Estimation in Constant-Partially Accelerated Life Test Plans for Linear Exponential Distribution with Progressive Type-II Censoring

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1 J. Stat. App. Pro. Lett. 4, No., (017) 37 Journa of Statistics Appications & Probabiity Letters An Internationa Journa Estimation in Constant-Partiay Acceerated Life Test Pans for Linear Exponentia Distribution with Progressive Type-II Censoring Abd EL-Baset A. Ahmad 1 and Mohammad A. Fawzy,3, 1 Math. Dept., Facuty of Science, Assiut University, Assiut, Egypt Math. Dept., Taibah University, madina munawara, Saudi Arabia 3 Math. Dept., Suez University, Suez, Egypt Received: 8 Nov. 016, Revised: 5 Mar. 017, Accepted: 17 Mar. 017 Pubished onine: 1 May 017 Abstract: In this paper, we consider the estimation probem in the case of constant-partiay acceerated ife tests using progressivey Type-II censored sampes. The ifetime of items under use condition foows the two-parameters inear exponentia distribution. The maximum ikeihood estimates of the parameters are obtained numericay. Approximate confidence intervas for the parameters, based on norma approximation to the asymptotic distribution of maximum- ikeihood estimators, studentized-t and percentie bootstrap confidence intervas are derived. A Monte Caro simuation study is carried out to investigate the precision of the maximum ikeihood estimators and to compare the performance of the confidence intervas considered. Finay, two exampes presented to iustrate our resuts are foowed by concusion. Keywords: Constant-stress partiay acceerated ife test, progressive Type-II censored, inear exponentia distribution, maximum ikeihood method, bootstrap method, Monte Caro simuation. 1 Introduction In some situations, the usua ife testing methods dose not obtain enough faiure data necessary to make the desired inference. So, one can use acceerated ife tests (ALT) to quicky obtain information about the ife time distribution of products. Such ALT or partiay ALT (PALT) resuts in shorter ives than that woud be observed under norma conditions. Test units are run ony at acceerated use conditions in ALT, whie they run at both norma and acceerated use conditions in PALT. According to Neson [8], the stress can be appied in various ways. One way to acceerate faiure is the constant stress where as each item runs at either use or acceerated conditions ony, see Bai and Chung [13]. Another way is stepstress, where as one increase the stress appied to test product in a specified discrete sequence. In other words, as indicated by Xiong and Ji [11], a test unit starts at a specified ow stress. If the unit does not fai at a specified time, stress on it is raised and hed a specified time. Stress is repeatedy increased and hed, unti the test unit fais or a censoring time is reached. Recenty, the estimation of parameters from different ifetime distributions based on progressive Type-I and Type- II censored sampes are studied by severa authors incuding Chids and Baakrishnan [6], Baakrishnan and Kannan, Ai Mousa and Jaheen [17], Baakrishnan, et a. [] and Soiman ([5]). In the step stress PALT, a test unit is first run at use condition and if it does not fai in a specified time, then it is run at acceerated condition unti faiures occurs or the observation censored see, for exampe, Abde-Ghani [19], Ismai and Abd-Efattah et a. [8]. Bayes and maximum ikeihood methods of estimation are appied on constant ALT s to finite mixtures of distributions by AL- Hussaini and Abde-Hamid ([14]-[15]). Based on progressive Type II censored sampes, Abde-Hamid [7] considered constant stress PALT when the ifetime of units under use condition foows Burr XII distribution. Aso, Ismai and Sarhan Corresponding author e-mai: mohfawzy180@yahoo.com

2 38 A. A. Ahmad, M. A. Fawzy: Estimation in constant-partiay... [4] study optima design of step-stress ife test with progressivey Type- II censored exponentia data, Srivastava and Neha [13] consider optimum step-stress PALT for the truncated ogistic distribution with censoring. Ismai estimating the parameters of Weibu distribution and the acceeration factor from hybrid PALT, Ismai [] study to inference in the generaized exponentia distribution under PALT with progressive Type-II censoring. The inear exponentia (LE) distribution with the parameters a > 0 and b > 0, wi be denoted by LE(a,b) distribution. Tow specia cases of LE(a, b) distribution are exponentia distribution (when b = 0) and Rayeigh distributions (when a = 0). The corresponding cumuative distribution (CDF), probabiity density function (PDF), surviva function(sf) and the hazard rate function(hrf) are given for t > 0, a, b > 0, respectivey, by f 1 (t)=(a+bt)exp { (at+ bt ) }, F 1 (t)=1 exp { (at+ b t ) }, F 1 (t)=exp { (at+ b t ) }, h 1 (t)=a+bt. (1.1) The LE(a, b) distribution has many appications in appied statistics and reiabiity anaysis. Broadbent [7], uses the LE(a,b) distribution to describe the service of mik bottes that are fied in a dairy, circuated to customers, and returned empty to the dairy. The LE(a,b) distribution was aso used by Carbone et a. to study the surviva pattern of patients with pasmacytic myeoma. The Type-II censored data is used by Bain [16] to discuss the east square estimates of the parameters a and b and by Pandey et a. [9] to study the Bayes estimators of (a,b). There are many situations in ifetesting and reiabiity studies in which the experimenter may be unabe to obtain compete information on faiure times for a experimenta items. There are aso situations wherein the remova of items prior to faiure is pre-panned in order to reduce the cost and time associated with testing. The conventiona Type-I, and Type-II censoring schemes do not have the fexibiity of aowing remova of items at points other than the termina point of the experiment. We consider here a more genera censoring scheme, known as progressivey Type-II censoring, detais of the scheme are discussed in Baakrishnan and Aggarwaa [0]. This paper considers the constant stress PALT appied to items whose ifetimes under design condition are assumed to foow LE(a, b) distribution under a progressive Type-II censoring scheme. The rest of the paper is organized as foows, in Section the mode description and MLEs of the invoved parameters are derived using progressive Type II censoring scheme. Aso, approximate and parametric bootstrap confidence intervas are derived. In Section 3, simuation studies are made using Monte Caro method. Finay, some concuding remarks are presented in Section 4. Mode description and maximum ikeihood estimation According to constant PALT, group 1 consists of n 1 items randomy chosen among n test items is aocated to use condition and group consists of n = n n 1 remaining items are subjected to an acceerated condition. Progressive Type-II censoring is appied as foows: In group j, j = 1,, at the time of the first faiure, R j1 items are randomy withdrawn from the remaining n j1 surviving items. At the second faiure, R j items from the remaining n j R j1 items are randomy withdrawn. The test continues unti the m th j faiure at which time, a remaining R jm j = n j m j R j1 R j R j(m j 1) items are withdrawn. The R ji are fixed prior to the study, m j n j and n 1 m j = np j R ji, (.) where p 1 (0< p 1 < 1) and p = 1 p 1, are the proportions of sampe units aocated to acceerated condition. It is cear that the compete sampe and Type-II censored sampes are specia cases of this scheme. i= j.1 Basic assumptions 1. The ifetime of an item tested at use condition foows LE(a,b) distribution.

3 J. Stat. App. Pro. Lett. 4, No., (017) / The HRF of an item tested at acceerated condition is given by h (t) = ρh 1 (t), where ρ is an acceeration factor satisfying ρ > 1. So, the HRF, CDF, PDF and SF under acceerated condition are given, for t > 0,a,b>0 and ρ > 1, respectivey, by f (t)=ρ(a+bt)exp { ρ(at+ b t ) }, F (t)=1 exp { ρ(at+ b t ) }, F (t)=exp { ρ(at+ b t ) } (.3), h (t)=ρ(a+bt). 3. The identicay distributed ifetimes T ji, j = 1,, i = 1,...,n j of items aocated to use condition ( j = 1) and acceerated condition( j = ) are mutuay independent.. Maximum ikeihood estimation The maximum ikeihood method is used to obtain the estimates of the parameter of the popuation distribution. Let T (R j1,...,r jm j ) j1:m j :n j < T (R j1,...,r jm j ) j:m j :n j <...<T (R j1,...,r jm j ) jm j :m j :n j, j= 1,, are the progressive Type-II censored data from two popuations whose SFs and PDFs are given by (1.1) and (.3), with(r j1,...,r jm j ) being the two progressive censoring schemes. In the seque, we wi use T jm j :m j :n j instead of T (R j1,...,r jm j ) jm j :m j :n j. Aso, et t j1:m j :n j < t j:m j :n j <...<t jm j :m j :n j be the corresponding observed vaues. The ikeihood function is given by [ ] L(θ;t)= = j=1 [ A j m j i=1 m j A j j=1 i=1 ( f j (t ji:m j :n j ))[F(t ji:m j :n j )] R ji [ ρ j 1 (a+bt ji:m j :n j ) exp { ρ j 1 ( at ji:m j :n j + b t ji:m j :n j )}] R ji +1 ] (.4) where θ =(a,b,ρ), t =(t 1,t ), t j =(t j1,,t jm j ), and A j = n j (n j 1 R j1 )(n j R j1 R j )(n j m j + 1 R j1 R j... R j(m j 1)). Hence, the og-ikeihood function, (θ;t) is (θ;t)=a+m nρ+ ) where η ji (a,b) = (R ji + 1) (at ji:m j :n j + b t ji:m j :n j U(θ)=(U a,u b,u ρ ) T are and where U a = U b = m j j=1i=1 m j j=1i=1 m j j=1i=1 { n(a+bt ji:m j :n j ) ρ j 1 η ji (a,b) } (.5) and A =na 1 +na. The components of the score vector { } 1 ρ j 1 (R ji + 1)t ji:m a+bt j :n j, (.6) ji:m j :n j { } t ji:m j :n j 1 a+bt ji:m j :n j ρ j 1 (R ji + 1)t ji:m j :n j, (.7) U ρ = m ρ S(a,b), (.8) S(a,b)= m i=1 η i (a,b). Making use of U(θ)=0, we obtain three noninear equations. The system of these noninear equations cannot be soved anayticay. So, we can appy numerica soution via iterative techniques such as Newton Raphson method, to get the MLEs, ˆθ =(â, ˆb, ˆρ ) T of θ.

4 40 A. A. Ahmad, M. A. Fawzy: Estimation in constant-partiay... On the other hand, setting U ρ = 0 in (.8), we can obtain ˆρ = m S(â, ˆb). (.9) substituting from (.9) about ˆρ in U a = 0 and U b = 0 in (.6) and (.7), we can sove them numericay to obtain â, ˆb and then substituting in (.9), we get ˆρ..3 Observed Fisher information matrix The observed Fisher information matrix (FI) for the MLEs of the parameters a, b and ρ is the 3 3 observed information matrix I(θ) = / θ θ T, which are given in (.11). The eements of the mutivariate norma N 3 (0,I( ˆθ) 1 ) distribution can be used to construct asymptotic confidence intervas for the parameters. The observed FI matrix is I( ˆθ)= U aa U ab U aρ U ba U bb U bρ θ = ˆθ (.10) U ρa U ρb U ρρ where the eements of the matrix are given, from (.6), (.7) and (.8), by U aa = U ab = m 1 j=1i=1 m 1 j=1i=1 1 (a+bt ji:m j :n j ), t ji:m j :n j (a+bt ji:m j :n j ), m U aρ = (R i + 1)t i:m :n, (.11) U bb = U bρ = 1 i=1 m 1 j=1i=1 m i=1 U ρρ = m ρ. t ji:m j :n j (a+bt ji:m j :n j ), (R i + 1)t i:m :n, 3 Simuation studies In order to obtain the MLEs of(a,b,ρ) and study the properties of their estimates through the mean squared errors(mse) and reative absoute biases(rab), a simuation study is performed, for j = 1,, according to the foowing steps: 1.Based on the vaues of n j and m j (1 m j n j ), generate two independent random sampes of sizes m 1 and m from Uniform(0,1) distribution,(u j1,u j,...,u jm j )..Determine the vaues of the censored scheme R ji, i=1,,...,m j. 3.Set E ji = 1/(i+ m j d=m j i+1 R jd). 4.Set V ji = U E ji ji. 5.Construct the two progressive Type-II censored sampes, (U j1,u j,...,u jm j ), from the Uniform (0,1) distribution, where Uji = 1 m j d=m j i+1 V jd. 6.Using Step 5, generate two random sampes(t j1,t j,...,t jm j ), from CDFs and F 1 (t) and F (t) given in(1.1) and(.3) respectivey, as foows t ji = [( aρ j 1 + a ρ ( j 1) b og(1 Ui ))/bρ ] and hence obtain the two ordered sampes (t j1:m j :n j,t j:m j :n j,...,t jm j :m j :n j ) which represent two progressive Type-II censored sampes from LE(a,b) distribution under constant PALT.

5 J. Stat. App. Pro. Lett. 4, No., (017) / From the ordered observations, obtained in Step 6, compute the MLE(â) of the parameter a by soving noninear U a = 0 and hence compute the MLE(ˆb) and MLE( ˆρ) from U b = 0 and U ρ = 0 respectivey. 8.Repeat Steps 1-7, r times representing r MLEs of(a, b, ρ) based on r different sampes. 9.If ˆΨ kd is a MLE of Ψ k, k = 1,,3 (where Ψ 1 a, Ψ b, Ψ 3 ρ) based on sampe d, d = 1,...,r, then the average of MLE, MSE and RAB, of ˆΨ k over the r sampes are given, respectivey, by ˆΨ k = 1 r r d=1 ˆΨ kd, MSE( ˆΨ k )= 1 r r d=1 ( ˆΨ kd Ψ k ), RAB( ˆΨ k )= r d=1 ˆΨ k Ψ k Ψ k. 10.From Step 9, compute ˆΨ k, MSE( ˆΨ k ) and RAB( ˆΨ k ). 3.1 Approximate confidence intervas For the genera asymptotic theory of MLE ˆθ for θ, the samping distribution of(( ˆθ θ)/ var( ˆθ)), can be approximated by a standard norma distribution, where var( ˆθ k ) is cacuated from I( ˆθ) 1. The asymptotic 100(1 α)% confidence intervasof ˆθ k are( ˆθ k ± z α SE( ˆθ k )), where θ 1 = a, θ = b, θ 3 = ρ and z α is the quantie(1 α ) of the standard norma distribution and SE(.) is the square root of the diagona eement of I( ˆθ) 1 corresponding to each parameter. A two-sided 100(1 α)% norma approximation CI for the parameter θ k can be constructed as( ˆθ k ± z α var( ˆθ k ), k=1,,3). 3. Bootstrap CIs For comparison purposes, two confidence intervas based on the parametric bootstrap methods are proposed. It was observed by Kundu et a. that the nonparametric bootstrap method does not work we. So, two parametric bootstrap methods are are: Studentized-t (Stud-t) bootstrap CI (boot t) suggested by Ha [4]. Percentie (Percen) bootstrap CI (boot p) suggested by Efron ([10]). Ha [4] showed that the Stud-t CI is better than the Percen bootstrap CI from an asymptotic point of view, athough the finite sampe properties are not yet known. In order to obtain the Stud-t and Percen CIs bootstrap methods of(a, b, ρ), a simuation study is performed, for j = 1,, according to the foowing steps: 1.Foow the same Steps 1-5 of the agorithm described above to generate two progressive Type-II right censored sampes from the Uniform(0,1) distribution of the form(v j1,v j,...,v jm j ), j = 1,, as shown in Step 5..Use (a) and Step 7 in the above agorithm to generate two random sampes(t j1,...,t jm j ), j = 1, from CDFs F1(t) and F(t) given in (1.1) and (.3), respectivey, as foows t ji = [( â ˆρ j 1 + â ˆρ ( j 1) bog(1 Vji ))/ ˆρ j 1ˆb], where i = 1,,...,m j,where â, ˆb and ˆρ are as obtained in Step 7 in the above agorithm, and hence obtain the two ordered sampes(t j1:m j :n,...,t j jm ), which represent two j :m j :n j progressive Type-II right-censored bootstrap sampes from LE(a, b) distributionunder constant PALT. The vaues of n j and m j, have been taken to be equa n j and m j, respectivey. 3.From the ordered observations obtained in (b), we can get the bootstrap estimate â of the parameter a, by soving U a = 0 and hence the bootstrap estimates ˆb and ˆρ can be obtained from U b = 0 and U ρ = 0. 4.Repeat Steps(a) (c), r times representing r bootstrap MLEs of(a,b,ρ). 5.Arrange a the vaues of â s, ˆb s and ˆρ s in an ascending order to obtainthe bootstrap sampes ( ˆψ, ˆψ [],..., ˆψ [r ] ), = 1,, 3, (where ψ1 a,ψ b,ψ3 ρ ) Stud-t bootstrap CIs The Stud-t bootstrap CIs can be constructed as foows:

6 4 A. A. Ahmad, M. A. Fawzy: Estimation in constant-partiay... 1.Find the order statistics δ,...,δ [r ], where δ [d] = ˆψ [d] ˆψ, d = 1,,...,r. Var( ˆψ [d] ).Consider a possibe 100(1 α)% CIs of the form(δ [h] the interva for which the width is minimum, say(δl,δ U ). 3.A( two-sided 100(1 α)% Stud-t bootstrap CI for ψ is either ) ˆψ δu var( ˆψ ), ˆψ δl var( ˆψ ). or ( ˆψ δ [(1 α/)r ] var( ˆψ ), ˆψ δ ) [αr /] var( ˆψ ).,δ [(1 α)r +h] where var( ˆψ ) can be estimated by the asymptotic variance from N 3 (0,I( ˆθ) 1 ). ), where h=1,...,αr, = 1,,3 and choose 3.. Percen bootstrap CIs The Percen bootstrap CIs can be constructed as foows: 1.Based on the order statistics ( ˆψ < ˆψ [] <... < ˆψ [r ] ( ) ), consider a possibe 100(1 α)% of the form, ˆψ [(1 α)r +h], h=1,...,αr, and choose the interva with minimum width, say( ˆψ L, ˆψ U ). ˆψ [h].a ( two-sided 100(1 α)% ) Percen bootstrap CI for ψ is either( ˆψ L, ˆψ U ) or ˆψ [αr /], ˆψ [(1 α/)r ]. 3.3 Simuation procedure A Monte Caro simuation study is carried out in order to cacuate the MLEs, MSEs, RABs and 95% approximate (Approx) CIs of the mode parameters, based on r = 1000 Monte Caro simuations. Based on r = 1000 bootstrap repications, ower and upper bounds of 95% Stud-t and Percen bootstrap CIs are cacuated and compared with Approx CIs through their engths. The ower and upper vaues of the bootstrap CIs are taken to represent the average ower and upper vaues over 1000 bootstrap CIs that correspond to 1000 Monte Caro simuations. 1. Tabe 1 ists the different censoring schemes, used in the simuation study, for different choices of sampe sizes, n 1 = n = n, and observed faiure times m 1 = m = m which represent 40%, 70% and 100% of the sampe size. The MLEs, MSEs and RABs for the parameters with(a, b, ρ)=(0.5, 1.5, 1.) and(a,b,ρ)=(0., 1, ) under progressive censoring schemes isted in Tabe 1 are shown in Tabe 3. The comparisons of engths and the coverage percentages(cp) of the 95% CIs are shown in Tabe 4.. Tabe ists the censoring schemes used in the simuation study for another exampe based on different vaues of n 1,n,m 1 and m. Based on censoring schemes isted in Tabe and simiary as in Tabes 3 and 4, Tabes 5 and 6 show the resuts of the second exampe (using the same generating vaues(a, b, ρ) =(0.5, 1.5, 1.) and(a, b, ρ) =(0., 1, )). The number of items to be tested in group, shoud be seected to be greater than those in group 1, since they are subjected to an acceerated condition (as in the data of Tabe ). Now, we sha consider the foowing two tabes, Tabe 1 and Tabe :

7 J. Stat. App. Pro. Lett. 4, No., (017) / 43 Tabe 1 : Progressive censoring schemes used in the Monte Caro simuation study at n 1 = n = n and m 1 = m = m. n m Sc (R 1,...,R m ) Sc (R 1,...,R m ) Sc (R 1,...,R m ) 0 8 R 1 =1 R i =0, i 1 [] R 5 =1 R i =0, i 5 R 8 =1 R i =0, i R 1 =6 R i =0, i 1 [] R 8 =6 R i =0, i 8 R 14 =6 R i =0, i R 1 =4 R i =0, i 1 [] R 9 =4 R i =0, i 9 R 16 =4 R i =0, i R 1 =1 R i =0, i 1 [] R 15 =1 R i =0, i 15 R 8 =1 R i =0, i R 1 =4 R i =0, i 1 [] R 15 =4 R i =0, i 15 R 8 =4 R i =0, i R 1 =1 R i =0, i 1 [] R 5 =1 R i =0, i 5 R 49 =1 R i =0, i 49 Tabe : Progressive censoring schemes used in the Monte Caro simuation study at n 1 n = n and m 1 m = m. n 1 n m 1 m scheme (R 1,...,R m1 ) (R 1,...,R m ) scheme (R 1,...,R m1 ) (R 1,...,R m ) scheme (R 1,...,R m1 ) (R 1,...,R m ) { R1 =6 R i =0, i 1 { R1 =1 R j =0, j 1 [] { R3 =6 R i =0, i 3 { R5 =1 R j =0, j 5 { R4 =6 R i =0, i 4 { R8 =1 R j =0, j { R1 =3 R i =0, i 1 { R1 =6 R j =0, j 1 [] { R4 =3 R i =0, i 4 { R8 =6 R j =0, j 8 { R7 =3 R i =0, i 7 { R14 =6 R j =0, j { R1 =18 R i =0, i 1 { R1 =4 R j =0, j 1 [] { R7 =18 R i =0, i 7 { R9 =4 R j =0, j 9 { R1 =18 R i =0, i 1 { R16 =4 R j =0, j { R1 =9 R i =0, i 1 { R1 =1 R j =0, j 1 [] { R11 =9 R i =0, i 11 { R15 =1 R j =0, j 15 { R1 =9 R i =0, i 1 { R8 =1 R j =0, j { R1 =36 R i =0, i 1 { R1 =4 R j =0, j 1 [] { R13 =36 R i =0, i 13 { R15 =4 R j =0, j 15 { R4 =36 R i =0, i 4 { R8 =4 R j =0, j { R1 =18 R i =0, i 1 { R1 =1 R j =0, j 1 [] { R =18 R i =0, i { R5 =1 R j =0, j 5 { R4 =18 R i =0, i 4 { R49 =1 R j =0, j 49 4 Concuding Remarks The subject of progressive censoring has received considerabe attention in the past few years, due in part to the avaiabiity of high speed computing resources, which mak it both a feasibe topic for simuation studies for researchers and a feasibe method of gathering ifetime data for practitioners. It has been iustrated by Viveros and Baakrishnan [6] that the inference is feasibe, and practica when the sampe data are gathered according to a Type-II progressivey censored experimenta scheme. In this paper a constant stress PALT mode when the observed faiure times come form LE(a, b) distribution under progressivey Type-II censored data have been considered. The MLEs of the considered parameters are obtained and the performance of this estimates are studied through their MSEs and RABs. Aso, We have constructed approximate and bootstrap CIs for the parameters. A simuation study, based on two different exampes (according to

8 44 A. A. Ahmad, M. A. Fawzy: Estimation in constant-partiay... choices of nj and m j, j = 1,), has been made to examine the performance of the MLEs and compare three different methods for the CIs. Three cases of censoring schemes are appied at fixed sampes and faiure time sizes. The first and the third assume that censoring occurs ony at the first and the ast observed faiure, whie the second scheme assumes censoring occurs ony at the midde observed faiure. It can be notice, from the simuation study, that: 1.The MSE and RAB of the MLEs are computed over different combinations of the censored schemes..for fixed vaues of the sampe size, by increasing the faiure times the MSEs and RABs of the considered parameters decrease. 3.For fixed vaues of the sampe and faiure time sizes, the schemes in which the censorin occurs after the first observed faiure gives more accurate resuts through the MSE and RABs than the other two schemes, and this coincides with Theorem [.] by Burkschat et a. [18]. 4.The bootstrap CIs give more accurate resuts than the approximate CIs since the engths of the former are ess than the engths of atter, for different sampe sizes, observed faiures and schemes. 5.For sma sampe sizes, the Stud-t bootstrap CIs are better than the Percen bootstrap CIs in the sense of having smaer widths. However, the differences between the engths of CIs using both methods decrease when sampe sizes increase. Tabe 3 : MLEs, MSEs and(rabs) for the parameters with(a,b,ρ)=(0.5, 1.5, 1.) and(a,b,ρ)=(0., 1, ). (0.5, 1.5, 1.) (0., 1, ) (n, m) SC a b ρ a b ρ (0,8) (0.016) (0.39) 0.91 (0.8) 0.18 (0.038) (0.338) (0.14) [] (0.050) (0.534) (0.1) (0.097) (0.765) (0.314) (0.0) (0.803) 1.18 (0.39) 0.34 (0.178) (0.971) 1.18 (0.543) (0,14) (0.068) (0.16) (0.14) (0.05) (0.0981) (0.04) [] (0.054) (0.66) (0.105) 0.8 (0.041) (0.007) ( 0.050) (0.087) (0.37) (0.14) (0.074) 0.41 (0.131) (0.07) (0,0) (0.065) (0.199) (0.074) (0.156) 0.87 (0.173) (0.061) (40,16) (0.016) (0.05) 0.53 (0.090) (0.037) 0.69 (0.019) 0.86 (0.04) [] (0.034) 1.07 (0.34) (0.111) 0.07 (0.003) (0.7) (0.311) (0.145) 0.86 (0.55) (0.133) (0.365) (0.36) (0.371) (40,8) (0.031) (0.094) 0.33 (0.030) 0.30 (0.079) (0.080) 0.88 (0.14) [] (0.046) (0.09) 0.35 (0.064) (0.088) 0.35 (0.134) (0.351) (0.038) (0.10) (0.044) 0.16 (0.095) (0.157) 0.54 (0.158) (40,40) (0.04) (0.05) 0.96 (0.038) (0.186) (0.081) (0.131) (70,8) (0.013) 0.56 (0.094) (0.040) 0.0 (0.0894) 0.38 (0.009) 0.639, (0.95) [] (0.0) (0.15) (0.04) (0.073) (0.138) 0.58 (0.071) (0.054) 0.85 (0.164) 0.36 (0.051) (0.106) (0.007) 0.5 (0.05) (0.3681) (70,49) (0.09) (0.057) 0.57 (0.03) 0.11 (0.005) 0.05 (0.079) (0.014) [] (0.044) (0.064) 0.76 (0.038) (0.087) 0. (0.081) (0.045) (0.08) (0.05) 0.6 (0.037) (0.065) 0.04 (0.085) (0.050) (70,70) (0.0) 0.33 (0.059) (0.004) (0.08) (0.034) (0.080)

9 J. Stat. App. Pro. Lett. 4, No., (017) / 45 Tabe 4 : Comparisons of engths and(cp) of 95% CIs for the parameters with (a,b,ρ)=(0.5, 1.5, 1.) and(a,b,ρ)=(0., 1, ). (0.5, 1.5, 1.) (0., 1, ) MLE boot t boot p MLE boot t boot p a a a a a a b b b b b b (n, m) SC ρ ρ ρ ρ ρ ρ (0,8) 1.36 (0.943) (0.964) 1.0 (0.967) 0.84 (0.933) (0.96) (0.937).678 (0.947).34(0.986) (0.948).061 (0.943) 1.97 (0.953).45 (0.977).989 (0.957) (0.97) (0.938) (0.944) (0.96) 4.76 (0.965) [] (0.954) 1.06 (0.960) 1.13 (0.965) (0.958) (0.943) (0.964) 3.77 (0.957).870 (0.979) 4.0 (0.934).708 (0.964) (0.976) (0.94).984 (0.894).975 (0.906) 3.8 (0.919) (0.95) (0.97) (0.918) 1.6 (0.938) 1.11 (0.949) (0.938) (0.957) (0.984) (0.976) (0.938) (0.956) (0.977) (0.975).86 (0.984) (0.946) (0.99) 3.8 (0.918) (0.96) (0.95) (0.914) (0.903) (0,14) (0.936) (0.935) (0.964) (0.96) (0.908) (0.938) (0.964) (0.944).11 (0.955) (0.976) 1.8 (0.977) (0.967).043 (0.894).056 (0.904).41 (0.934) (0.904) (0.946) (0.96) [] 1.00 (0.939) 1.05 (0.963) (0.95) (0.94) (0.981) (0.976).004 (0.99) (0.97).389 (0.943) (0.905) (0.96) (0.966).004 (0.95).06 (0.938).419 (0.957) 3.54 (0.958) 3.0 (0.946) (0.986) (0.94) (0.959) (0.933) (0.944) (0.988) (0.967).398 (0.915).008 (0.99).669 (0.949) 1.68 (0.954) 1.53 (0.964) (0.986).061 (0.939).070 (0.954).51 (0.904) 3.68 (0.938) 3.15 (0.965) 3.65 (0.978) (0,0) (0.97) (0.943) (0.966) (0.934) (0.945) (0.976) (0.94) (0.909) (0.937) 1.15 (0.95) 1.34 (0.957) 1.64 (0.969) 1.68 (0.939) (0.979) (0.954).707 (0.970).11 (0.975) (0.948) (40,16) 0.97 (0.943) (0.967) (0.976) (0.947) (0.977) (0.954) (0.961) (0.958).00 (0.946) (0.968) 1.40 (0.971) (0.957) (0.94) (0.975).145 (0.944) (0.948) (0.959) (0.974) [] 1.36 (0.911) (0.907) 1.0 (0.909) (0.917) (0.919) (0.93).678 (0.95).34 (0.939) (0.956) (0.946) 1.56 (0.936) (0.949).985 (0.908) (0.919) (0.99) 3.03 (0.904) (0.897) (0.96) (0.963) (0.974) (0.961) 0.57 (0.946) (0.963) (0.953) (0.90).519 (0.938) (0.944).116 (0.934) (0.97).645 (0.94) (0.963) (0.991).41 (0.984) (0.969).74 (0.978) (0.988) (40,8) (0.944) (0.946) (0.953) (0.958) 0.75 (0.96) (0.947).010 (0.971) 1.94 (0.976).165 (0.945) (0.97) (0.965).16 (0.975) (0.918) 1.34 (0.94) (0.91).745 (0.919).685 (0.904).805 (0.895) [] (0.939) (0.96) (0.953) 0.44 (0.953) (0.946) 0.4 (0.938).093 (0.974).031 (0.983).9 (0.990) (0.979) 1.71 (0.976) (0.986) 1.35 (0.99) (0.97) (0.98).771 (0.953).863 (0.964).930 (0.94) (0.944) (0.977) (0.935) (0.96) 0.43 (0.94) 0.53 (0.98).418 (0.949).51 (0.955).593 (0.937) 1.31 (0.939) 1.18 (0.947) (0.938) (0.934) 1.36 (0.946) (0.951) (0.959) (0.9) (0.948)

10 46 A. A. Ahmad, M. A. Fawzy: Estimation in constant-partiay... Tabe4: (continued) (0.5, 1.5, 1.) (0., 1, ) MLE boot t boot p MLE boot t boot p a a a a a a b b b b b b (n, m) SC ρ ρ ρ ρ ρ ρ (40,40) (0.93) (0.951) 0.74 (0.956) (0.946) (0.98) (0.9) (0.91) (0.93) ( 0.938) 1.11 (0.949) (0.939) (0.945) (0.947) 1.136(0.935) (0.934) (0.99) (0.941) (0.934) (70,8) (0.90) (0.896) 0.8 (0.919) (0.90) (0.889) 0.4 (0.910) (0.950) (0.98).169 (0.971) 0.95 (0.960) 0.97 (0.966) (0.977) 1.30 (0.965) (0.963) 1.49 (0.947).79 (0.958).777 (0.96).83 (0.968) [] (0.977) 0.73 (0.958) (0.986) (0.971) (0.963) (0.973).40 (0.936).07 (0.947).58 (0.919) (0.917) (0.903) (0.933) 1.34 (0.957) (0.957) (0.963).35 (0.956).171 (0.954).6 (0.966) (0.966) (0.936) (0.938) (0.949) (0.943) (0.955) (0.933).838 (0.937) (0.943) 1.35 (0.941) (0.949) (0.951) (0.905) (0.918) 1.50 (0.918) 3.141(0.901) (0.885) (0.89) (70,49) (0.955) (0.959) 0.66 (0.948) (0.960) (0.963) (0.947) 1.48 (0.946) (0.956) (0.949) 0.85 (0.955) (0.977) (0.966) (0.99) (0.937) (0.958) (0.955) (0.968) (0.959) [] 0.61 (0.977) (0.97) (0.983) (0.974) 0.34 (0.973) (0.985) 1.55 (0.936) (0.939) (0.99) (0.940) (0.945) (0.93) (0.949) 1.05 (0.967) (0.959) (0.968) 1.44 (0.97) (0.980) (0.934) (0.946) (0.949) 0.44 (0.966) (0.957) (0.939) 1.78 (0.91) (0.916) (0.910) (0.98) 0.961(0.9) (0.90) (0.9) 1.06 (0.934) (0.90) 1.56 (0.918) (0.93) 1.50 (0.919) (70,70) (0.946) (0.959) (0.947) 0.331(0.953) 0.39 (0.937) (0.944) 1.39 (0.934) 1.33 (0.938) 1.93 (0.945) (0.947) (0.951) (0.948) (0.94) (0.91) (0.9) (0.919) (0.896) (0.9)

11 J. Stat. App. Pro. Lett. 4, No., (017) / 47 Tabe 5 : MLEs, MSEs and(rabs) for the parameters with(a, b, ρ) =(0.5, 1.5, 1.) and(a,b,ρ)=(0., 1, ). (0.5, 1.5, 1.) (0., 1, ) (n1,n) (m1,m) SC a b ρ a b ρ (10,0) (4,8) (0.035) (0.687) (0.43) (0.097) (0.168) 1.75 (0.08) [] (0.00) 1.48 (0.796) 1.3 (0.333) 0.69 (0.137) 0.75 (0.038) (0.404) (0.16) (1.64) (1.64) (0.091) (0.057) (0.1067) (7,14) (0.013) (0.38) (0.14) (0.14) (0.36) (0.096) [] (0.017) 1.94 (0.73) 0.74 (0.158) (0.040) (0.049) (0.176) (0.093) 1.00 (0.51) 0.74 (0.153) (0.170) 0.39 (0.001) (0.03) (10,0) (0.06) (0.48) 0.53 (0.077) (0.01) (0.311) (0.08) (30,40) (1,16) (0.007) (0.184) (0.083) 0.3 (0.06) (0.057) (0.8) [] (0.034) 1.07 (0.34) (0.11) 0.78 (0.03) 0.44 (0.063) (0.148) ,, (0.083) (0.387) (0.105) (0.090) 0.61 (0.087) (0.135) (1,8) (0.019) (0.105) (0.045) (0.04) 0.5 (0.056) (0.163) [] (0.033) (0.11) (0.05) (0.055) (0.06) 0.71 (0.156) (0.044) 0.70 (0.16) 0.45 (0.076) 0.11 (0.06) 0.83 (0.088) (0.15) (30,40) (0.05) (0.080) (0.03) (0.059) 0.38 (0.0733) (0.03) (60,70) (4,8) (0.003) (0.075) (0.055) 0.13 (0.097) (0.073) (0.065) [] (0.046) (0.18) (0.081) (0.06) (0.157) 0.87 (0.005) (0.091) (0.05) (0.061) 0.11 (0.085) 0.43 (0.106) (0.030) (4,49) (0.018) (0.053) 0.55 (0.01) (0.08) (0.051) 0.73 (0.086) [] (0.01) 0.49 (0.059) 0.79 (0.018) (0.033) 0. (0.135) (0.004) (0.031) (0.078) 0.78 (0.08) (0.018) 0.63 (0.015) (0.047) (60,70) (0.03) (0.033) 0.38 (0.08) (0.038) (0.019) (0.170)

12 48 A. A. Ahmad, M. A. Fawzy: Estimation in constant-partiay... Tabe 6 : Comparisons of engths and(cp) of 95% CIs for the parameters with (a,b,ρ)=(0.5, 1.5, 1.) and(a,b,ρ)=(0., 1, ). (0.5, 1.5, 1.) (0., 1, ) MLE boot t boot p MLE boot t boot p a a a a a a b b b b b b (n1,n) (m1,m) SC ρ ρ ρ ρ ρ ρ (10,0) (4,8) (0.9573) (0.947) (0.963) 1.46 (0.987) 1.96 (0.973) (0.974) 4.7 (0.933) (0.941) (0.99) (0.968) 4.18 (0.971) (0.990) (0.93) 4.33 (0.930) 4.4 (0.9) 4.79 (0.93) (0.941) (0.9) [].175 (0.973).571 (0.95) (0.968) (0.91) (0.919) (0.93) (0.947) (0.964) (0.944).975 (0.961).5 (0.944) (0.943) 4.35 (0.966) 4.80 (0.975) (0.964) 7.01 (0.954) (0.961) 7.13 (0.97) (0.97) (0.919) (0.93) (0.941) (0.939) (0.96) (0.99) (0.948) (0.934) (0.91) (0.941) 4.34 (0.97) (0.984) 4.19 (0.979) (0.988) (0.919) (0.90) (0.89) (7,14) (0.959) (0.966) 1.8 (0.944) (0.96) (0.95) (0.944).634 (0.963).334 (0.954) (0.976).339 (0.975).14 (0.97).661 (0.961).497 (0.98).704 (0.978).95 (0.979) (0.916) (0.931) 4.13 (0.9) [] (0.917) (0.933) (0.91) (0.971) (0.977) (0.983) 4.36 (0.945) 3.86 (0.971) (0.956) (0.99) (0.891).95 (0.895).581 (0.973).73 (0.983).966 (0.989) (0.931) (0.90) (0.914) (0.94) 1.30 (0.98) (0.96) (0.94) (0.951) (0.99) 3.45 (0.95).69 (0.958) (0.94).00 (0.94) (0.964).404 (0.976).587 (0.91).779 (0.90) (0.893) (0.963) 3.73 (0.951) (0.944) (10,0) (0.948) 1.18 (0.96) (0.953) (0.961) (0.97) (0.966).019 (0.978) (0.986).479 (0.984).08 (0.96) (0.949).97 (0.959) (0.963).101 (0.949).5 (0.943) (0.904).938 (0.931) 3.15 (0.9) (30,40) (1,16) (0.93) 1. (0.918) (0.98) (0.978) (0.981) (0.983) (0.958).904 (0.963) (0.956) (0.943) (0.934) (0.96) (0.971).0 (0.983).38 (0.981) (0.881) 3.80 (0.89) 4.53 (0.911) [] (0.968) 1.05 (0.961) (0.953) (0.945) (0.958) 0.64 (0.95) 3.56 (0.96) 3.30 (0.94) 4.17 (0.911) (0.967) 1.51 (0.98) (0.973).033 (0.954).068 (0.97).385 (0.977) (0.908) (0.91) (0.930) (0.918) (0.99) (0.961) (0.961) (0.97) (0.944) 5.1 (0.946) 4.07 (0.957) (0.94) (0.98) 1.87 (0.944).171 (0.96).046 (0.961).056 (0.97).445 (0.934) (0.98) (0.905) (0.913) (1,8) 0.11 (0.958) (0.944) (0.93) (0.964) (0.943) 0.65 (0.95) 0.06 (0.961).155 (0.971).494 (0.988) 1.37 (0.991) 1.4 (0.987) 1.40 (0.984) (0.94) (0.959) (0.95).087 (0.946).0 (0.971).151 (0.96) [] (0.918) (0.99) (0.9) (0.94) (0.959) (0.957).369 (0.940).81 (0.951).648 (0.97) (0.938) (0.98) 1.8 (0.9) (0.987) (0.991) (0.989).7 (0.941).714 (0.970).891 (0.885) (0.918) (0.89) (0.90) (0.96) (0.977) (0.98).34 (0.961).01 (0.949).56 (0.938) (0.963) (0.97) 1.6 (0.939) (0.90) 1.56 (0.938) (0.97).906 (0.981) (0.99) 3.07 (0.966)

13 J. Stat. App. Pro. Lett. 4, No., (017) / 49 Tabe 6 : (continued) (0.5, 1.5, 1.) (0., 1, ) MLE boot t boot p MLE boot t boot p a a a a a a b b b b b b (n1,n) (m1,m) SC ρ ρ ρ ρ ρ ρ (30,40) (0.97) (0.961) (0.98) (0.96) (0.958) (0.953) (0.971) (0.96).009 (0.955) (0.937) (0.949) (0.945) 1.18 (0.904) 1.14 (0.901) 1.50 (0.889).005 (0.961).841 (0.957) (0.95) (60,70) (4,8) 0.88 (0.958) 0.93 (0.961) (0.954) (0.97) (0.951) 0.31 (0.94).01 (0.971) (0.95).183 (0.941) (0.983) 0.96 (0.989) (0.994) (0.959) 1.46 (0.945) (0.9).375 (0.90).56 (0.91).781 (0.90) [] (0.93) (0.931) (0.91) 0.41 (0.966) (0.957) (0.941).75 (0.946).38 (0.99).38 (0.904) 1.88 (0.97) (0.973) (0.957) 1.43 (0.95) (0.947) (0.911).69 (0.915).44 (0.90).454 (0.931) (0.949) (0.947) (0.951) (0.95) (0.906) (0.91) 3.63 (0.938).90 (0.93) 3.41 (0.97) 1.53 (0.96) (0.979) 1.40 (0.978) (0.971) (0.99) 1.64 (0.98).391 (0.958).371 (0.984).661 (0.994) (4,49) (0.958) 0.76 (0.99) (0.993) 0.45 (0.958) (0.951) (0.96) (0.941) (0.958) (0.955) (0.961) (0.959) (0.931) (0.910) (0.904) (0.90) (0.931) 1.46 (0.901) 1.64 (0.944) [] (0.96) (0.943) (0.951) (0.991) (0.986) 0.40 (0.981) 1.60 (0.949) (0.937) (0.93) (0.965) (0.973) (0.947) (0.957) (0.94) (0.911) (0.894) (0.910) 1.8 (0.887) (0.946) (0.954) (0.933) (0.983) (0.971) (0.979) (0.90) (0.895) (0.914) (0.96) 0.84 (0.958) (0.948) 1.07 (0.919) (0.97) (0.944) (0.94) (0.956) (0.951) (60,70) (0.97) (0.966) 0.59 (0.94) 0.86 (0.96) 0.63 (0.957) (0.941) 1.61 (0.941) 1.54 (0.959) (0.953) (0.974) (0.969) (0.963) (0.90) (0.96) (0.917) 1.65(0.948) (0.931) (0.9) References A.A. Ismai. Estimating the parameters of Weibu distribution and the acceeration factor from hybrid partiay acceerated ife test. Appied Mathematica Modeing 36, (01). [] A.A. Ismai. Inference in the generaized exponentia distribution under partiay acceerated tests with progressive Type-II censoring. Theoretica and Appied Fracture Mechanics. 59, (01). A.A. Ismai. The test design and parameter estimation of Pareto ifetime distribution under partiay acceerated ife tests, Ph.D. thesis, Department of Statistics, Facuty of Economics and Poitica Science, Cairo University, Egypt (004). [4] A.A. Ismai and A.M. Sarhan. Optima Design of Step-Stress Life Test with Progressivey Type-II Censored Exponentia Data. Internationa Mathematica Forum. 40, (009). [5] A.A. Soiman. Estimation of parameters of ife from progressivey censored data using Burr-XII mode. IEEE Trans. Reiab. 54, 34-4 (005). [6] A. Chids and N. Baakrishnanm. Conditiona inference procedures for the Lapace distribution when the observed sampes are progressivey censored, Metrika, 5, (000). [7] A.H. Abde-Hamid. Constant-partiay acceerated ife tests for Burr Type-XII distribution with progressive Type-II censoring. Computationa Statistics and Data Anaysis 53, (009). [8] A.M. Abd-Efattah, A.S. Hassan and S.G. Nassr. Estimation in step-stress partiay acceerated ife tests for the Burr Type XII distribution using Type I censoring. Statistica Methodoogy 5, (008). [9] A. Pandey, A. Singh and W.J. Zimmer. Bayes estimation of the inear hazard rate mode, IEEE Trans. Reiab., 4, (1993). [10] B. Efron. The Jackknife, the Bootstrap and Other Re-samping Pans. In:CBMS/NSF; Regiona Conference Series in Appied Mathematics, vo. 38. SIAM, Phiadephia, PA (198). [11] C. Xiong and M. Ji. Anaysis of Grouped and Censored Data from Step-stress Life Testing. IEEE Trans. Reiab., 53,1,-8 (004). D. Kundu, N. Kannan and N. Baakrishnan. Anaysis of progressivey censored competing risks data. In: Baakrishnan, N., Rao, C.R. (Eds.), Handbook of Statistics, vo. 3. Esevier, New York (004). [13] D.S. Bai and S.W. Chung. Optima design of partiay acceerated ife tests for the exponentia distribution under Type-I censoring. IEEE Trans. Reiab. 41, (199).

14 50 A. A. Ahmad, M. A. Fawzy: Estimation in constant-partiay... [14] E.K. AL-Hussaini and A.H. Abde-Hamid. Bayesian estimation of the parameters, reiabiity and hazard rate functions of mixtures under acceerated ife tests. Commun. Stat. - Simu. Comput. 33 (4), (004). [15] E.K. AL-Hussaini and A.H. Abde-Hamid. Acceerated ife tests under finite mixture modes. J. Stat. Comput. Simu. 76 (8), (006). [16] L.J. Bain. Statistia Anaysis of Reiabiity and Life Testing Modes, Mace Dekker (1978). [17] M.A.M. Ai Mousa and Z.F. Jaheen. Statistica inference for the Burr mode based on progressivey censored data. Comput. Math. App. 43, (00a). [18] M. Burkschat, E. Cramer and U. Kamps. On optima schemes in progressive censoring. Statist. Probab. Lett. 76, (006). [19] M.M. Abde-Ghani. Investigation of some ifetime modes under partiay acceerated ife tests, Ph.D. thesis, Department of Statistics, Facuty of Economics and Poitica Science, Cairo University,Egypt (1998). [0] N. Baakrishnan and R. Aggarwaa. Progressive Censoring: Theory, Methods and Appications. Birkhauser Pubishers, Boston (000). N. Baakrishnan and N. Kannan. Point and interva estimation for parameters of the ogistic distribution based on progressivey Type-II censored sampes, in Handbook of Statistics N. Baakrishnsn and C.R. Rao, Eds. Amsterdam: North-Hoand, 0, (001). [] N. Baakrishnan, N. Kannan, C.T. Lin and H. Ng. Point and interva estimation for gaussian distribution based on progressivey Type-II censored sampes, IEEE Trans. Reiab. 5, (003). P. Carbone, L. Keerthouse and E. Gehan. Pasmacytic Myeoma: Astudy of the reationship of surviva to various cinica manifestations and anomaous protein Type in 11 patients, American Journa of Medicine, 4, (1967). [4] P. Ha. Theoretica comparison of bootstrap con dence intervas. Ann. Statist.16, (1988). [5] P.W. Srivastava, and N. Mitta. Optimum step-stress partiay acceerated ife tests for the truncated ogistic distribution with censoring. Appied Mathematica Modeing. 34, (010). [6] R.Viveros and N. Baakrishnan. Interva estimation of parameters of ife from progressivey censored data. Technometrics 36, (1994). [7] S. Broadbent. Simpe mortaity rates, Journa of Appied Statistics, 7, 86 (1958). [8] W. Neson. Acceerated Testing: Statistica Modes, Test Pans and Data Anaysis. Wiey, NewYork (1990). Abd EL-Baset A. Ahmad is Professor of Mathematica statistics at Assuit University, Egypt. He received his M.Sc. in 1993 from the Department of Mathematics, Assuit University, Egypt, and Ph.D. in 1995 in Mathematica statistics from the Department of Mathematics, Assuit University, Egypt. His current research interests are Bayesian estimation, acceerated ife test, theory, competing risk and prediction. He supervised severa M.Sc. and Ph.D.. He has about 34 scientic research artices, pubished in internationa refereed journas. M. Fawzy is Assistant Professor of Mathematica statistics at suez University, Egypt, and at Taibah University, Saudi Arabia. He received his M.Sc. in 00 from the Department of Mathematics, South Vaey University, Egypt, and Ph.D. in 010 in Mathematica statistics from the Department of Mathematics, Sohag University, Egypt. His current research interests are Bayesian estimation, acceerated ife test, theory, competing risk and prediction. He has about 8 scientific research artices, pubished in internationa refereed journas.