Statistical inference for generalized Pareto distribution based on progressive Type-II censored data with random removals

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1 Internatonal Journal of Scentfc World, 2 1) 2014) 1-9 c Scence Publshng Corporaton do: /jsw.v Research Paper Statstcal nference for generalzed Pareto dstrbuton based on progressve Type-II censored data wth random removals Reza Azm, Bahman Fash, Faramarz Azm Sarkhanbaglu Department of Statstcs, Parsabad Moghan Branch, Islamc Azad Unversty, Parsabad Moghan, Iran *Correspondng author E-mal: azmreza1365@gmal.com Copyrght c 2014 Reza Azm et. al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract In ths artcle, We consder the estmaton problems of the parameter and relablty functon of the generalzed Pareto dstrbuton based on a progressvely type-ii censored sample wth random Bnomal) removals. we use the method of maxmum lkelhood and Bayesan estmaton to estmate parameter and relablty functon. Bayesan estmates are derved under squared error and LINEX loss functons. we also construct the confdence nterval for the parameter of generalzed Pareto dstrbuton based on a progressvely type-ii censored sample wth random removals. The comparsons between dfferent estmators are made based on smulaton study. Keywords: Generalzed Pareto dstrbuton, progressve Type-II censored, random removals, Bayesan estmates, relablty functon 1. Introducton In varous lfe-testng and relablty studes, experments must often termnate before all unts on test have faled. In such cases, one has complete nformaton only on part of the sample. On all unts whch have not faled, one has only partal nformaton. Such data are called censored. There are several types of censored tests. One of the most common censored tests s progressve type II censorng. In progressve type II censorng, Suppose that n unts are placed on a lfe test and the expermenter decdes beforehand quantty m, the number of unts, to be faled. Now at the tme of the frst falure, R 1 of the remanng n 1 survvng unts are randomly removed from the experment. Contnung on, at the tme of the second falure, R 2 of the remanng n R 1 2 unts are randomly drawn from the experment. Fnally, at the tme of the mth falure, all the remanng R m = n m R 1 R 2... R m 1 survvng unts are removed from the experment. Note that, n ths scheme, R 1, R 2,..., R m are all pre-fxed. However, n some practcal stuatons, these numbers may occur at random. for example, n some relablty experments, an expermenter may decde that t s napproprate or too dangerous to carry on the testng on some of the tested unts even though these unts have not faled. In such cases, the pattern of removal at each falure s random Zenab [3], Yen and Tse [2]). Ths leads to progressve censorng wth random removals llustrated by Table 1 ). There have been several references about the statstcal nference on lfetme dstrbutons under progressve censorng wth random removals; for example, we refer to Yuen and Tse [2], Tse and Yuen [5], Tse et al. [10], Shuo and Tao [12], Wu [11], Wu and Chang [4] and Wu et al. [9]. Based on a progressvely type-ii censored sample wth random removals, we consder the problem of estmatng

2 2 Internatonal Journal of Scentfc World Table 1: A schematc representaton of the progressve type-ii censorng wth bnomal removals Process The number Falures Bnomal Removals Remans n lfe testng 1 n 1 R 1 Bn m, p) n 1 R 1 2 n 1 R 1 1 R 2 Bn m R 1, p) n 2 R 1 R m 1 n m 2) m 2 R j 1 R m 1 Bn m m 2 R j, p) n m 1) m 1 R j m n m 1) m 1 R j 1 R m = n m m 1 R j 0 parameter and relablty functon of the two parameter generalzed Pareto dstrbuton wth the shape parameter and the scale parameter, proposed by Castllo and Had [7]. under both classcal and Bayesan wth dfferent loss functons) contexts. The cumulatve dstrbuton functon, probablty densty and relablty functons of two parameter generalzed Pareto dstrbuton are respectvely gven by F x, ) = 1 1 x ) 1 ; > 0, 0 < x < 1) and fx, ) = 1 1 x ) 1 1 ; > 0, 0 < x < 2) and Rx, ) = 1 x ) 1 ; > 0, 0 < x < for more detal about two parameter generalzed Pareto dstrbuton see Castllo and Had [7]. two parameter generalzed Pareto dstrbuton were wdely used by several authors, Among others, we refer to Grmshaw [6], Castllo and Had [7], and Castllo et al. [8]. 2. Maxmum lkelhood estmaton Let X 1 < X 2 <... < X m be the ordered falure tmes out of n randomly selected tmes, where m s predetermned before the test. At the th falure, R tems are removed from the test. For progressve censorng wth pre-determned number of removals R = R 1 = r,..., R m 1 ) = r m 1, the lkelhood functon can be defned as the followng form L,, x R) = c m fx, )[1 F x, )] r 3) where c = nn 1 R 1 ) n m 1 ) R + 1). Equaton 3) s derved condtonal on R. Each R can be of any nteger value between 0 and n m 1 R j ). It s dfferent from progressve censorng wth fxed removal that R s a random number and s assumed to follow a bnomal dstrbuton wth parameter p. It means that each unt leaves wth equal probablty p and the probablty of R unts leavng after the th falure occurs s P R 1 = r 1 ) = p r 1 1 p) n m r 1 1 P R = r R 1 = r 1,..., R 1 = r 1 ) = n m r j )p r 1 p) n m m 1 r j

3 Internatonal Journal of Scentfc World 3 where 0 r n m 1 r j = 1,..., m 1). Furthermore, we assume that R s ndependent of X for all. The jont lkelhood functon of X = X 1, X 2,..., X m ) and R = r 1, r 2,..., r m ) can be found as L,, x, p) = L,, x R)P R, p) 4) where P R, p) s the jont probablty dstrbuton of R = r 1, r 2,..., r m ) and n partcular P R, p) = P R m = r m R m 1 = r m 1,..., R 1 = r 1 )... Therefore P R 2 = r 2 R 1 = r 1 )P R 1 = r 1 ) n m)! P R, p) = p n m m 1 r j )! m 1 r j m 1 r j 1 p) m 1)n m) m 1 m j)r j Substtutng 1) and 2) nto 4), the lkelhood functon takes the followng form { L,, x, p) m m 1 m exp r + 1) log 1 x ) } exp { m log 1 x ) } p The frst partal dervatves of log-lkelhood functon wth respect to s log L,, x, p) = m m 1 m r + 1) log 1 x ) r j m 1)n m) m 1 m j)r j 1 p) I0,) x ) 5) 2 = 0 therefor we get the MLE of as n the followng form m ˆ = r + 1) log 1 x ) m By the nvarant property of the MLE, the MLE of the relablty functon, S = Rt), wth fxed t > 0. s gven by Ŝ MLE = 1 t ) 1 ˆ MLE 6) 7) 3. Exact nterval estmaton Let X 1 < X 2 < < X m be a progressvely type II censored sample from the generalzed Pareto dstrbuton. Furthermore, let Y = 1 ln ) 1 x, = 1,..., m. It s easy to show that Y1,, Y m s a progressvely Type II censored sample from the standard exponental dstrbuton. For a fxed set of R = r 1,..., r m ), let us consder the followng transformaton: Z 1 = ny 1 Z 2 = n r 1 1)Y 2 Y 1 ).

4 4 Internatonal Journal of Scentfc World Z m = n r 1 r m m + 1)Y m Y m 1 ) 8) Balakrshnan and Aggarwala [1] showed that the progressvely type II rght censored spacngs Z 1, Z 2,..., Z m as defned n equaton 8), are ndependent and dentcally dstrbuted as a standard exponental dstrbuton. Hence, 2Z 1 has a ch-square dstrbuton wth 2 degrees of freedom. Now, let m W = 2 Z = 2 m r + 1) log 1 x ) = 2mˆ It s easy to see that W has a χ 2 dstrbuton wth 2m degrees of freedom. Confdence nterval for can be obtaned through the pvotal quantty 2mˆ χ 2 2mˆ 2m). Snce the pvotal quantty χ 2 2m), then we have ) 1 α = P χ 2 2mˆ 1 α 2m) < < χ 2 α 2 2m) 2 = P χ 2 1 α 2 2mˆ 2m) < < ) 2mˆ 2m) χ 2 α 2 m 2 = P r + 1) log 1 x ) χ 2 1 2m) α 2 < < 2 m r + 1) log 1 x 2m) χ 2 α 2 )) 9) 4. Bayesan estmaton In the followng, we present Bayes estmators of the shape parameter and relablty functon of generalzed Pareto dstrbuton when samples are drawn from progressvely Type-II censorng data wth bnomal removals For ths purpose we assume the parameters and p behave as ndependent random varables. In ths paper, for parameter we consder nverted-gamma pror dstrbuton of the form π 1 ) = βα Γα) α+1) e β ) ; α, β) > 0 Independently from parameter, p has a beta pror dstrbuton wth parameters a and b of the form π 2 p) = 1 Ba, b) pa 1 1 p) b 1, 0 < p < 1; a, b) > 0 Based on the pror π 1 ) and π 2 p), the jont pror PDF of, p) s π, p) = π 1 )π 2 p), > 0, 0 < p < 1 = β α Γα)Ba, b) pa 1 1 p) b 1 α+1) e β ) ; > 0, 0 < p < 1 10) It follows, from 3) and10), that the jont posteror dstrbuton of, p) s π, p x, r) = β α Γα )Ba, b ) α +1) e β p a 1 1 p) b 1 11) Where α = m+α, β = β m r + 1) log )) 1 x,a = a+ m 1 r j b = b+m 1)n m) m 1 m j)r j Therefore, the margnal posteror PDFs of and p are gven respectvely by π x, r) = β α Γα ) α +1) e β and πp x, r) = 1 Ba, b ) pa 1 1 p) b 1 12)

5 Internatonal Journal of Scentfc World Bayesan Estmaton Under squared error loss functon Under SE loss functon symmetrc), the estmator of a parameters s the posteror mean. Thus, Bayes estmators of the parameter s obtaned by usng the posteror densty 12) ˆ S = E x,r) = β m r + 1) log ) 1 x m + α 1 13) 4.2. Bayesan Estmaton under LINEX loss functon The LINEX loss functon for can be expressed as the followng proportonalsee Basu and Ebrahm [13]) L ) expc ) c 1; c 0 where = ˆ and ˆ s an estmate of. The Bayes estmator of, denoted by ˆ L under the LINEX loss functon s the soluton of the followng equaton. [ ) ] [ ] 1 E exp cˆ L 1 x,r = e c E x,r Therefore we have ˆ L = β m r + 1) log 1 x ) c ) 1 e c α+m+1 14) 5. Bayesan estmaton of relablty functon S = Rt) Other problems of nterest are those of estmatng the relablty functon Rt), wth fxed t > 0. Let the relablty S = Rt) be a parameter tself. replacng = ln t ln S n terms of S by that of equaton 12), we obtan the posteror densty functon S as π S X) = νx, t) α Γα ln s) α 1 s νx,t) 1 15) ) where, νx, t) = ln β t By usng posteror densty functon S 15), the Bayes estmate of the S = Rt) relatve to quadratc loss s Ŝ S = νx, t) ) α νx, t) ) Under LINEX loss functon, the Bayes estmate of S = Rt) usng equaton 15) s [ Ŝ L = 1 c ln c) l l=0 l! νx, t) ) α ] νx, t) + l 17) 6. Numercal study In ths secton, a Monte Carlo smulaton study s conducted wth varous choces of sample szes.we frstly generate the numbers of progressve censorng wth bnomal removals r = 1, 2,..., m), and progressve censorng wth bnomal removals samples generated from generalzed Pareto dstrbuton by usng the algorthm descrbed n Balakrshnan and Aggarwala [1]. We used the followng steps to generate a progressve censorng wth bnomal removals samples generated from generalzed Pareto dstrbuton

6 6 Internatonal Journal of Scentfc World 1. Generate a group values 1 r Bnomaln m r j, p), m 1 r m = n m r, = 1, 2,..., m 1 accordng to the relevant value of p. 2. Smulate m ndependent exponental random varables Z 1, Z 2,..., Z m. Ths can be done usng nverse transformaton Z = ln1 U ) where U are ndependent unform0, 1) random varables. 3. Set X = Z 1 n + Z 2 n R Z 3 n R 1 R Z n R 1 R 2 R for = 1, 2,..., m. Ths s the requred progressvely type-ii censored sample wth bnomal removals from the standard exponental dstrbuton. 4. Fnally, we set Y = F 1 1 exp X )),for = 1, 2,..., m, where F 1.) s the nverse cumulatve dstrbuton functon of the generalzed Pareto dstrbuton. Then Y 1, Y 2,, Y m s the requred progressvely type-ii censored sample wth bnomal removals from the generalzed Pareto dstrbuton. 5. We compute the MLE of and Rt) = S by 6) and 7). 6. We compute the the Bayes estmates and Rt) = S by respectvely, usng 13), 14), 16) and 17). 7. We compute the confdence nterval of by usng 9). 8. We repeat the above steps 2000 tmes. We then obtan the means and the MSEs mean squared error), where 2000 MSE = φ ˆφ) 2 and ˆφ s the estmator of φ In all above cases the pror parameters chosen as α = 2, β = 1) whch yeld the generated value of = 2 as the true value. The true values of Rt) n t = 0.5 s obtaned R0.5) = The results are summarzed n Tables Concluson Ths paper presents dfferent methods of estmaton to estmate parameter and relablty functon of two parameter generalzed Pareto dstrbuton based on a progressvely type-ii censored sample wth random removals. Our observatons about the results are stated n the follow: Table 2 and 4 shows that the Bayes estmates under squared error loss functon has the smallest MSE s as compared wth other estmates Bayes estmates under the LINEX loss functon and maxmum lkelhood estmator. However, maxmum lkelhood estmator method relatvely more accurate estmators as compared wth the Bayes estmaton. It s mmedate to note that the MSE s decrease as sample sze n ncreases. Table 3 and 5 shows that the Bayes estmates of relablty functon under squared error loss functon has the smallest estmated MSE s as compared wth Bayes estmates under the LINEX loss functon and maxmum lkelhood estmator. However, Bayes estmates under the LINEX loss functon relatvely more accurate estmators as compared wth the maxmum lkelhood estmator and Bayes estmates under squared error loss functon. Also, the MSE s decrease as n ncreases. Table 2 and 4 shows that for dfferent sze n and m, the wdth of the Confdence nterval for, decrease as n and m ncreases.

7 Internatonal Journal of Scentfc World 7 Table 2: Averaged values of MSEs for estmates of, P = 0.1) n m Generated R ˆMLE ˆSE ˆLI c = 0.1) ˆLI c = 0.1) %95CI MSE MSE MSE MSE Wdth , 1, 0, 0, 3) ,6.1896) , 2, 1) ,5.0021) ,1,2,2,0*5,5) ,4.2008) ,1,0*8,1,0,1,0,2) ,3.5692) ,0,1,0* ,3.2989),1,0,1,0*5,1,2) *3,1,2,0, ,3.1254),1,0*16,1) ,2,0*5,1,0,2,0, ,2.9838),0*4,1,0*4,1,0*7,1) ,0,1,0,1,0, ,2.8804) 1,0,0,2,0*25) Table 3: Averaged values of MSEs for estmates of the relablty functon,p = 0.1 n m Ŝ MLE Ŝ S Ŝ L c = 0.1) Ŝ L c = 0.1 MSE MSE MSE MSE e e e e e e e e-05

8 8 Internatonal Journal of Scentfc World Table 4: Averaged values of MSEs for estmates of, P = 0.6) n m Generated R ˆMLE ˆSE ˆLI c = 0.1) ˆLI c = 0.1) %95CI MSE MSE MSE MSE Wdth ,0,1,0,0) , ) ,0*6) , ) ,2,1,0*7) , ) ,1,2,0*12) , ) ,3,1,0,1,2,0*14) , ) ,2,1,0*22) , ) ,2,2,2,0*26) , ) ,0,1,1,0*31) , ) Table 5: Averaged values of MSEs for estmates of the relablty functon, P = 0.6 n m Ŝ MLE Ŝ S Ŝ L c = 0.1 Ŝ L c = 0.1 MSE MSE MSE MSE e e e e e e e e-05

9 Internatonal Journal of Scentfc World 9 References [1] N. Balakrshnan, R. Aggarwala, Progressve Censorng: Theory, Methods, and Applcatons, Boston, MA: Brkhauser, [2] H.K. Yuen, S.K. Tse, Parameters estmaton for Webull dstrbuted lfetmes under progressve censorng wth random removals Journal of Statstcal Computaton and Smulaton, 55, 1, 1996), [3] H.A. Zenab, Bayesan nference for the pareto lfetme model under progressve censorng wth bnomal removals, Journal of Appled Statstcs, 35, 11, 2008), [4] S.J. Wu, C.T. Chang, Inference n the Pareto dstrbuton based on progressve censorng wth random removals, Journal of Appled Statstcs, 30, 2, 2003), [5] S.K. Tse and H.K. Yuen, Expected Experment Tmes for the Webull Dstrbuton under Progressve Censorng wth Random Removals, Journal of Appled Statstcs, 25, 1, 1998), [6] S.D. Grmshaw, Computng maxmum lkelhood estmates for the generalzed Pareto dstrbuton,technometrcs 35,1993), [7] E. Castllo, A.S. Had, Fttng the generalzed Pareto dstrbuton to data Journal of the Amercan Statstcal Assocaton,92, 1997), [8] E. Castllo, A.S. Had,N. Balakrshnan,J.M. Saraba, Extreme Value and Related Models wth Applcatons n Engneerng and Scence, Wley, NewYork, [9] C.C. Wu, S.F. Wu, H.Y. Chan, MLE and the estmated expected test tme for Pareto dstrbuton under progressve censorng data, Internatonal Journal of Informaton and Management Scences, 15, 3, 2004), [10] S.K. Tse,C. Yang,H. K. Yuen, Statstcal analyss of Webull dstrbuted lfetme data under Type II progressve censorng wth bnomal removals, Journal of Appled Statstcs, 27, 8, 2000), [11] S.J. Wu, Estmaton for the two-parameter Pareto dstrbuton under progressve censorng wth unform removals, Journal of Statstcal Computaton and Smulaton, ), [12] W. Shuo-Jye and C. Chun-Tao, Inference n the Pareto Dstrbuton Based on Progressve Type II Censorng wth Random Removals,Journal of Appled Statstcs, 30, 2, 2003), [13] A. Basu, N. Ebrahm, Bayesan approach to lfe testng and relablty estmaton usng asymmetrc loss functon, Journal of Statstcal Plannng and Inference 29, 1991),