PARAMETER ESTIMATION IN WEIBULL DISTRIBUTION ON PROGRESSIVELY TYPE- II CENSORED SAMPLE WITH BETA-BINOMIAL REMOVALS

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1 Econoy & Busness ISSN , Volue 10, 2016 PARAMETER ESTIMATION IN WEIBULL DISTRIBUTION ON PROGRESSIVELY TYPE- II CENSORED SAMPLE WITH BETA-BINOMIAL REMOVALS Ilhan Usta, Hanef Gezer Departent of Statstcs, Faculty of Scence, Anadolu Unversty, Esksehr, Turkey Abstract In ths artcle, the estaton of paraeters based on progressvely type-ii censored saple wth rando reovals fro the Webull dstrbuton s studed. The nuber of unts reoved at each falure te s assued to follow a Beta-bnoal dstrbuton. Based on ths type of censored saple, the axu lkelhood (ML) and approxate axu lkelhood (AML) estators for the paraeters of the Webull dstrbuton are derved. A Monte Carlo sulaton study s also conducted to copare the perforance of ML and AML estators under progressvely type-ii censorng wth the dfferent rando schees. Key words: Webull dstrbuton, axu lkelhood estator, approxate axu lkelhood estator, progressvely censorng type-ii wth rando reovals, beta-bnoal reovals 1. INTRODUCTION The Webull dstrbuton s one of the ost coonly used dstrbutons n the feld of relablty and lfe-testng analyss where saples are usually censored. Aong the dfferent censorng schees, the progressvely Type-II censorng schees has drawn attenton by any authors durng recent years. Because the tradtonal Type-I, Type-II and hybrd censorng schees do not allow the experenter to reove unts before the ternaton of the experent. Therefore, the progressve Type-II censorng schees has been wdely used n relablty and lfe-testng experents, etc. The advantage of ths censorng schees can be explaned (descrbe) as follows. The experenter places n dentcal unts on test at te zero and copletely observe only falures. When the frst falure s observed, R 1 of the reanng n 1 survvng unts are randoly selected and reoved. Then after the second observed falure, R 2 of the reanng n 1 R 1 survvng unts are randoly selected and reoved, and so on. Fnally, the experent ternates untl the th falure s observed and reanng R = n 1 R survvng unts all reoved. If R 1 = R 2 = = R 1 = 0 then R = n that corresponds Type-II censorng. If R 1 = R 2 = = R = 0, then n = that corresponds the coplete saple. For a coprehensve recent revew of progressve censorng, see (Balakrshnan & Aggarwala 2000) and (Balakrshnan 2007). The R 1, R 2,, R are all prefxed n ths censorng schee. However, n soe practcal stuatons, the R nubers randoly occur and so they cannot be prefxed. For nstance, (Yuen & Tse 1996) ndcated that the nuber of patents that wthdraw fro a clncal test at each stage s rando and cannot be prefxed. (Kaushka et al. 2015) also pont out that the probablty of reoval ay vary fro patent to patent and reans unknown to the experenter. Thus, recently, the statstcal nference for dfferent lfete dstrbutons under progressve type II censorng wth rando reovals has been nvestgated by varous authors such as (Yuen & Tse 1996), (Wu et al. 2007), (Yan et al. 2011), and (Usta & Gezer 2015). However, there are few studes n the lterature concernng the estaton of paraeters fro the Webull dstrbuton based on progressve type II censorng wth rando reovals. For exaple, (Yuen & Tse 1996) consdered the estaton proble when lfetes are Webull dstrbuted and are collected under a Type-II progressve censorng wth rando reovals follows a unfor dscrete dstrbuton. They derved the axu lkelhood (ML) estator of the paraeters and ther asyptotc varances. Tse and Yuen (1998) provded the expected experent tes for Webulldstrbuted lfetes under Type- II progressve censorng, wth the nubers of reovals dstrbuted bnoal. Tse et. al. (2000) studed the analyss of Webull dstrbuted lfete data observed under Type II progressve censorng wth bnoal reovals. The ML estators of the paraeters and ther asyptotc varances are derved. Tse and Xang (2003) explored the proble of nterval estaton for Page 505

2 Econoy & Busness ISSN , Volue 10, 2016 paraeters of Webull dstrbuton based on Type-II progressvely censored wth bnoal rando reovals. In ths paper, seven dfferent confdence nterval-estaton procedures were consdered. Sarhan and Al-Ruzazaa (2010) dscussed statstcal nference for the paraeters of Webull dstrbuton odel, usng Type-II progressvely censored data wth bnoal rando schee. They also used the ML ethod to derve both pont and nterval estates of the paraeters. In ths paper, we consder the estaton proble for two paraeters of Webull dstrbuton under progressve Type II censored saple wth rando reovals where the nuber of unts reoved at each falure te follows a Beta-bnoal dstrbuton. We derve the ML estators of the unknown paraeters. Snce the ML estators cannot be derved n explct for, we obtan approxate axu lkelhood (AML) estators, whch have explct expressons, for the paraeters of Webull. Moreover, a Monte Carlo sulaton study s conducted to copare the perforance of ML and AML estators under progressvely type-ii censorng wth the dfferent rando schees. The reander of ths paper s organzed as follows. In Secton 2, the Webull dstrbuton and the notaton and defnton used n ths (throughout the) paper are presented. The ML and AML estators are derved under Type-II progressve censorng wth Beta-bnoal reovals n Sectons 3 and 4, respectvely. The results of the sulaton study are presented n Secton 5. The conclusons of the paper are provded n Secton THE LIKELIHOOD FUNCTION OF THE MODEL It s assued that the lfete X of a unt has a Webull dstrbuton wth the shape (α) and scale (β) paraeters. The probablty densty functon (pdf) and the cuulatve dstrbuton functon (cdf) of X are gven as, respectvely, and f(x; α, β) = αβx α 1 e βxα ; x > 0, α > 0, β > 0 (2.1) F(x; α, β) = 1 e βxα ; x > 0, α > 0, β > 0 (2.2) Let X 1 < X 2 < < X denote a progressvely Type-II censored saple fro a Webull dstrbuton, where < n s predeterned before the test. For progressve type II censorng wth a pre-deterned nuber of reovals R = (r 1, r 2,, r ), the condtonal lkelhood functon can be wrtten as (Cohen 1963): L 1 (Ө; x R = r) = C f(x )[1 F(x ] r (2.3) where, C = n(n r 1 ) (n r + 1). Substtutng Eqs. (2.1) and (2.2) nto Eq. (2.3), the condtonal lkelhood functon s derved as: L 1 (α, β; x R = r) = C αβx α 1 e βxα (e βxα ) r (2.4) Suppose that the nuber of unts reoved at each falure te R, follows a bnoal dstrbuton wth paraeters n r j and p. Thus, for gven p, the probablty of R unts leavng after the th falure s gven by: P(R = r p) = ( n r j) pr(1 p) r n r j ; = 1,2,, 1 (2.5) Page 506

3 Econoy & Busness ISSN , Volue 10, 2016 Also, slar to (Sngh et al. 2013) and (Kaushk et al. 2015), we assued that the probablty of reovals (p) s not fxed and p s a rando varable followng a beta dstrbuton wth paraeters a and b: g(p a, b) = 1 B(a, b) pa 1 (1 p) b 1 ; a, b > 0, 0 < p < 1 (2.6) Then, the dstrbuton of R can be obtaned as follows 1 P(R = r, a, b) = P(R = r p)g(p a, b) dp (2.7) 0 Substtutng Eqs. (2.5) and (2.6) n Eq. (2.7) and after splfcaton, we get P(R = r, a, b) = ( n r B(a + r j, b + n ) B(a, b) r where, B(a, b) = Γ(a)Γ(b) Γ(a+b) r j ) (2.8), a, b > 0 and r = 0,1,, n r j, = 1,2,, ( 1). The probablty ass functon gven n Eq. (2.8) s known as Beta-bnoal dstrbuton and t s denoted by Beta Bno(n, a, b) (Sngh et al. 2013). The saplng procedure for generatng a progressvely Type-II censored saple wth Beta-bnoal reovals fro a test, s llustrated n Table 1. The jont probablty of R 1 = r 1, R 2 = r 2,, R = r s gven by then P(R = r, a, b) = P[R 1 = r 1 ] x P[R 2 = r 2 R 1 = r 1 ] x x P[R 1 = r 1 R 2 = r 2,, R 1 = r 1 ] (2.9) 1 1 P(R = r, a, b) = A (B(a, b)) 1 B(a + r, b + n r j ) where A = (n )! 1 1 r!(n r j )! Moreover, we presue that be expressed as R. s ndependent of X (2.10) for all th. Accordngly, the lkelhood functon can L(α, β, a, b; x, r) = L 1 (α, β; x R = r)p(r = r, a, b) (2.11) Page 507

4 Econoy & Busness ISSN , Volue 10, 2016 Table 1: Saplng procedure for a lfe test under progressve Type-II censorng schee wth Betabnoal reovals 3. MAXIMUM LIKELIHOOD ESTIMATIONS In ths secton, we derve the axu lkelhood estators (MLEs) of the paraeters α, β and a, b under progressvely Type II censorng saple wth rando reovals where the nuber of unts reoved at each falure te follows a Beta-bnoal dstrbuton. It s clear that L 1 (α, β) does not nclude a and b.thus, the MLEs of α and β can be derved by axzng Eq. (2.4) drectly. The log lkelhood of L 1 (α, β) s wrtten as: l 1 (α, β; x R = r) = lnc + lnα + lnβ β x α (1 + r ) + (α 1) lnx The MLEs of α and β, can be obtaned by solvng the followng noral equatons: (3.1) l 1 (α, β; x R = r) = β β x α (1 + r ) = 0 (3.2) l 1 (α, β; x R = r) = α α β x α (1 + r )lnx + lnx = 0 (3.3) It should be noted that there s no explct soluton for Eqs. (3.2) - (3.3), and thus, teratve ethods should be used [34, 36]. In ths paper, we use the well-known Newton-Rapson to obtan the MLEs α and β of α and β. Slarly, because of P(R = r, a, b) does not nvolve α and β, the MLEs of a and b can be derved by axzng Eq. (2.10) drectly. The log lkelhood of P(R = r, a, b) s wrtten as log P = lna ( 1)[ln Γ(a) + lnγ(b) lnγ(a + b)] + lnγ(a + r ) 1 + lnγ(n r j b) lnγ (n r j + a + b) The MLEs a and b of a and b can be obtaned by solvng the noral equatons d log P db = 0. However, we wll not concerned wth that n ths paper. d log P da = 0 and (3.4) Page 508

5 Econoy & Busness ISSN , Volue 10, APPROXIMATE MAXIMUM LIKELIHOOD ESTIMATION Snce the ML estators of α and β do not obtan as explct forulas for the Webull dstrbuton based on progressve Type II censored saple wth rando reovals, we derve the approxate axu lkelhood (AML) estators developed by Balakrshnan (1989) and (Balakrshnan & Vardan 1991) to estate the paraeters α and β. As entoned before, snce the probablty P(R = r, a, b) does not depend on the paraeters α and β, so we gnore t and f we consder the transforaton V = ln X, = 1,2,, and α = 1 and σ μ = 1 lnβ. Then, the lkelhood functon can be wrtten as: α L 1 (σ, μ R = r) = C 1 μ σ ev σ v μ e σ where C = n(n r 1 ) (n r + 1). Let z = v μ σ (e v μ σ ) r (4.1), g(z ) = e z e z and G (z ) = e ez, then the Eq. (4.1) can be rewrtten as: L 1 (σ, μ R = r) = c 1 g(z σ )(G (z )) r (4.2) wth the log-lkelhood functon l 1 (σ, μ; x R = r) = lnc lnσ + g(z ) + r G (z ) (4.3) Takng partal dfferentaton of l 1 (σ, μ; x R = r) wth respect to μ and σ, the noral equatons are gven followng as: dl 1 (σ, μ; x R = r) = 1 dμ σ g (z ) + r g(z ) g(z ) σ G (z ) = 0 (4.4) dl 1 (σ, μ; x R = r) = dσ σ g (z ) z g(z ) σ + r g(z ) G (z ) z σ = 0 (4.5) Clearly, the noral equatons (4.4) and (4.5) do not have explct solutons. Thus, we consder a frst order Taylor expanson to the functon g (z ) and g(z ) around the g(z ) G (z ) G 1 (p ) = ln( lnq ) = μ where p =, q +1 = 1 p, = 1,2,, see (Balakrshnan & Aggarwala 2000) and (Hashe & Ar 2011). Then, we can consder the followng approxatons: g (z ) g(z ) α β z (4.6) g(z ) G (z ) 1 α + β z (4.7) where α = 1 + lnq (1 ln( lnq )) and β = lnq. By usng Eqs. (4.6) and (4.7) nto Eqs. (4.4) and (4.5), we get ( α β z ) + r (1 α + β z ) = 0 (4.8) Page 509

6 Econoy & Busness ISSN , Volue 10, 2016 ( α β z )z + r (1 α + β z )z = 0 (4.9) Fro Eqs. (4.8) and (4.9), we can obtan μ = B D σ (4.10) σ 2 + Eσ F = 0 (4.11) where, B = (r + 1)β z (r + 1)β, D = α r (1 α ), (1 + r )β E = α (x B) r (1 α ) (x B) 2D (1 + r ) β (x B), F = (1 + r ) β (x B) 2. Thus, AML estators of σ and μ are, respectvely, σ = E+ E2 +4F 2 obtan α = 1 σ and β = e α μ as the AMLE of α and β. and μ = B Dσ. Then, we 5. SIMULATION In ths secton, a Monte Carlo sulaton study s conducted to copare the perforance of the ML and AML estates derved n the prevous sectons for progressvely Type II censorng data wth the dfferent Beta-bnoal rando schees. 5.1 Algorth for generatng progressvely type-ii censored saples fro Webull dstrbuton By applyng the algorths gven n (Balakrshnan & Aggarwalla 2000) and (Sngh et al. 2013), the followng steps are used generate progressvely Type II censored data wth the Beta-bnoal reovals fro Webull dstrbuton. The steps are: 1. Fx the values of n,,. 2. Fx the values of paraeters α, β and a, b. 3. Generate a rando nuber r 1 fro Beta Bno(n, a, b). 4. Generate a rando nuber r fro Beta Bno(n r j, a, b); = 2,, 1 5. Set r = { n 1 r 1 j f n r j > 0 } 0 otherwse. 6. Generate ndependent w ; = 1,2,, fro Unfor(0,1). 1/(+ j= +1 r 7. Set V = w j ) = 1,2,,. 8. Set U = 1 V V 1 V +1 = 1,2,,. Then, U 1, U 2,, U are ranked progressve censored saple of sze fro Unfor(0,1) wth Beta-bnoal reovals. 9. Fnally, for fxed values of paraeters α and β, we set X = 1 F 1 (U ) = ( ln (1 U ) β ) 1/α. Then, (X 1, X 2,, X ) s the progressve type-ii censored saple fro the Webull dstrbuton wth the Beta-bnoal reovals. Page 510

7 Econoy & Busness ISSN , Volue 10, Sulaton desgn Gven the values of paraeters β = 1; α = 1.5, 2 and β = 2; α = 2, saple sze n, the nuber of falure and the values of paraeters a and b, generate a progressvely type-ii censored saple usng the algorths n Secton 5.1. The saple sze n s taken as 20, 30 and 50, the value of percentles falures are taken as ( ) x100 = 40% and 80% and the values of paraeters (a, b) n are consdered as (1,1) and (2,2). Copute the ML and AML estates of paraeters α and β. Repeat frst and second steps 5000 tes. Copare the perforance of the ML and AML estators by coputng bas and MSE. 5.3 Sulaton results The obtaned results fro the sulaton study are suarzed n Tables 2-3. Frst, table 2 provdes the bas and MSE of the ML and AML estators for β = 1, α = 1.5 and a = 1, b = 1, a = 2, b = 2. Table 2: Sulaton results for β = 1, α = 1.5 The results gven n Table 2 pont out that for β and α, the MSEs of MLEs and AMLEs decrease when the saple sze n and the falure nforaton ncrease under all censorng rando schees. The MLEs overestate β and α. However, the AMLE underestate α except for n=50,=40 whle β s overestated by the AMLE. For β, the MLE provdes nu bases when all the saple szes and censorng rando schees. The AMLE exhbts the best perforance for α n ters of bases. Wth respect to the MSEs, for all the saple szes and censorng rando schees, whle the MLE gves the best estates for β, the AMLE s the best estator for α. Table 3 reports the bas and MSE of the ML and AML estators for β = 1, α = 2 and a = 1, b = 1, a = 2, b = 2. Page 511

8 Econoy & Busness ISSN , Volue 10, 2016 Table 3: Sulaton results for β = 1, α = 2 It can be seen fro Table 3 that slar to results of β = 1, α = 1,5, when β = 1, α = 2, the MLE perfors better than the AMLE for β accordng to the bases and MSEs when all the saple szes and censorng rando schees. For α, the bases and MSEs of the AMLE are saller than the MLE for all consdered censorng rando schees and saple szes. Table 4 provdes the bas and MSE of the ML and AML estators for β = 2, α = 2 and a = 1, b = 1, a = 2, b = 2. Page 512

9 Econoy & Busness ISSN , Volue 10, 2016 Table 4: Sulaton results for β = 2, α = 2 It can be concluded fro Table 4 that the MLEs underestate β, but t overestate α for when all the saple szes and censorng rando schees. The AMLE underestate β except for n = 50, = 40 whle α s underestated by the AMLE when the rate of falures n the n saple s %40. Accordng to bases, for β, the AMLE shows the best perforance for all the saple szes and censorng rando schees. However, the MLE provdes nu bases for α when the rate of falures n the n saple s 0.4. Wth respect to the MSEs, the AMLE gves the best result for β when all the saple szes and censorng rando schees, but the MLE works well for α when n = 30, = 12 and n = 50, = CONCLUSION In ths study, we have consdered the proble of paraeter estaton for Webull dstrbuton n presence of progressvely Type-II censored saple wth Beta-Bnoal reovals. For ths purpose, the axu lkelhood (ML) and approxate axu lkelhood (AML) estators for the paraeters of the Webull dstrbuton based on ths type of censored saple are derved. Furtherore, the perforance of ML and AML estators s copared n ters of bas and MSE for dfferent saple szes and rando schees va a Monte Carlo sulaton. As a consequence, the overall sulaton results reveal the followng: When the values of paraeters are β = 1; α = 1.5, 2, for all the saple szes and censorng rando schees, the MLE perfors better than the AMLE n ters of bas and MSE for β and the AMLE provdes nu bases and MSEs for α. If the values of paraeters are β = 2; α = 2, consderng bas and MSE, the AMLE s exhbts best perforance for β when all the saple szes and censorng rando schees. the AMLE gves ostly good estates for α. However, the MLE provdes nu bases and MSE for only α when the rate of falures n the n saple s 0.4. Page 513

10 Econoy & Busness ISSN , Volue 10, 2016 Based on ths, the AML s generally ore effcent than ML to estate the paraeters β and α of Webull dstrbuton under progressvely Type-II censored saple wth Beta-Bnoal reovals. REFERENCES Balakrshnan, N & Vardan, J 1991, 'Approxate MLEs for the Locaton and Scale Paraeters of the Extree Value Dstrbuton wth Censorng', IEEE Transactons on Relablty, pp.40, Balakrshnan, N 2007, 'Progressve Censorng Methodology: An Apprasal', Test (2007), 16: Balakrshnan, N 1989, 'Approxate MLE of the scale paraeter of the Raylegh dstrbuton wth censorng', IEEE Transactons on Relablty, 38, Balakrshnan, N & Aggarwala, R 2000, 'Progressve censorng: Theory, Methods and Applcaton', Brkhauser,Boston. Cohen, AC 1963, 'Progressvely censored saples n the lfe testng', Technoetrcs, 5, pp.pp Hashe, R & Ar, L 2011, 'Analyss of progressve Type-II censorng n the Webull odel for copetng rsks data wth bnoal reovals', Appled Matheatcal Scences, 5(22), pp Huang, SR & Wu, SJ 2011 Bayesan estaton and predcton for Webull odel wth progressve censorng, Journal of Statstcal Coputaton and Sulaton, 82:11, pp Kaushk, A, Sngh, U, & Sngh, SK 2015, 'Bayesan Inference for the Paraeters of Webull Dstrbuton under Progressve Type-I Interval Censored Data wth Beta-bnoal Reovals', Councatons n Statstcs - Sulaton and Coputaton, ISSN: (Prnt) Kaushka, A, Sngh, U & Sngh, SK 2015 Bayesan Inference for the Paraeters of Webull Dstrbuton under Progressve Type-I Interval Censored Data wth Beta-bnoal Reovals Councatons n Statstcs - Sulaton and Coputaton 2015 (Accepted anuscrpt) Lee, P.M. 2012, 'Bayesan Statstcs: An Introducton', wley. Mubarak, M 2012, Paraeter estaton based on the Frechet progressve type II censored data wth bnoal reovals, Internatonal Journal of Qualty Statstcs and Relablty, Vol. 2012, Artcle ID , 5 p. Pareek, B, Kundu, D & Kuar, S 2009, 'On progressvely censored copetng rsks data for webull dstrbutons', Coputatonal Statstcs and Data Analyss, pp.53, Sarhan, AM & Al-Ruzazaa, A 2010, Statstcal nference n connecton wth the Webull odel usng type-ii progressvely censored data wth rando schee, Pakstan Journal of Statstcs, Vol.26(1), pp Sngh, SK, Sngh, U & Shara, VK 2013, 'Expected total test te and Bayesan estaton for generalzed Lndley dstrbuton under progressvely Type-II censored saple where reovals follow the Beta-bnoal probablty law' Appled Matheatcs and Coputaton, 222, pp Sultan, KS, MahMoud, MR & Saleh, HM 2007, 'Estaton of Paraeters of the Webull Dstrbuton Based on Progressvely Censored Data', Internatonal Matheatcal Foru, 2, 2007, no. 41, Tse SK & Xang, L 2003, Interval estaton for Webull-dstrbuted lfe data under type II progressve censorng wth rando reovals, Bopharaceutcal Statstcs, vol. 13, no. 1, pp Tse, SK, Yang, C & Yuen, HK 2000, 'Statstcal analyss of Webull dstrbuted lfete data under Type II progressve censorng wth bnoal reovals', Journal of Appled Statstcs, (27), pp Tse SK & Yuen, HK 1998, Expected experent tes for the Webull dstrbuton under progressve censorng wth rando reovals, Journal of Appled Statstcs, 25, pp Page 514

11 Econoy & Busness ISSN , Volue 10, 2016 Usta, I & Gezer, H 2015, Relablty estaton n Pareto-I dstrbuton based on progressvely type II censored saple wth bnoal reovals,journal of Scentfc Research and Developent,2(12), (2015) (ISI Index), 05/09/2015 Wu, CC, Wu, SF & Chan, HY 2006, MLE and the estated expected test te for the two-paraeter Gopertz dstrbuton under progressve censorng schee wth bnoal reovals, Appled Matheatcs and Coputaton, 181 (2): pp Wu, SJ 2010, Estaton for the two-paraeter Pareto dstrbuton under progressve censorng wth unfor reovals, Journal of Statstcal Coputaton and Sulaton, 73:2, Yuen, HK & Tse SK 1996, Paraeter estaton for Webull dstrbuted lfetes under progressve censorng wth rando reovals, Journal of Statstcal Coputaton and Sulaton, 55:1-2, Page 515