lastng customer dssatsfacton that can play havoc wth the company s market poston. It takes a longtme for a company to buld up a reputaton for relablty and only a short tme to be branded as unrelable after shppng a flawed product. Contnual assessment of new product, relablty and ongong 1

11 control of the relablty of everythng shpped are vtal requrements n today s compettve busness arena. The everyday usage term qualty of a product s taken for granted to mean ts nherent degree of excellence. In ndustry, ths s made more precse by defnng qualty to be conformance to requrements at the start of use. Assumng that the product specfcatons adequately capture customer requrements, the qualty level can now be precsely measured by the fracton of unts that meet specfcatons after a week of operaton? Or after a month or at the end of a one-year warranty perod? That s where relablty steps n. Qualty s a snapshot at the start of lfe and relablty s a moton pcture of day-by-day operaton. The qualty level mght be descrbed by a sngle fracton defectve. To descrbe relablty fallout a probablty model that descrbes the fracton fallout over tme s needed. Ths s known as lfe dstrbuton model. A lfe dstrbuton does fnd ts frequent applcaton n the engneerng and bomedcal scences. The tmes to the occurrences of events, whch are of nterest for some populaton of ndvduals, are termed as lfe tmes. Some tmes the events of nterest are deaths of ndvdual or may be a survval tme measured from some partcular startng pont. In some nstances lfe tme s used n a fguratve sense. Mathematcally, one can thnk of lfe tme as merely meanng non-negatve valued varable. For e.g. manufactured tems such as mechancal or electronc components are often subjected to lfe tests n order 2

12 to obtan nformaton on ther endurance. Ths nvolves subjectng tems n operaton, often n a laboratory settng and observng them untl they fal. In such stuaton t s common to refer to lfe tmes as falure tmes, snce when an tem ceases operatng satsfactorly, t s sad to have faled. The theoretcal populaton models that are used to descrbe unt lfe tmes are known as lfetme dstrbuton models. The populaton s generally consdered to be all of the possble unt lfe tmes that could be manufactured based on a partcular desgn, choce of materals and manufacturng process. A random sample of sze n from ths populaton s the collecton of falure tmes observed for a randomly selected group of n unts. A lfetme dstrbuton model can be any probablty densty functon f (t) defned over the range of tme from t = 0 to t =. The correspondng cumulatve dstrbuton functon F (t) s a very useful functon as t gves the probablty that a randomly selected unt wll fal by tme t. The data, to whch statstcal methods are appled n order that parameters of nterest can be estmated n ther relablty context, usually result from lfe tests. A typcal lfe test s one n whch prototypes of the tem or organsm of nterest s subjected to stresses and envronmental condtons demonstrate the ntended operatng condtons. Durng the test successve tmes to falures are noted. Snce the falures occur n order, the theory of order statstcs plays an mportant role n the analyss of the lfe test data. 3

13 Lterature related to statstcal methods used n the analyss of lfe test data les scattered n a number of professonal journals and books. Relablty studes frequently nvolve testng of tems (say n n number) that are desgned to last for long perods of tme. In such studes, constrants n the form of truncaton and / or censorng would be deemed essental as means of obtanng nformaton wthn reasonable tme lmtatons; whle there are several means of censorshp (see Gajjar and Khatr (1969)) two types are commonly used. These are referred as Type-I and Type-II censorshps. Type-I censorshp or censorng occurs when the researcher sets a tme lmt on termnatng the lfe test, even though some of the test tems reman operatonal. Type-II censorng occurs when the lfe test s termnated at the partcular (the r th, say r < n) falure. In Type-I censorng the number of falures and all the falure tmes are random varables, the number of falures beng consdered fxed. Type-II censorng has the advantage of provdng more or less unform amount of nformaton n repeated samplng wth the dsadvantage that the length of testng tme vares from test to test. Type-I censorng provdes a constant length of testng tme n repeated samplng wth amount of nformaton varyng from test to test. One advantage of Type-I censorng s that t smplfes the problem of test schedulng n a producton process where nformaton from perodc producton of lots has to be obtaned at regular ntervals. 4

14 1.2 PRE-WORK There s an extensve body of lterature concernng propertes of several estmators that are proposed for estmatng parameters of probablty models commonly used n relablty studes under Type-II censorng. Though some work n the area of relablty and lfe testng has been done under Type-I censorng but t s not as extensve as that under Type-II censorng. The early work concernng estmaton of parameters from contnuous lfe tme dstrbutons such as Normal, Exponental, Webull, Extreme Value dstrbutons and dscrete lfe tme dstrbuton partcularly Geometrc dstrbuton based on sngle stage Type-I and Type-II censorng was ntated by Gupta(1952), Epsten and Sobel (1953, 1954), Leblen and Zelen (1956), Bartholomew (1957, 1963), Cohen (1965), Tku (1967) and others. Recently, rather extensvely the work has been studed by Yaqub and Khan (1981), Patel and Gajjar (1990), Cohen (1991), Balakrshnan and Cohen (1991). These authors have all consdered lfetme studes n ndustral as well as actuaral (human lfe tme) contexts, n parametrc and non-parametrc cases. In several stuatons, the ntal censorng results only n wthdrawal of a porton of the survvng tems. Those whch reman on test contnue under further observaton untl an ultmate falure or untl a subsequent stage of censorng s performed. For suffcently large samples censorng s done 5

15 through several stages. Ths leads to progressve censorng of Type-I or Type-II. Progressve censorng can be adopted for several reasons. Progressvely censored sample arse, for nstance, when certan tems must be wthdrawn from a lfe test pror to falure for use as test objects n related expermentaton. They may also result from a compromse between the need for more rapd testng and the desre to nclude at least some extreme lfe spans n the sample data. When the test facltes are lmted and when prolonged lfe tests are cost-prohbtve, the early censorng of a substantal number of tems from the test frees facltes for other tests whle tems whch are allowed to contnue on test untl subsequent falures provde nformaton on extreme sample values. Cohen (1963) consdered Type-I progressvely censored samples n case of Normal and Exponental dstrbutons and obtaned maxmum lkelhood estmates of the parameters of these dstrbutons wth the assumpton that the parameters reman the same at each stage of censorng. But there are stuatons where t mght be reasonable to assume that the parameters of a dstrbuton under consderaton mght change at each stage of censorng. The justfcaton of ths reasonng lkes n the fact that the survvng tems enterng the subsequent stage are checked and overhauled elmnatng or reparng mnor defects wherever possble. It may be noted that due to dfferent parameters at dfferent stages of censorng t leads to estmatng parameters from truncated censored dstrbutons. Srvastava (1967), Gajjar and Khatr 6

16 (1969), Patel and Gajjar (1979) and Patel (1991) have consdered Type-I progressvely censored and group-censored samples from Exponental, Webull, Inverse Gaussan, Log-normal, Power seres and Logstc dstrbutons wth dfferent parameters at dfferent stages of censorng and obtaned maxmum lkelhood estmates of the parameters. The maxmum lkelhood estmates or estmatng equatons obtaned by Gupta (1952) and Cohen (1963) can be deduced as specal cases from these results. 1.3 PRESENT WORK In order to obtan nformaton about the relablty or warranty perod of manufactured tems such as electrcal or electroncs, components are often put on lfe tests and lfe tmes are observed perodcally. A model s specfed to represent the dstrbuton of lfe tmes and statstcal nferences are made on the bass of ths model. The lfetme models may be dscrete or contnuous. The wdely-used contnuous lfetme models are Exponental, Webull, Raylegh, Lognormal dstrbutons etc, whereas Geometrc dstrbuton, a dscrete analogue of Exponental dstrbuton, s used as dscrete lfetme falure model. In lfe testng experments, usually the tems are checked by destroyng them and/or are very costly. Ths lmts the number of tems we can test. In these stuatons the lfe test may be termnated at the pre-determned number 7

17 of falures. For nstance, we may put N tems on the test and termnate the experment when a pre-assgned number of tems, say r (< N) have faled. The samples obtaned from such an experment are called rght Type-II censored samples. Another way to get censored data s to observe largest lfetmes. The lfetmes of frst (N-s) components are mssng; such a censorng s called left Type-II censorng. Moreover, f left and rght Type-II censorng stuatons arse together, ths s known as doubly Type-II censorng scheme. Estmaton based on classcal nferences has been found to be extremely useful for a varety of problems. Ths thess s concerned wth the problem of estmaton under progressve Type-I and Type-II, and progressve Type-I nterval censorng schemes. Suppose an tem wth falure rate X follows the dstrbuton F(X θ) wth densty functon f(x θ) for θ s a vector valued parameter n a real parameter space Ω. Suppose X has the dstrbuton functon F(X θ ) n the tme nterval (N -1,N ] for =1,2,,k (k>1) wth N 0 = 0 and N k =. Let n tems are placed on a lfe test wthout replacement and let n be the number of tems that wthdrawn from the test mmedately after the censorng tme N -1, =2,3,,k so that (k) (k) = k ; where n (k) r n n denotes the number of tem enterng the k th stage of an experment. Also, let () () () 1 2 n X X... X be 8

18 the tmes of falure for =1,2,,k (k>1) then the lkelhood functon for k-stage Type-I progressve censorng wthout replacement s gven by k n k () Lα f( xj ) 1 F( N). =1 j= 1 1= 1 r Takng X as non-negatve nteger valued random varable and N s can be chosen to be non-negatve ntegers, a problem of estmatng parameters at dfferent stages of censorng can be consdered. The method of maxmum lkelhood can be employed to estmate the propertes of dfferent types of estmators lke MLE, shrnkage estmator, mnmum mean square error estmator, and almost unbased estmator can be nvestgated. Patel and Patel (2003, 2005a, 2005b, 2005c, 2006) have consder estmaton of parameters of geometrc lfe tme dstrbuton under progressve Type-I and Type-II censorng wth mxture as well as competng rsk models. A generalzaton of Type-II censorng s progressve Type-II censorng. Accordng to Balakrshnan and Aggrawala (2000) under progressve Type-II censorng scheme a total of n unts are placed on a lfe test, only m are completely observed untl falure. At the tme of frst falure, R 1 of the n-1 survvng unts are randomly wthdrawn from the test. At the tme of next falure R 2 of the n-2- R 1 survvng unts are censored, and so on. Fnally, at the tme of n th falure all the remanng R m = n-m-σr survvng unts are censored. 9

19 Agan a more generalzaton of such progressve Type-II censorng scheme s dscussed by Lawless (1982). In ths scheme, the frst n 1 falures n a sample of n tems are observed. Then r 1 of the remanng n-n 1 workng tems are wthdrawn from the experment, leavng n- n 1 - r 1 on the test. When further n 2 tems have faled r 2 of the stll workng tems are wthdrawn and so on. Fnally, the experment s termnated at the end of n th k falure. Let () () () X 1, X 2,..., Xn are the falure tmes durng the th stage of censorng = 1, 2,,k and (1) (2) (k) n1 n2 nk X, X,..., X are the censorng tmes for k-stage respectvely. Then the lkelhood functon for k-stage Type-II progressve censorng wthout replacement s gven by k () n k n! () () L= f( x ) j 1 F( xn ). =1 n r! j= 1 1= 1 () ( ) r where f (.) and F(.) are composte probablty densty functon and cumulatve dstrbuton functon of lfe tme random varable respectvely. Usng the method of maxmum lkelhood estmaton of the parameters expected watng tme of the test, expected total tme of the test, sample sze 10

20 to mnmze the total cost of the test can be consdered for dscrete or contnuous lfetmes models. Patel and Patel (2007) have used progressve Type-II censored sample for geometrc lfe tme model. Gajjar and Patel (2008) have consdered estmaton for a mxture of exponental dstrbuton based on progressve Type-II censored sample. In ths thess the length based exponental dstrbuton, recprocal exponental dstrbuton, generalzed half logstc dstrbuton are used as lfe tme models. The thess may be dvded nto three categores vz: (1) Estmaton of the parameters under Type-I and Type-II progressve censorng scheme when samples are drawn from (a) Length based exponental dstrbuton (b) Recprocal exponental dstrbuton (c) Generalzed half logstc dstrbuton (2) Estmaton of the parameters under progressve nterval Type-I censorng scheme when samples are drawn from (a) Recprocal exponental dstrbuton (3) Bayesan estmaton for parameters for (a) Length based exponental dstrbuton (b) Type-II generalzed half logstc dstrbuton 11

21 Detal ndex s gven n chapter-1 Chapter-2 deals wth the study of some basc results and characterzatons of Length Based Exponental dstrbuton. Length Based samplng was ntroduced by Cox (1962) (see Patl 2002). It has varous applcatons n bomedcal area such as famly hstory and dsease, survval and ntermedate events and latency perod of AIDS due to blood transfuson (Gupta and Akman 1995). Patl and Rao (1978) wrote an artcle on The study of human famles and wldlfe populatons The most common forms of all weght functon useful n scentfc and statstcal lterature are some basc theorems for weghted dstrbuton and sze-based. As specal case they arrved at a concluson that the length based verson of some dscrete dstrbuton arses as mxture of the length based verson of these dstrbutons. A lot of work has been done by Khatree (1989) to derve relatonshp between orgnal dstrbutons and ther length based versons. A very useful result gvng a relatonshp between orgnal random varable X and ts length based verson Y when X s ether nverse Gaussan or Gamma dstrbuton. He also proved that length based random varable Y can be wrtten as a lnear combnaton of the orgnal random varable X and a ch-square random varable Z and nversely the orgnal random varable can be characterzed through ths relatonshp. 12

22 Several authors such as Patl et al. (1986), Jan et al. (1989), Gupta and Krman (1990) and recently by Olyede and George (2002) treated relatonshps n the perspectve of relablty. In these works the survval functon, the falure rate, and the mean resdual lfe functon of the length-based dstrbuton were expressed n relaton wth the orgnal dstrbuton. If a random varable X follows any dstrbuton wth probablty densty functon f(x) then the probablty densty functon of length based dstrbuton of X xf( x) s defned as gx ( ) =. EX ( ) We have consdered estmaton related to parameters of the length based exponental dstrbuton based on progressvely Type-II censored samples. Maxmum lkelhood estmators as well as approxmate Bayes estmators of the parameters are developed. A smulaton study s consdered for dfferent patterns of censorng. The results based on ths chapter are publshed by Bhman, Arora and Patel (2008). Chapter -3 s consdered wth the estmaton of parameters of recprocal exponental dstrbuton based on progressve nterval Type-I censored samples. Maxmum lkelhood estmator along wth ts asymptotc varance s derved and compared for dfferent censorng patterns. Confdence nterval estmaton s 13

23 consdered based on bootstrap and r - level lkelhood rato, under the three censorng patterns. Non parametrc as well as parametrc estmate of the survval functon are obtaned wth ther asymptotc varances. Usng the method suggested by Kendall and Anderson (1971) expected duraton of lfe test s derved and computed for dfferent choce of tme ntervals. In most applcatons, the data may be nterval-censored. By ntervalcensored data, we mean that a random varable of nterest s known only to le n an nterval, nstead of beng observed exactly. In such cases, the only nformaton we have for each ndvdual s that ther event tme falls n an nterval, but the exact tme s unknown. Generally statstcan faces lot of problem n the analyss of tme-to-event data such as falure tme data, ncubaton tme data etc. Such data arses n lot of felds such as medcne, engneerng, economcs. For example doctor may be nterested to know the tme of convergence to AIDS for HIV postve ndvdual, the tme to the death for cancer patents, lfetme of a devce etc. The analyss of tme-to-event later becomes more complcated on account of censorng. Interval censorng also known as group censorng arses when observatons occur n some nterval of tme a and b. Such data occurs n varety of crcumstances but generally t s encountered n medcal studes where patents are only montored at regular ntervals (e.g. weekly or quarterly 14

24 checkup). Thus, the exact tme of occurrence of some changed response may only be known to have some tme between two vsts. Samuelson and Kongerud (1994); Kokasa et al (1993); Farrngton (1996); Odell et al (1992), Sun (1997); Lndsey and Ryan (1998) and Scallan (1999) have dscussed applcaton of nterval censorng n clncal, medcal, bomedcal and engneerng studes. Rao (1998) gave standard methods for analyzng nterval censored data and dscussed effcences of estmators derved from censorng over conventonal Type-I and Type-II censorng schemes. Estmaton related to the parameters of recprocal exponental dstrbuton s dscussed for progressvely Type-II censored samples. A maxmum lkelhood estmator for the parameters s developed. A smulaton study s consdered for dfferent pattern of censorng. These results are presented n Chapter-4. Chapter-5 deals wth progressve Type-II censored sample for a Type-II generalzed half logstc dstrbuton. Classcal nference s carred out usng smulaton of such a censored sample. Maxmum lkelhood estmator as well as approxmate Bayes estmator of the parameter along wth ther asymptotc varances and MSE s are derved and compared for dfferent censorng patterns. Confdence nterval estmaton s consdered based on bootstrap and r- level lkelhood rato under the three censorng patterns. 15

25 Half logstc model obtaned as the dstrbuton of the absolute standard logstc varate s probablty model consdered by Balakrshnan (1985). Balakrshnan and Puthenpura (1986) obtaned best lnear unbased estmator of locaton and scale parameters of the half logstc dstrbuton through lnear functons of order statstcs. Balakrshnan and Wong (1991) obtaned approxmate maxmum lkelhood estmates for the locaton and scale parameters of the half logstc dstrbuton wth Type-II Rght-Censorng. Olapade (2003) proved some theorems that characterzed the half logstc dstrbuton. The half logstc dstrbuton has not receved much attenton from researchers n terms of generalzaton. A generalzed verson of half logstc dstrbuton namely Type-I and Type II generalzed half logstc dstrbutons are consdered by Ramakrshna (2008). In chapter-6 we have dscussed the maxmum lkelhood estmators of the generalzed half logstc dstrbuton under Type-I progressve censorng wth changng falure rates s consdered. The numercal evaluaton of ther relatve performance s made for selected values of n and p. MLE and ts asymptotc varance are obtaned usng a smulaton study based on 1000 random samples. Further results ncludng total expected watng tme are obtaned n case of nterval censorng schemes also. 16

26 CHAPTER 2* Estmaton under Progressve Type-II censorng for Length Based Exponental Dstrbuton 2.1 INTRODUCTION Relablty studes frequently nvolve testng of tems that are desgned to last for a long perod of tme. In such studes constrants are n the form of truncaton and / or censorng would be deemed essental as a mean of obtanng nformaton wthn reasonable tme lmtatons. Whle there are several types of censorshp, two are of common usage. These are commonly referred to as Type-I and Type-II censorng. Type-I censorng occurs when the researcher sets a tme lmt on termnatng the lfe test even though some of the test tems reman operatonal. * A paper on the bass of ths chapter s publshed n the journal IAPQR Transactons, Vol. 33(2), page no ,

27 Type-II censorng occurs when the lfe test s termnated at the partcular (say, r < n) falure. Progressve Type-II censorng defned by Cohen (1963) s as follows. Before conductng a lfe test the expermenter fxes a sample sze n, a number of complete observaton m and a censorng scheme (R 1, R 2, R m ), n = m+ R. The n unts are placed on a lfe test. Immedately after the frst falure, R 1 survvng unts are randomly chosen and removed from the experment. Then after second falure, R 2 unts are wthdrawn and so on. The procedure s contnued untl all R m remanng unts are removed after the m th falure. If R 1 = R 2 = = R m = 0, then n = m whch corresponds to a complete sample. If R 1 = R 2 = = R m-1 = 0 then R m = n m corresponds to conventonal Type-II rght censorng scheme. Balakrshnan and Aggarwala (2000) provded a comprehensve reference on progressve censorng, ts applcaton and technques for analyzng data from progressve Type-II censorng schemes. 18

28 2.2 Length Based Exponental Dstrbuton Consder a group of subjects who experence some event (say, the onset of dsease) at tmes [x.sub.], followed by some other event (say, death) at endponts [x.sub.y]. In epdemology studes t s often the am to estmate the dstrbuton of the ntervals from ntaton to the endponts or to compare the dstrbutons of these survval tmes for two or more well-defned groups. When t s possble to follow all subjects n a group prospectvely, standard technques of survval analyss are applcable. Frequently, however, subjects are dentfed to have experenced ntaton through a cross-sectonal study at some fxed tme pont; hence those who have survved to that tme are recruted nto the study, whereas those who have not wll not be ncluded n ths ntal recrutment phase, and ndeed wll not even be dentfed. Thereafter, the group of recruted subjects s followed untl a second tme pont, correspondng to the end of the study. Of course, some of these subjects wll have censored falure tmes for varous reasons, ncludng ther survval untl the end of the study. We assume that for every subject ncluded, an ntaton date s recorded. Therefore, the data on each subject nclude the dates of onset and falure/censorng (as well as censorng ndcators) for those subjects who have been recruted. 19

29 The ntervals from ntaton to falure/censorng are well known to be "length based," whch means that those tme ntervals actually observed tend to be longer than those arsng from the true underlyng falure (censorng dstrbutons). The phenomenon of length bas was systematcally studed by McFadden (1962), Blumenthal (1967), and later by Cox (1969) n the context of estmatng the dstrbuton of fber lengths n a fabrc. Length based samplng has varous applcatons n bomedcal area such as famly hstory and dsease, survval and ntermedate events and latency perod of AIDS due to blood transfuson (Gupta and Akman 1995). Patl and Rao (1978) wrote an artcle on The study of human famles and wldlfe populatons They arrved at a concluson that the length based verson of some dscrete dstrbuton arses as a mxture of the length based verson of these dstrbutons. A lot of work has been done by Khatree (1989) to derve relatonshp between orgnal dstrbutons and ther length based versons. A very useful result gvng a relatonshp between orgnal random varable X and ts length based verson Y when X s ether nverse Gaussan or Gamma dstrbuton. He also proved that length based random varable Y can be wrtten as a lnear combnaton of the orgnal random varable X and a ch-square random varable Z and nversely the orgnal random varable can be characterzed through ths relatonshp. 20

30 Several authors such as Patl et al. (1986), Jan et al. (1989), Gupta and Krman (1990) and recently by Olyede and George (2002) treated relatonshps n the perspectve of relablty. In these works the survval functon, the falure rate, and the mean resdual lfe functon of the length-based dstrbuton were expressed n relaton wth the orgnal dstrbuton. If a random varable X follows any dstrbuton wth probablty densty functon f(x) then the probablty densty functon of length based dstrbuton of X xf(x) s defned as g(x)=. E(X) 2.3 Maxmum Lkelhood Estmaton The probablty densty functon and cumulatve densty functon of a length based exponental dstrbuton wth parameter θ s gven by, -x x1 θ gx ( ) = e, x> 0, θ > 0. (2.3.1) θθ and -x -x x θ θ G(x) = 1- e +e. θ (2.3.2) 21

31 If n tem are put on test, then the lkelhood functon under Progressve Type-II censorng scheme as dscussed n the secton 2.1 s gven by, m m = 1 = 1 R [ ] L = constant g(x θ ) 1 -G (x θ ). Usng (2.3.1) and (2.3.2) the lkelhood functon becomes m -x m -x -x x x θ θ θ L = constant e e +e. 2 =1 θ =1 θ R (2.3.3) The log lkelhood functon s gven by m m m -x θ ln L = ln c + ln x 2mlnθ + R ln e +1. = 1 =1 θ =1 θ m m m x (1+R ) x = ln c + ln x 2mlnθ- + R ln +1. = 1 =1 θ =1 θ x x 22

32 Dfferentatng ln L wth respect to θ and equatng to zero we obtan m m 2m x (1+R ) R x = x θ =1 θ =1 +1 θ θ (2.3.4) Hence we obtan the mle ˆθ as m x (1+R ) =1 ˆθ =. m xr 2m+ =1 x +θˆ (2.3.5) Solvng the equaton (2.3.5) by any teratve method lke Newton-Raphson for ˆθ, maxmum lkelhood estmate of θ can be obtaned. Now agan dfferentatng (2.3.4) we get 23

33 ( ) ( ) ln L 2m x (1+R ) x R x +2θ -2 - θ θ θ x +θ 2 m m = θ =1 =1 ( ) Hence observed asymptotc varance of ˆθ s gven by (Due to Cohen1963) ˆ 1 V(θ) =. 2 ln L 2 θ ˆ θ=θ 2.4 Bayes Estmate Snce last three decades lot of work has been developed n the feld of relablty usng Bayesan approach. Under certan lmtatons, the maxmum lkelhood estmators have a number of desrable propertes and are extensvely used n preference to other classcal estmators. A Bayesan, however, nterprets probablty as a person s degree of belef n a certan proposton based on pror knowledge (or current knowledge) about parameter θ and ths degree of belef s successvely revsed or updated as new nformaton s accumulated about the proposton. 24

34 In Bayesan framework the parameter s justfably regarded as a random varable and the data once obtaned, s gven or fxed. Also t s realstc to assume that lfe parameter s stochastcally dynamc. Martz and Waller (1982) have done lot of work regardng Bayes estmaton n the feld of lfe testng and relablty. In ths secton the Bayesan approach s used to derve estmate of the parameterθ, assumng we are n the stuaton where very less s known about a pror about the values ofθ. Pror dstrbuton s an essental component of Bayesan nference. There s no sngle answer to the queston, What should be the rght pror? For much of the tme the pror nformaton s subjectve and s based on a person s own experence and judgement. Dfferent types of prors lke non-nformatve pror, unform pror, Jeffreys pror, Hartgan s pror, natural conjugate pror, mnmal nformatve pror and Drchlet s pror are used n Bayesan nference. To avod the complexty nvolved n solvng Bayes estmates. Here we consder pror dstrbuton of θ as exponental dstrbuton wth meanβ. That s ( ) θ - β 1 π θ = e, θ>0, β>0. β (2.4.1) 25

35 Usng the lkelhood functon gven n (2.3.3) and the pror defned n (2.4.1), the posteror dstrbuton of θ s gven by: h ( θ x) α Lπ( θ ) m x R m m x θ x x θ - x =1 θ θ 1 β 2m =1 θ =1 θ β α e e +e e m x R - + θ m m x x x θ β - - =1 x θ θ 2m =1 βθ =1 θ. (2.4.2) α e e +e Now under squared error loss functon, the Bayes estmator of θ can be obtaned as * θ = E(θ x) m x θ m - + m x x x θ β x - - =1 θ θ = c θ e 2m e +e dθ 0 =1 βθ =1 θ R where c s normalzng constant. Here t s not possble to get * θ n closed form, so we refer to numercal ntegraton to fnd a soluton. Lndley (1980) gave an alternatve method to approxmate the ntegrals that occur n Bayesan statstcs. Accordng to Lndley (1980), the Bayes estmator * θ s approxmated as 26

36 * θ = E(θ x) θ ˆ + ( u 2 +2u 1ρ 1) σ + l3u 1σ. 2 2 ˆ θ = θ (2.4.3) where dθ u1 = = 1 dθ 2 θ u2 = = 0 2 θ θ ρ = lnπ( θ ) = -lnβ- β dρ -1 ρ 1 = = dθ β l = ln L l l 2 l = θ 3 l = θ ( l ) 2-1 σ =

37 2.5 Smulaton Study In ths secton we consder a smulaton study to observe behavor of ML and Bayes estmate of θ under dfferent censorng patterns. Here we generate 8000 random samples of sze 15, 25 and 50 from length based dstrbuton defned n (2.3.1) forθ = 0.2, 0.8 and 1. To generate a sample (x) under progressve Type-II censorng wth m = 5 we have used the followng method as dscussed by Aggarwala and Balakrshnan (2002). Step 1:- Generate U, where Step 2:- Z = -ln (1-U ) U s a set of random number = 1, 2, 3, 4, 5 Step 3:- Z Z Z Y = n- R n n-r 1-1 j j=1 Step 4:- G(x )=1-exp(-Y ).e. -x -x x θ θ 1- e +e =1-exp(-Y ) θ By solvng the equaton n Step 4 we wll get the values of x. On the bass of smulated samples ML estmates of θ, as gven (2.3.5) along wth ts asymptotc varance are demonstrated n Table-1, where as Table-2 represents Bayes 28

38 estmates of θ, as gven n (2.4.3) wth ts smulated varance for the three censorng patterns. Here we have consdered the followng three censorng patterns for smulaton. n = 15 n= 25 n = 50 R 1 : (3, 3, 2, 0, 2) R 1 : (3, 3, 2, 0, 12) R 1 : (3, 3, 2, 0, 37) R 2 : (1, 2, 3, 3, 1) R 2 : (1, 2, 3, 3, 11) R 2 : (1, 2, 3, 3, 36) R 3 : (0, 0, 0, 0, 10) R 3 : (0, 0, 0, 0, 20) R 3 : (0, 0, 0, 0, 45) 29

39 For n=15, for 8000 teratons. Table-1 Estmator of θ under Maxmum Lkelhood Estmaton ˆθ Mnθ Maxθ Asy V( θ ˆ ) Sm V( θ ˆ ) arthmetc mean R θ = 0.2 R R R θ =0.8 R R R θ = 1 R R

40 Table-2 Estmator of θ under Bayesan Analyss β=3 β=6 β=10 θ * * * SmV(θ ) θ * * SmV(θ ) θ * SmV(θ ) arthmetc arthmetc arthmetc mean mean mean R θ= 0.2 R R R θ =0.8 R R R θ = 1 R R

41 For n=25, for 8000 teratons. Table-3 Estmator of θ under Maxmum Lkelhood Estmaton ˆθ Mn θ Max θ Asy V( θ ˆ ) Sm V( θ ˆ ) arthmetc mean R θ = 0.2 R R R θ =0.8 R R R θ = 1 R R

42 Table-4 Estmator of θ under Bayesan Analyss β=3 β=6 β=10 θ * * * SmV(θ ) θ θ * * * SmV(θ ) θ arthmetc arthmetc arthmetc arthmetc mean mean mean mean R θ= 0.2 R R R θ =0.8 R R R θ = 1 R R

43 For n=50, for 8000 teratons. Table-6 Estmator of θ under Maxmum Lkelhood Estmaton ˆθ arthmetc Mnθ Maxθ Asy V( θ ˆ ) Sm V( θ ˆ ) mean R θ = 0.2 R R R θ =0.8 R R R θ = 1 R R

44 Table-7 Estmator of θ under Bayesan Analyss β=3 β=6 β=10 θ * * * SmV(θ ) θ θ * * * SmV(θ ) θ arthmetc arthmetc arthmetc arthmetc mean mean mean mean R θ= 0.2 R R R θ =0.8 R R R θ = 1 R R

45 2.6 Conclusons and Suggestons:- 1) For a small sample sze (n = 15) ˆ * Sm V(θ)< Sm V(θ ).e. the MLE s better than the Bayes estmator for a gven θ and β n the case of all the three censorng schemes. 2) For any sample sze (n= 15, n=25, n=50) smulated varance of MLE and Bayes estmator decreases n case of all the three censorng schemes. 3) For the fxed values of θ and β smulated varance of MLE and Bayes estmator decreases accordng to the selecton of the censorng schemes R 1, R 2 and R 3 respectvely. 4) As n ncreases ˆ Sm V(θ) as well as * Sm V(θ ) decreases for fxed values of θ and β for the three censorng schemes. 36

46 CHAPTER 3 Estmaton under Progressve Interval Type-I Censorng for Recprocal Exponental Dstrbuton 3.1 INTRODUCTION In most applcatons, the data may be nterval-censored. By ntervalcensored data, we mean that a random varable of nterest s known only to le n an nterval, nstead of beng observed exactly. In such cases, the only nformaton we have for each ndvdual s that ther event tme falls n an nterval, but the exact tme s unknown. Generally statstcan faces lot of problem n the analyss of tme-to-event data such as falure tme data, ncubaton tme data etc. Such data arses n lot of felds such as medcne, engneerng, economcs. For example doctor may be nterested to know the tme of convergence to AIDS for HIV postve ndvdual, 37

47 the tme to the death for cancer patents, lfetme of a devce etc. The analyss to tme-to-event later becomes more complcated on account of censorng. Interval censorng also known as group censorng arses when observatons occur n some nterval of tme a and b. Such data occurs n varety of crcumstances but generally t s encountered n medcal studes where patents are only montored at regular ntervals (e.g. weekly or quarterly checkup). Thus, the exact tme of occurrence of some changed response may only be known to have some tme between two vsts. Samuelson and Kongerud (1994); Kokasa et al (1993); Farrngton (1996); Odell et al (1992), Sun (1997); Lndsey and Ryan (1998) and Scallan (1999) have dscussed applcaton of nterval censorng n clncal, medcal, bomedcal and engneerng studes. Rao (1998) gave standard methods for analyzng nterval censored data and dscussed effcences of estmators derved from censorng over conventonal Type-I and Type-II censorng schemes. In many lfe test studes, t s common that the lfetmes of test unts may not be recorded exactly. An expermenter may termnate the lfe test before all n products fal n order to save tme or cost. Hence, the test s sad to be censored n whch data collected are the exact falure tmes on those functonal (none faled) unts. Moreover, some of the test unts may have to be removed at dfferent stage(s) of censorng related study for varous other reasons; whch 38

48 leads to progressve censorng. For example some products are wthdrawn for more thorough nspecton or are saved so that t can be used as test specmens n other studes, or patents who for some reasons do not turn up n a clncal study would also result n progressve removal. Accordng to the current trend Type-I and Type-II progressve censorng schemes are becomng qute popular for analyzng hghly relable data. Cohen (1963) had ntroduced progressve Type-II censorng. Mahmond et al (2006) consdered progressve Type-II censorng samples for many contnuous lfe tme models. Balakrshnan and Aggarwala (2000) gve an nsght on ths method and the applcatons of ths scheme. Aggarwala (2001) ntroduced progressve Type-I nterval censorng scheme for exponental lfe tme model. In ths type of censorng n unts are put on test at tme 0 and each unt s kept on lfe test untl the unt fals or s censored. All the unts are observed durng pre-set tmes T 1, T 2,, Tm where m s a fxed nteger. Thus the tme axs s parttoned nto nterval I = (T -1, T ] where = 1, 2,, m+1 and T 0 = 0, T m+1 =, T m s the tme at whch we wll termnate the experment. Let n denote the number of unts whch fal n the nterval I. The values R 1, R 2 R m may be specfed as postve ntegers or percentages p 1, p 2 p m wth p m = 100 of remanng functonal unts and the number of unts whch are functonng at tme T 1, T 2,,T m are random varables. 39

49 In case when R 1, R 2,, R m are pre- specfed postve ntegers, the number of unts removed at tme T s R obs = mn( R, no. of unts remanng) = 1, 2,, m-1 and R obs m = all the remanng unts at tme Tm, when lfe test experment s termnated. In ths chapter we have consdered recprocal exponental dstrbuton as a contnuous lfetme model and apply progressvely Type-I nterval censorng wthout changng the parameters at dfferent stages of censorng. Secton 2 states the propertes and applcatons of Recprocal Exponental dstrbuton; n Secton 3 the method of maxmum lkelhood estmaton s descrbed. Smulaton of progressve Type-I nterval censored samples s carred out n Secton 4. Secton 5 deals wth nterval estmaton. Expected duraton of the lfe test s dscussed n Secton 6. Comparson between Non-parametrc and Parametrc estmaton of survval functon and ts confdence nterval are consdered n Secton 7. The methods are llustrated usng numercal examples for dfferent censorng pattern. 40

50 3.2 Recprocal Exponental Dstrbuton A random varable X follows a Recprocal Exponental dstrbuton f ts recprocal 1/X follows an Exponental dstrbuton wth scale parameter θ, θ>0. The probablty densty functon (pdf) and cumulatve dstrbuton functon (cdf) of recprocal exponental dstrbuton are as follows, θ g(x,θ)= e, x>0, θ>0. -θ/x 2 x (3.2.1) and - θ /x G(x,θ )= e. (3.2.2) Recprocal Exponental dstrbuton s a specal case of Inverted Gamma dstrbuton, havng pdf α α 1 β/x β x e f(x;α,β) = x 0, α,β 0 wth β = θ α and α = 1. The Recprocal Exponental dstrbuton appears n Bayesan nference n a natural way as the posteror dstrbuton of the varance n normal samplng when reference or conjugate dstrbutons on the parameters are used. Recprocal Exponental dstrbuton s especally used n relablty applcatons (see Barlow 41

51 and Proschan (1981)). It s also hdden among the Pearson curves, specfcally Pearson V and Vnc (1921) should be credted for hs ncome dstrbuton applcatons. In actuaral lterature, Cumms et al. (1990) used the Inverse Gamma dstrbuton for approxmatng the fre loss experences of a major unversty. The dstrbuton turns out to be one of the best two parameter models; n fact the data are approxmately modeled by one parameter specal case where α = 1, an Inverse Exponental dstrbuton. 3.3 Maxmum Lkelhood Estmaton Suppose a progressve Type-I nterval censored sample s collected as descrbed n Secton1, begnnng wth a random sample of n unts havng probablty densty dstrbuton functon gven by (3.2.1). Based on the observed data, the lkelhood functon L s proportonal to the expresson. n [ ( ) ( )] [ ( )] m =2 1 R1 x 1 x 0 x 1 L α G T -G T 1-G T [ ( ) ( )] [ ( )] n x x -1 x R G T -G T 1-G T. m -n 1 1θ/T -θ/t R R 1 -θ/t -θ/t-1 -θ/t L α e 1-e e -e 1-e. =2 n 42

52 Here for the sake of smplcty we consder equal length tme nterval.e. T -T =t -1 Thus T =t, =1,2,...,m Thus the lkelhood functon reduces to, m ( ) -θ/t R1 -θ/t R -n1θ/t -θ/(-1)t -θ/(-1)t L α e 1-e e e -1 1-e. =2 n (3.3.1) The lkelhood equaton for estmatng θ s obtaned by lnl θ = 0. whch gves -θ/t ( ) -n R -1 n + -e - 1-e =2 m 1 1 -θ/t t t (-1)t n -θ/t ( ) m m -θ/(-1)t e + -e =0. -θ/(-1)t -θ/t e -1 t(-1) t =2 =2 R 1-e (3.3.2) 43

53 Under ths stuaton the MLE of θ can be obtaned by usng any teratve procedure lke Newton-Raphson and solvng the equaton (3.3.2). Hence we get maxmum lkelhood estmator of θ, denoted byθ $. Now agan dfferentatng (3.3.2) we get, -θ/t -θ/t 2 m m -θ/(-1)t lnl -R 1e R e n e 2 = θ/t -θ/t θ/(-1)t =2 =2 ( ) ( ) ( ) θ t 1-e t 1-e t (-1) e -1 Hence observed asymptotc varance of θ $ s gven by (Due to Cohen 1963). -1 V(θ) ˆ. 2 lnl 2 θ ˆ θ=θ 3.4 Comparson of censorng patterns va smulaton In ths secton consderng equal nterval length, the Recprocal Exponental dstrbuton defned n (3.2) as the lfetme model from whch 1000 samples were generated usng the values θ = 3 and 5, t = 2, m = 5 and sample sze n = 20 and 50 respectvely, under the followng censorng patterns. 44

54 n = 20 n = 50 S 1 : (3, 3, 2, 1, n-n 1 -n 2 -n 3 -n 4 -R 1 -R 2 -R 3 -R 4 ) S 1 : (12, 10, 8, 6, n-n 1 -n 2 -n 3 -n 4 -R 1 -R 2 -R 3 -R 4 ) S 2 : (1, 2, 3, 3, n-n 1 -n 2 -n 3 -n 4 -R 1 -R 2 -R 3 -R 4 ) S 2 : (6, 8, 10, 12, n-n 1 -n 2 -n 3 -n 4 -R 1 -R 2 -R 3 -R 4 ) S 3 : (0, 0, 0, 0, n-n 1 -n 2 -n 3 -n 4 -R 1 -R 2 -R 3 -R 4 ) S 3 : (0, 0, 0, 0, n-n 1 -n 2 -n 3 -n 4 -R 1 -R 2 -R 3 -R 4 ) We have used the smulaton algorthm gven by Aggarwala (2001) to generate samples from progressve Type-I ntervals censorng scheme. Here we have specfed the fxed number of unts nstead of proporton of survvng unts to be removed at fve montorng and censorng ponts. The removng unts from the survvng unts at fve stages are decreasng n pattern S 1 whle ncreasng n pattern S 2. In pattern S 3, a convectonal Type-I nterval censorng scheme s employed. Steps for Smulaton:- Consder n 1 ~ Bnomal (n, G(T 1 )). and ( ) ( ) G T -G T Bnomal n- n +R, n n -1,...,R -1,...,R 1~ ( j j) j=1 1-G ( T-1 ) 45

55 Table-1 gves the summary statstcs of the maxmum lkelhood estmators for the three censorng patterns; wth ts observed asymptotc varance AV ( θ ˆ ) and smulated varance SV( θ ˆ ), n case of 1000 random samples generated for n = 20 and 50, θ = 3 and 5, m= 5 and t = 2. Here ˆθ s the average of smulated MLE. Table-1: Summary Statstcs For θ = 3, n = 20 Scheme Mn ˆθ Max ˆθ ˆθ AV ( θ ˆ ) SV( θ ˆ ) S S S For θ =5, n = 20 Scheme Mn ˆθ Max ˆθ ˆθ AV ( θ ˆ ) SV( θ ˆ ) S S S For θ = 3, n = 50 Scheme Mn ˆθ Max ˆθ ˆθ AV ( θ ˆ ) SV( θ ˆ ) S S