Exact Inference for the Two-Parameter Exponential Distribution Under Type-II Hybrid Censoring

Transcription

1 Exact Inference for the Two-Parameter Exponential Distribution Under Type-II Hybrid Censoring A. Ganguly, S. Mitra, D. Samanta, D. Kundu,2 Abstract Epstein [9] introduced the Type-I hybrid censoring scheme as a mixture of Type-I and Type-II censoring schemes. Childs et al. [5] introduced the Type-II hybrid censoring scheme as an alternative to Type-I hybrid censoring scheme, and provided the exact distribution of the maximum likelihood estimator of the mean of a one parameter exponential distribution based on Type-II hybrid censored samples. The associated confidence interval also has been provided. The main aim of this paper is to consider a two-parameter exponential distribution, and to derive the exact distribution of the maximum likelihood estimators of the unknown parameters based on Type-II hybrid censored samples. The marginal distributions and the exact confidence intervals are also provided. The results can be used to derive the exact distribution of the maximum likelihood estimator of the percentile point, and to construct the associated confidence interval. Different methods are compared using extensive simulations and one data analysis has been performed for illustrative purposes. Key Words and Phrases: Hybrid censoring; two-parameter exponential distribution; joint moment generating function; maximum likelihood estimators; bootstrap confidence interval. Department of Mathematics and Statistics, IIT Kanpur, Kanpur, Pin 20806, India. 2 Corresponding author. kundu@iitk.ac.in Introduction In life testing experiments, often the data are censored. Among the different censoring schemes, Type-I and Type-II censoring schemes are the two most popular censoring schemes. In Type-I censoring scheme, the experimental time is fixed, but the number of failures is random, whereas Part of this work has been supported by grants from DST and CSIR, Government of India.

2 2 in Type-II censoring scheme, the experimental time is random but the number of failures is fixed. A mixture of Type-I and Type-II censoring scheme is known as the hybrid censoring scheme and it can be described as follows: Suppose a total of n units is placed on a life testing experiment, and the lifetimes of the sample units are independent and identically distributed i.i.d. random variables. Let the ordered lifetimes of these items are denoted by T :n,..., T n:n respectively. The test is terminated when r, a pre-chosen number, out of n items are failed, or when a pre-determined time T on test has been reached, i.e., the test is terminated at the time point T = min{t r:n, T. This particular censoring scheme was introduced by Epstein [9], and it is popularly known now as the Type-I hybrid censoring scheme. It has been used quite extensively in reliability acceptance test in MIL-STD-78C [2]. Like conventional Type-I censoring scheme, the main disadvantage of Type-I hybrid censoring scheme is that most of the inferential results are obtained under the condition that the number of observed failures is at least one, and moreover, there may be very few failures at the termination point of the experiment. In that case the efficiency of the estimators may be very low. Due to this reason, Childs et al. [5] introduced the Type-II hybrid censoring scheme, as an alternative to the Type-I hybrid censoring scheme, that would terminate the experiment at the time point T = max{t r:n, T. It has the advantage of guaranteeing that at least r failures are observed at the end of the experiment. Childs et al. [5] considered the exact distribution of the maximum likelihood estimator MLE of the mean of an one-parameter exponential distribution based on a Type-II hybrid censored samples. The exact confidence interval of the mean also has been obtained based on the exact distribution. Recently, Banerjee and Kundu [3] considered the statistical inference of the two-parameter Weibull distribution based on Type-II hybrid censored samples. Very recently, Childs et al. [6] obtained the exact distributions of the MLEs of the scale and location parameters of a two-parameter exponential distribution, when the data are Type-I hybrid censored. The purpose of this paper is to consider a Type-II hybrid censoring scheme when the lifetime distributions of the n experimental units are assumed to be i.i.d. two-parameter exponential distribution. It is assumed that the probability density function of the lifetime of an experimental unit,

3 3 for > 0, is ft; µ, = e t µ if t > µ, and 0 otherwise. First we obtain the MLEs of the unknown parameters of µ R and > 0, and provide the joint moment generating function. Based on the joint moment generating function MGF, we obtain the marginal MGFs, and the marginal distribution functions of the MLEs. From the marginal distribution functions, using the same idea as in Chen and Bhattacharya [4] the confidence interval of is obtained by solving non-linear equations. Since the confidence interval based on the MLEs is quite difficult to implement, we propose to use different bootstrap confidence intervals, whose implementation are quite straight forward. Rest of the paper is organized as follows. In Section 2, we present the maximum likelihood estimators MLEs of the unknown parameters, and the joint and marginal moment generating functions of the MLEs are derived in the same section. Different confidence intervals are proposed in Section 3. Simulation results and the analysis of a data set are presented in Section 4. Finally we conclude the paper in Section 5. Proof of the theorems are provided in the appendix. 2 Maximum Likelihood Estimators and Their Marginal Distributions Let N be the number of units failed before the time T, then the likelihood of the observed data is given by L µ, Data = n! e P N n N! N i= t i:n µ n NT µ if t r:n < T n! e P r n r! r i= t i:n µ n rtr:n µ if t r:n > T. 2 From 2, it is clear that MLE of does not exist when r = and N = 0. So for r =, the conditional MLEs, conditioning on the event {N is given by µ = T :n, { = N T i:n n T :n + n N T. N i=2 3

4 4 For r 2, MLEs of the unknown parameters exist for all values of N and they are given by µ = T :n, = { N T i:n n T :n + n N T N i=2 { r T i:n n T :n + n r + T r:n r i=2 if T r:n < T if T r:n > T. 4 Next we consider the distributions of the MLEs. When r =, MLE of does not exist if N = 0. So for r =, we consider the conditional distribution of conditioning on the event {N. For r 2, MLE of exists for all values of N. In order to find the distribution of µ and, we first find the joint moment generating function MGF of and µ, and then inverting it to get the distribution of the MLEs. Theorem 2.. For r =, the conditional joint MGF of, µ conditioning on the event {N exists for all < ω < and < ω 2 < and is given by E[e ω b +ω 2 bµ N ] = q n n + c ij i=2 + c nj c 0 e µ 0ω +µ ω 2 + ω λ 0 ν 0 d 0 e µ ijω +µ ω 2 e T ω 2 + ω λ 0 ν 0 + ω λ i αi + ω λ ij d ij e µ njω +µ ω 2 ω αn + ω j d nj e µ ω 2 q n e T ω 2 ω αn ν n e T ω 2 ω λ i αi + ω λ ij e T ω 2 ω αn + ω j.

5 5 For r 2, the joint MGF of and µ exists for ω < r and ω 2 < + n r E[e ω b +ω 2 bµ ] ω and is given by = q n e T ω 2 e ω α µ ω 2 q n e T ω 2 + λ ν n ω αn ν n n e µ ij ω +µ ω 2 + c ij i= ω αi λ i + ω λ ij n e T ω 2 d ij ν i= j ω αi λ i + ω λ ij e µ nj ω +µ ω 2 + c nj ω αn + ω when µ < T and for µ T where q = e T µ j E[e ω b +ω 2 bµ ] = n j T µ r µ ij = n j T µ i r α i = i e T ω 2 d nj ν j ω αn + ω e µ ω 2 ω α, λ ν n j for j = 0,, i, i =,, r for j = 0,, i, i = r,, n. for i =,, r for i = r,, n 5 6 ν i = i + r λ i = λ ij = i r j + n j i j + n j c ij = i j n i d ij = i j n i i q n j j + i q n j + for i = 0,, n for i =,, r for i = r,, n for j = 0,, i, i =,, r for j = 0,, i, i = r,, n for j = 0,, i, i =,, n for j = 0,, i, i =,, n 7

6 6 Remark. Note that as T, the joint MGF of and µ at ω, ω 2 reduces to e µ ω 2 ω αn ν n which is the joint MGF of and µ in case of complete sample. That means as T, 2n / is distributed as χ 2 random variable with 2n 2 degrees of freedom, n µ µ/ is a standard exponential random variable, and they are independently distributed. Remark 2. For r 2, if T is smaller than the left end point of the support µ, i.e., T µ, the joint MGF of and µ at ω, ω 2, as given in 5, becomes the joint MGF of and µ in case of ordinary Type-II censoring scheme. In this case 2r / is distributed as χ 2 random variable with 2r 2 degrees of freedom, n µ µ/ is a standard exponential random variable, and they are independently distributed. Remark 3. It is obvious from the expression of the joint MGF of and µ that the marginal distributions belong to the generalized mixtures of the well known distributions. From the MGF of, PDF of can be obtained by using the inversion technique as suggested by Chen and Bhattacharya [4], and it is as follows: Theorem 2.2. For r =, conditional PDF of conditioned on {N for < t < is f b t = g t; α n, + q n [c 0 g t + µ 0 ;, λ 0 d 0 g t;, λ 0 n + {c ij g 2 t µ ij ; α i, λ i, λ ij d ij g 2 t; α i, λ i, λ ij i=2 ] + {c nj g 2 t µ nj ; α n,, j d nj g 2 t; α n,, j. 8 For r 2, the PDF of for < t < is g 3 t f b t = g t; α, λ if µ < T if µ T 9

7 7 where, g t; α, λ = g 2 t; α, λ, λ 2 = α λ α Γ α e λt t α if t 0, 0 otherwise α p k g t; α k, λ + p k g t,, λ 2 for t R, p k = λ 2 λ k. λ +λ 2 λ +λ 2 g 3 t = q n g t; α, λ + q n g t; α n, n + {c ij g 2 t µ ij ; α i, λ i, λ ij d ij g 2 t; α i, λ i, λ ij i= + {c nj g 2 t µ nj ; α n,, j d nj g 2 t; α n,, j. Note that g 3 t depends on µ ij, α i, λ i, and λ ij ; i, j =,..., n, however, for brevity we do not write it explicitly. Since the integration of density function over the whole range is one, we have, from the Theorem 2.2, the following identity. n c ij d ij + c nj d nj = 0. i= Theorem 2.3. When < t <, for r =, conditional PDF of µ conditioning on {N is given by f bµ t = q n [c 0 g t µ;, ν 0 d 0 g t T ;, ν 0 + g t µ;, ν n n q n g t T ;, ν n + {c ij g t µ;, d ij g t T ;, i=2 ] + {c nj g t µ;, d nj g t T ;,,

8 8 and for r 2, PDF of µ is where f bµ t = g 4 t µ if µ < T g t µ;, ν n if µ T, n g 4 t = g t;, ν n + {c ij g t;, d ij g t + µ T ;, i= + {c nj g t;, d nj g t + µ T ;,. Note that g 4 t depends on ν i ; i =,..., n, however, for brevity we do not write it explicitly. From PDFs of, the corresponding moments can be easily obtained. The first two moments of are as follows. For r =, E[ ] = + A µ, + q n B µ,, E[ 2 ] = C µ, + D µ, + q n E µ,. For r 2, E[ ] = + A 2 µ, + B µ, if µ < T if µ T, r E[ 2 ] = C 2 µ, + D 2 µ, + E µ, if µ < T 2 if µ T, r

9 9 where A µ, = [ n q n i=2 j n c ij d ij { i 2 r c 0 d 0 n qn n p jk i k i ], i 2 p jk n j ij + B µ, = C µ, = n c ij µ ij, i= [ n i= j n c ij d ij D µ, = 2 q n [ n i= { i 2 +2n c 0 d 0 qn n +n c 0 µ 0 ], j n i k p jk i k i + 2 i 2 2 { i 2 c ij µ ij ] q n, p jk i k i p jk i 2 p jk n j i 2 j + 2 n j ij + E µ, = n c ij µ 2 ij, i= A 2 µ, = q n n r n r + c ij d ij C 2 µ, = + i= n i=r j n r c ij d ij i= + n i=r j n qn r qn, n c ij d ij { r 2 c ij d ij { r 2 { i 2 p jk r k r p jk i k i r 2 p jk i 2 p jk r k p jk r 2 r k + 2 r 2 p jk { i 2 p jk i k i 2 i k + 2 i 2 p jk n j rj + n j ij + n j 2 r 2 j + 2, n j 2 i 2 j + 2

10 0 { r r 2 r k D 2 µ, = 2 c ij µ ij p jk r i= +2 n i=r j n { i 2 c ij µ ij p jk i k i r 2 p jk i 2 n j rj + p jk n j ij +. The CDFs of and µ can be easily obtained from their respective PDFs. The CDFs are as follows. When < t <, the CDF of for r = is F b t = Γ t, α n + q n [ c 0 e λ 0 µ 0 t d 0 e λ 0 t and for r 2, the CDF is F b t = n + {c ij Gt; µ ij, α i, λ i, λ ij d ij G t; 0, α i, λ i, λ ij i=2 ] + {c nj Gt; µ nj, α n,.j d nj G t; 0, α n,, j, q n Γ λ t, α + q n Γ t, α n n + {c ij Gt; µ ij, α i, λ i, λ ij d ij G t; 0, α i, λ i, λ ij i=2 n 2 + {c nj Gt; µ nj, α n,.j d nj G t; 0, α n,, j if µ < T Γ λ t, α if µ T, where, x denotes the max{0, x, and 0 if t 0 Γ t; α = t Γα e x x α dx if t > 0, Gt; µ, α, λ, λ 2 = 0 α p k Γ λ 2 t; if t 0 α p k Γ λ t; α k if t 0. The CDF of µ for r = is F bµ t = 0 if t < µ q n [ e n t µ] if µ t < T if t T,

11 and for r 2, the same is F bµ t = { 0 if t < µ e n t µ if t µ. Note that the p th quantile of the exponential distribution given in is η p = µ + a p, where a p = ln p. Therefore the MLE of η p can be found by replacing µ and by their respective MLEs. The moment generating function of the MLE of η p, say η p can be obtained easily from the joint MGF of µ and in Theorem Confidence Intervals: Note that the distribution of µ is same as that of the lowest order statistic and this has been extensively studied in the literature, hence we do not pursue it here. In this section, we present different methods for construction of confidence intervals CIs for the unknown parameter. From the Theorem 2.2, we can find the approximate CI for. However, as PDF of is quite complicated, we also present the bootstrap CI for the same parameter. 3. Approximate Confidence interval: From CDF of, approximate CI can be found, based on the assumption that CDF of is a strictly decreasing function of. Several authors including Chen and Bhattacharyya [4], Gupta and Kundu [0], Kundu and Basu [], Childs et al. [5], and Balakrishnan et al. [2] used this method to find the CI of. Though it is difficult to verify the assumption, an extensive numerical study supports the monotonicity assumption. Suppose obs is the estimate of. Then a two-sided 00 α% approximate CI, say L, U, for can be constructed by solving the equations F L obs = α 2 and F U obs = α 2 for L the lower bound of and U the upper bound of replacing µ by its MLE. However, they are nonlinear equations, and we need to solve them by some numerical procedure, e.g., bisection method or Newton-Raphson method.

12 2 3.2 Bootstrap Confidence Interval: The exact CIs presented in the previous section are computationally quite complicated, specially when sample size is large. So we consider the bootstrap CIs. Here we consider two types of bootstrap CIs, viz., percentile bootstrap CI and Bias adjusted percentile BCa interval; see Efron and Tibshirani [8] for details Bootstrap sample: Step. Given T, r, n, and the original Type-II sample, µ and are obtained from 3 or 4. Step 2. Based on T, r, n, µ, and, a random sample of size n from Uniform0, distribution is generated, and order them to get U :n,..., U n:n. Step 3. Let t i:n = µ log U i:n Step 4. If t r:n < T, then find N such that t N :n < T t N +:n, and set N = Now, {t :n,..., t N :n is the bootstrap sample. { N if t r:n < T r if t r:n T. Step 5. Based on n, T, r, and the bootstrap sample, µ and are obtained form 3 or 4. Step 6. Step -5 are repeated B times, and s are ordered in ascending order to obtain the bootstrap sample { [], [2],..., [B] Percentile bootstrap CI: A two-sided 00 α% bootstrap confidence interval for is [ α 2 B], [ α 2 B], where, [x] denotes the largest integer less than or equal to x.

13 Bias adjusted percentile BCa interval: A two-sided 00 α% BCa bootstrap confidence interval for is [α B], [α 2 B], where, { ẑ 0 + z α/2 α = Φ ẑ 0 + aẑ 0 + z α/2 and { ẑ 0 + z α/2 α 2 = Φ ẑ 0 +. aẑ 0 + z α/2 Here Φ is the CDF of the standard normal distribution, z α is the upper α-point of standard normal distribution and ẑ 0 = Φ { # of [j] < B, j =,..., B. A estimate of the acceleration a is â = N i= { N 6 i= [ i] 3 [ i] 2 3/2, where i is the MLE of based on the original sample with the i th observation deleted, and = N N i= i. 4 Simulation Results and Data Analysis In this section the results of Monte Carlo simulation is presented to study the performance of the inference procedures described in the Sections 3. We choose the value of the location parameter µ to be zero and different values for the scale parameter, viz.,.00, 2.00, 3.00, 4.00 and We also take n = 20, r = 6 and different choices for T, viz.,.5, 2.5, 3.5. The coverage percentage of different confidence interval discussed in the section 3 are calculated based on the 0, 000 Monte Carlo simulations and B = 000. These values are presented in the Tables 2, 3 and 4. It is clear from the Table 2 that the approximate method of constructing confidence interval is always maintaining its coverage percentage to its pre-fixed nominal level. Among the bootstrap

14 4 methods for constructing confidence interval, adjusted percentile bootstrap method is better then percentile bootstrap method with respect to the coverage percentage. From the Table 3, we can see that the coverage percentage of the percentile bootstrap method is quite lower than its pre-fixed nominal level, while the same of the adjusted bootstrap method is somewhat close to its nominal level see Table 4. Though approximate method of construction of confidence interval is always better with respect to coverage percentage and average length, it is very complicated to calculate specially when n is large. So we suggest to use the BCa bootstrap CI for the large values of n. Next, we consider a data set to illustrate the procedures described in the previous sections. Here we consider the data provided in Bain []. Suppose a sample of 20 items are put on the test and the test is terminated after 50 hours. There are 3 failures within this time and the failures times are 3, 9, 23, 26, 37, 38, 4, 45,58, 84, 90, 09, and 38. To make an artificial hybrid Type-II data from this data set one can take any r less than or equal to 3. Here we take r = 2. Under the assumption that lifetime has two-parameters exponential distribution, MLEs of µ and are 3 and respectively. Different types of confidence intervals are reported in the Table. 5 Conclusion In this paper, we have considered the Hybrid Type-II censoring scheme, when life times have twoparameters exponential distribution, under frequentist approach. We have found the MLEs for both the parameters. We have considered different methods for construction of confidence interval. We have seen that the approximate and BCa bootstrap methods of construction of confidence interval are quite good. So we recommend to use BCa bootstrap method of construction of confidence interval specially when n is large. Acknowledgements The authors would like to thank the referees and the editor-in-chief Professor N. Balakrishnan for many valuable suggestions which had helped us to improve the previous version of the manuscript.

15 5 A Appendix We need the the following lemmas to prove the theorems. Lemma A.. Let T :n <... < T n:n be the order statistics of a random sample of size n from a continuous distribution with PDF fx and the corresponding CDF is F. Let T be a pre-fixed number such that F T is positive and N denote the number of order statistics less than or equal to T. The conditional joint PDF of T :n,..., T N:n conditioned on the event N = i is given by n! i f t, t 2,..., t i N = i = f t j { F T n i n i!p [N = i] j= where F is the distribution function of f. if t <... < t i < T, Proof: See [5]. Lemma A.2. Let T :n <... < T n:n be the order statistics of a random sample of size n from a continuous distribution with PDF fx and the corresponding CDF is F. Let T be a pre-fixed number such that F T is positive and N denote the number of order statistics less than or equal to T. Let r {, 2,..., n be a pre-fixed integer. Then PDF of T :n, T 2:n,..., T r:n conditioned on the event N = 0 is given by f t, t 2,..., t r N = 0 = n! n r!p [N = 0] r f t j { F t r n r j= if T < t <... < t r <. For i =, 2,..., r, PDF of T :n, T 2:n,..., T r:n conditioned on the event N = i is given by n! r f t, t 2,..., t r N = i = f t j { F t r n r n r!p [N = i] Proof: See [5]. j= if t <... < t i < T < t i+ <... < t r <. Lemma A.3. Let X be a Gammaα, random variable, Y be a standard Exponential random variable, and they are independently distributed. The PDF of X is Γα xα e x for x > 0 f x; α, λ = 0 otherwise.

16 6 The PDF of Y is e y for y > 0 f 2 y = 0 otherwise. Then for any arbitrary constant A, λ and λ 2 the MGF of A + λ X + λ 2 Y is given by M A+λ X+λ 2 Y ω = e ωa λ ω α λ 2 ω. This MGF exists if ω { λ 2, λ if λ 2 < 0, min{ λ, λ2 if λ 2 0. Proof: This lemma can be proved using the joint distribution of X, Y, and therefore the proof is omitted. Lemma A.4. Let X be a Gammaα, λ with α integer and Y be a Exponentialλ 2 random variable and they be independently distributed. Then the PDF of X Y is given by α α g 2 t; α, λ, λ 2 = p k g t; α k, λ + p k g t,, λ 2 for t R, where p k = λ 2 λ +λ 2 λ λ +λ 2 k. Proof: See [6]. Proof of Theorem 2.: Note that the number of unit failed before time T, N, is a non-negative random variable with the probability mass function PMF P [N = i] = n i e T µ i e n i T µ if i = 0,,..., n. 0 Case-I: r = The joint conditional MGF of, µ conditioned on the event {N can be written as E[e ω b +ω 2 bµ N ] = n E[e ω +ω b 2 bµ N = i] P [N = i N ]. i=

17 7 Using 3 and Lemma A., E[e ω b +ω 2 bµ N = ] = = e T µ e n T ω + µ T µ e ω n T t+ω 2 t t µ dt {e n µ ω +µ ω 2 µ e T n T ω +T ω 2 e T µ + n ω. Using the above expression and 0 E[e ω +ω b 2 bµ N = ] P [N = N ] [ = q n e µ 0 ω +µ ω 2 ] c 0 + n ω d e T ω 2 0, 2 + n ω where q, c 0, d 0 and µ 0 are as in 6 and 7. For i = 2, 3,..., n, using 3 and Lemma A., we have E[e ω b +ω 2 bµ N = i] = i! i i e T µ T T µ t... T t i e ω i { P i j=2 t j n t +n it+ω 2 t P i j= t j µ dt i... dt = T µ +T ω 2 T i! T e i i... i e T µ µ t = e i T µ +T ω 2 e T µ i T t i e + n i ω ω 2t T ω i P i j=2 t j T dt i... dt i i j j + i ω e j+ + n j Using the above expression and 0, one will get for i = 2,, n ω ω 2T µ i. + n j ω ij+ ω j+ 2 E[e ω b +ω 2 bµ N = i] P [N = i N ] T µ = e n +T ω 2 T µ n e n i j i i j + i ω e j+ + n j ω i ω 2T µ i + n j ij+ ω j+ ω 2.

18 8 So using the above expression and 2 in, we have E[e ωb N ] = q n n + c ij i=2 + c nj c 0 e µ 0ω +µ ω 2 + ω λ 0 ν 0 d 0 e µ ijω +µ ω 2 e T ω 2 + ω λ 0 ν 0 + ω λ i αi + ω λ ij d ij e µ njω +µ ω 2 ω αn + ω j d nj e µ ω 2 q n e T ω 2 ω αn ν n e T ω 2 ω λ i αi + ω λ ij e T ω 2 ω αn + ω j. where q, α i, λ i, µ ij, λ ij, c ij and d ij are as in 6 and 7. Now using Lemmas A.3 and A.4, one will get 8. Case-II: r 2 In this case, the joint MGF of and µ can be expressed as E[e ω b +ω 2 bµ ] = n E[e ω +ω b 2 bµ N = i] P [N = i]. 3 i=0 Now using 4 and Lemma A.2, for ω < r and ω 2 < + n r ω = E[e ω +ω b 2 bµ N = 0] n! r n r!p [N = 0] max{t,µ t... t r e ω r { P r j=2 t j n t +n r+t r+ω 2 t P r j= tj µ n r tr µ dtr... dt = e max{t,µ µ n +ω 2 max{t,µ P [N = 0] r ω r n ω 2. Hence for ω < r and ω 2 < + n r ω E[e ω +ω b 2 bµ max{t,µ µ n +ω N = 0] P [N = 0] = e 2 max{t,µ r ω r n ω 2. 4

19 9 Now for i =, 2,..., r and ω < r, using 4 and Lemma A.2 E[e ω b +ω 2 bµ N = i] = = n! n r!p [N = i] r T µ T... t i T... nµ n! e n r!p [N = i] r { T T T... µ t {... T t i+ it n i r T ω +ω 2 T t r e ω r { P r j=2 t j n t +n r+t r+ω 2 t P r j= tj µ t i e + n r ω ω 2t T ω r P i j=2 t j T dt i... dt t r e ω r P r j=i+ t j n r+ ω r t r dt r... dt i+ n r tr µ dtr... dt T µ = e n +T ω 2 P [N = i] n i j i Hence for i =, 2,..., r and ω < r, E[e ω b +ω 2 bµ N = i] P [N = i] T µ n +T ω = e 2 i j n i i j + i j + Now for i = r, r +,..., n, using 4 and Lemma A. E[e ω b +ω 2 bµ N = i] r ω r ω e j+ + n j ω r ω 2T µ r. + n j ω rj+ ω j+ 2 e j+ + n j ω r ω 2T µ r. 5 + n j ω rj+ ω j+ 2 = n! n i! i P [N = i] T T T µ t... T µ = e n +T ω 2 P [N = i] t i e ω i { P i j=2 t j n t +n it+ω 2 t P i j= n i j i Hence for i = r, r +,..., n, E[e ω b +ω 2 bµ N = i] P [N = i] T µ n +T ω = e 2 i j n i i j + i j + i ω i ω tj µ n i T µ dti... dt e j+ + n j ω i ω 2T µ i + n j ij+ ω j+ ω 2 e j+ + n j ω i ω 2T µ i + n j ij+ ω j+ ω 2..

20 20 So using the above expression, 4 and 5 in 3, we have for ω < r and ω 2 < n E[e ω b +ω 2 bµ ] = q n e T ω 2 e ω α µ ω 2 q n e T ω 2 + λ ν n ω αn ν n n e µ ij ω +µ ω 2 + c ij i= ω αi λ i + ω e µ nj ω +µ ω 2 + c nj ω αn + ω λ ij j n e T ω 2 d ij ν i= j ω αi λ i + ω e T ω 2 d nj ν j ω αn + ω j λ ij where q, µ ij, α i, λ i, λ ij, c ij and d ij are as in 6 and 7. Now using Lemmas A.3, A.4, one will get 9., Table : Lower and upper limits of different confidence intervals for Approx. CI Per Boot. CI BCa Boot. CI Level LL UL LL UL LL UL 90% % % Table 2: Coverage percentage and average length of approximate confidence intervals based on 0000 simulations with µ = 0, n = 20 and r = 6. 90% C.I. 95% C.I. 99% C.I. T Cov. Per. Ave. Len. Cov. Per. Ave. Len. Cov. Per. Ave. Len

21 2 Table 3: Coverage percentage and average length of bootstrap confidence intervals based on 0000 simulations with µ = 0, n = 20, r = 6 and B = % C.I. 95% C.I. 99% C.I. T Cov. Per. Ave. Len. Cov. Per. Ave. Len. Cov. Per. Ave. Len Table 4: Coverage percentage and average length of BCa bootstrap confidence intervals based on 0000 simulations with µ = 0, n = 20, r = 6 and B = % C.I. 95% C.I. 99% C.I. T Cov. Per. Ave. Len. Cov. Per. Ave. Len. Cov. Per. Ave. Len

22 22 References [] Bain, L. J. 978, Statistical analysis of reliability and life testing models, Marcel and Dekker Inc., New York. [2] Balakrishnan, N., Xie, Q. and Kundu, D. 2009, Exact inference for a simple step-stress model from the exponential distribution under time constraint, Annals of the Institute of Statistical Mathematics, vol. 6, [3] Banerjee, A. and Kundu, D. 2008, Inference based on Type-II hybrid censored data from a Weibull distribution, IEEE Transaction on Reliability, vol. 57, [4] Chen, S. and Bhattacharya, G.K. 988, Exact confidence bounds for an exponential parameter under hybrid censoring, Communications in Statistics - Theory and Methods, vol. 6, [5] Childs, A., Chandrasekar, B., Balakrishnan, N. and Kundu, D. 2003, Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Annals of the Institute of Statistical Mathematics, vol. 55, [6] Childs, A., Balakrishnan, N. and Chandrasekar, B. 20, Exact distribution of the MLEs of the parameters and of the quantiles of two-parameter exponential distribution under hybrid censoring, Statistics, DOI: 0.080/ [7] Draper, N. and Guttman, I. 987, Bayesian analysis of hybrid life tests with exponential failure times, Annals of the Institute of Statistical Mathematics, vol. 39, [8] Efron, B. and Tibshirani 998, An introduction to the bootstrap, Boca Ration: Chapman and Hall/CRC CRC Press. [9] Epstein, B. 954, Truncated life-test in exponential case, Annals of Mathematical Statistics, vol. 25, [0] Gupta R. D. and Kundu D. 998, Hybrid censoring scheme with exponential failure distribution, Communication in Statistics-Theory and Methods, vol. 27,

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