Minimal repair under step-stress test

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1 Minimal repair under step-stress test N Balakrishnan, U Kamps, M Kateri To cite this version: N Balakrishnan, U Kamps, M Kateri Minimal repair under step-stress test Statistics and Probability Letters, Elsevier, 2009, 79 3, pp548 <006/jspl > <hal > HAL Id: hal Submitted on 3 Dec 200 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not The documents may come from teaching and research institutions in France or abroad, or from public or private research centers L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés

2 Accepted Manuscript Minimal repair under step-stress test N Balakrishnan, U Kamps, M Kateri PII: S DOI: 006/jspl Reference: STAPRO 5382 To appear in: Statistics and Probability Letters Received date: 29 July 2008 Revised date: 7 January 2009 Accepted date: 7 March 2009 Please cite this article as: Balakrishnan, N, Kamps, U, Kateri, M, Minimal repair under step-stress test Statistics and Probability Letters 2009, doi:006/jspl This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain

3 Minimal Repair under Step-Stress Test N Balakrishnan a, U Kamps b, and M Kateri c a Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K b RWTH Aachen University, Institute of Statistics, Aachen, Germany c Department of Statistics and Insurance Science, University of Piraeus, 8534 Piraeus, Greece Abstract In the one- and multi-sample cases, in the context of life-testing reliability experiments, we introduce minimal repair processes under a simple step-stress test, based on exponential distributions and an associated cumulative exposure model, and then develop likelihood inference for such a model Keywords and Phrases: Minimal repair times, Record values, Nonhomogeneous Poisson process, Maximum likelihood estimation, Conditional moment generating function, Gamma distribution, Mixtures Introduction We consider the sequence of failure times of a technical system in the sense of minimal repair In such a scheme, with respect to just one component, upon failure, this component will instantaneously be repaired, and by this, put into the condition immediately prior to its failure The times to repair are considered to be low and so are neglected As we will indicate below, the minimal repair model is also used as an approximate description of complex systems Let Z i denote the operating time between the i th and i th repairs of some component, so that X i i j Z j corresponds to the i th repair time, i N Moreover, let F be the continuous distribution function of Z, and furthermore Z j, conditioned on j i Z i z, be distributed according to F +z Fz Fz, j 2, 3,, which is simply the distribution F truncated on the left at z Then, the minimal repair times form a Markov chain, and we find for x, t > 0, PX i+ X i > x X i t i PX i+ > x + t i X i t i i P Z i+ > x Z j t i Fx + t i, i N Ft i These transition probabilities coincide with those of record values; see, for example, Arnold et al 998, p and Kamps 995, p 32 Hence, the minimal repair times possess the same joint distribution as record values based on F which are usually denoted by X L, X L2, as well as epoch times of some nonhomogeneous Poisson process NHPP; see Gupta and Kirmani 988 A formal definition of the operating times Z, Z 2, can be given based on a sequence Y i i N of iid random variables with continuous distribution function F via j j ] j Z Y, Z j F [FY j F Z i } + F Z i Z i, j 2; i j i i Corresponding author, Tel: ; fax: ; address: udokamps@rwthaachende

4 see Kamps 995, p 45 Another interpretation of minimal repair is to say that, successively, the failed component is replaced by a component of equal age in contrast to the model of a renewal process A general scheme of life tests with replacement of failed items and censoring has been presented by Fairbanks et al 982 For further details on minimal repair processes we also refer to Kirmani and Gupta 995 The term minimal repair was introduced first by Barlow and Hunter 960 Other terms in vogue are bad-as-old model Ascher 968 and age-persistence model Balaban 978 For further details on terminology and pertinent references, we refer the readers to Ascher and Feingold 984, p 5-52, Moreover, in this reference, several concrete applications are outlined while dealing with reliability of automobiles and aircrafts, and inferential procedures are also addressed Some other applications can be found in Balaban and Singpurwalla 984 For more recent papers on bad-as-old models and minimal repair models as well as for an overview, we refer to Finkelstein 2004, Kirmani and Gupta 995, Langseth and Lindqvist 2006, Lugtigheid et al 2008, Raqab and Asadi 2008, and Wang and Pham 2006 A different way of viewing a minimal repair process is to consider iterations of the so-called relevation transform see Krakowski 973, Baxter 982, Lau and Prakasa Rao 990, and Cramer and Kamps 2003 Hence, the model of record values, the analysis of occurrence times of some NHPP, the iterative use of the relevation transform as well as the minimal repair model all are equivalent distributionwise Results derived for any of these models may therefore be used for the situation under consideration For a review of more advanced models for imperfect maintenance and repairable systems, one may consult Pham and Wang 996, Wang and Pham 2006 and Lindqvist 2006, and also the references contained therein For stochastic ordering results in the context of records and minimal repair, stochastic comparisons in terms of epoch times of NHPP, and some results on prediction intervals in terms of record values, one may refer to Khaledi and Shojaei 2007, Belzunce and Shaked 200, Belzunce et al 2003, and Raqab and Balakrishnan 2008 Up to now, we have discussed successive minimal repair of a single particular component within a system However, considering just one component which is successively minimally repaired is not a practical situation Ascher and Feingold 984, p 5 have explained the use of a minimal repair modelling in their discussion on probabilistic modelling with NHPP Understanding minimal repair as described above, we may likewise argue in terms of occurrence times of some corresponding NHPP as mentioned above In particular, such a modelling may be appropriate when considering successive repairs of a system when only a very small fraction of components is either repaired or replaced by new components In these cases, it is reasonable to assume that, upon restart, the reliability of the complex system after some minimal repair is approximately the same as it was immediately prior to its failure We also refer to Love and Guo 99 for a justification of using a bad-as-old model on the system level Methods of accelerated life-testing cf Bagdonavicius and Nikulin 2002, Meeker and Escobar 998, Nelson 990, in particular step-stress methods, are a common approach in life-time experiments and are applied in general to reduce experimental time, when technical systems tend to have quite long life times Under normal operating conditions, lifetime tests would be time consuming and expensive Therefore, an accelerated testing is adopted, wherein experimental units are exposed to increasing stress levels higher than the normal one Moreover, since the number of minimal repair times or records that occur would be fairly small [see Arnold et al 998, p 24], a problem arises in making inference based on lifetimes observed from such experiments We consider a Type-II censored experiment which terminates as soon as the r th failure is observed for some r Thus, the mean time to the r th failure of the system under test may be quite large, even too large to complete the experiment within a reasonable period of time By applying the step-stress set-up, we develop a methodology to shorten experimental time in life-tests for complex systems, in the sense that, successively upon failures of components, the system is restored to operating status by repairing or replacing respective components and so 2

5 may be regarded as having been minimally repaired For details on step-stress models, we refer to Nelson 990, Gouno and Balakrishnan 200, Bagdonavicius and Nikulin 2002, Gouno 2006, Balakrishnan et al 2007 and Balakrishnan 2009 We consider here the simple step-stress setup which means that there is only one change in the stress levels; however, the results can be generalized to the case of multiple stress levels as well Besides the consideration of a step-stress experiment as one form of a planned experiment, one may also think of situations facing an unavoidable change in the underlying life-time distribution of the test units during an experiment at some change point τ for some technical reason Let X, X 2, denote minimal repair times or records, respectively from some absolutely continuous distribution function F with density function f Then, the joint density function of X,, X r, for some r N, is given by [see Arnold et al 998, p 0 and Kamps 995, p 3] r f X,,X r fx i x,, x r fx r, 0 x x r Fx i i In the simple step-stress model, we assume to start the lifetime experiment at the first stress level with an underlying distribution function F and then to switch to the second stress level at some pre-fixed time τ > 0 with an underlying distribution function F 2, where F i t exp t µ }, t µ 0, θ i > 0, i, 2 θ i By applying the cumulative exposure model which chooses s such that F τ F 2 s, we have the distribution function G to be [Nelson 990] F t, µ t < τ 2 Gt F 2 s + t τ exp t τ τ µ }, t τ for some pre-fixed τ > µ and s τ µ + µ The corresponding density function g is given by gt θ exp t µ }, exp t τ and the hazard rate g/ G is consequently τ µ gt Gt θ, µ t < τ, t τ µ t < τ },, t τ Here, we consider minimal repair times X, X 2, in the step-stress context based on the cumulative exposure distribution G in 2, and for convenience we denote the survival function G by Ḡ In order to increase precision of inferential procedures, one may wish to combine different stepstress experiments which were conducted at different locations or at different times or even under different testing conditions For this kind of meta-analysis we present MLEs to handle multi-sample situations The paper is organized as follows Some preliminary results are stated and proved in Section 2 Section 3 presents the maximum likelihood estimates MLEs of the model parameters and in the case of one-sample simple step-stress minimal repair system, as well as their exact distributions, conditional on the fact that the MLEs exist In Section 4, the multi-sample case is introduced and dealt with and the corresponding results are developed Specifically, the MLEs ˆ and ˆ and their exact conditional distributions by means of their conditional moment generating functions are derived in this section 3

6 2 Preliminaries Let X i i N be a sequence of minimal repair times or record values or epoch times of some NHPP based on a continuous distribution function G Moreover, let R denote the random number of minimal repairs before time τ > 0, ie, R ρ X ρ τ < X ρ+, ρ N 0, with X 0 0 The following theorem states that R is distributed as Poisson This result, in fact, holds true for any continuous distribution function G Theorem 2 With the above notation, we have R Poisson log Ḡτ, ie, P X j τ < X j+ j! Ḡτ logḡτj, j N 0, τ > 0 Proof Since record values possess a nice distributional structure, we use it to establish the required d result It is known cf Arnold et al, 998 that X j G j U i, j N, where U i are iid Unif0, Moreover, if Y j denote record values from a standard exponential distribution, then d j Y j Z i, where Z i are iid Exp Using these, we find, with t log Ḡτ, i PX j τ < X j+ P G PY j logḡτ < Y j+ P i j+ j U i τ < G U i where N denotes the Poisson-process associated with j Z i j N i i j j+ Z i t < Z i PN t j t j e t, j N 0, j! i Furthermore, it is well-known [see Arnold et al 998, p ] that for minimal repair times or records X, X 2, based on some continuous distribution function G, the distribution of such quantities, conditioned on a previous one, is distributed as the unconditioned minimal repair times from G truncated on the left More precisely, 2 P X ρ+,,x r X ρ y P Y,,Y r ρ, ρ < r, where Y, Y 2, are minimal repair times based on distribution function H with Hz Gz Gy Gy, z y A similar result, which is also valid for an arbitrary G, holds true by conditioning on the number ρ of minimal repairs up to time τ, ie, conditioning on the event X ρ τ < X ρ+ Theorem 22 Let X, X 2, be minimal repair times or record values based on some absolutely continuous distribution function G with density function g Then, i i P X r X ρ τ<x ρ+ P Y r ρ, ρ r, where Y r ρ is the r ρ th minimal repair time based on G truncated on the left at τ, ie, based on distribution function Hz Gz Gτ Gτ, z τ 4

7 Proof We first consider the joint distribution of X r and the event X ρ τ < X ρ+, ρ < r : PX r τ < X ρ+ τ say Then, upon using 2, we have PX r x, X ρ τ < X ρ+ X ρ y d P X ρ y PX r x, X ρ+ > τ X ρ y d P X ρ y Py PY r ρ x, Y > τ x v τ τ τ f Y,Y r ρ u, v du dv; Py d P X ρ y, since the joint density of Y and Y r ρ is [Arnold et al 998, p and Kamps 995, p 68] we obtain f Y,Y r ρ u, v Py r ρ! log Hu r ρ 2 hu r ρ 2! Hv Hu hv log Ḡu r ρ 2 gu r ρ 2! Ḡv Ḡu Ḡy Thus, by interchanging the integrals, we find τ Py dp X ρ y ρ!v ρ! logḡτρ ρ!r ρ! x τ τ x τ x τ log Ḡτ r ρ gv dv Ḡv gv Ḡy, log Ḡτ r ρ gv log Ḡy ρ gy dvdy Ḡv Ḡy log Ḡτ r ρ gv dv Ḡv From this expression and Theorem 2, the distribution function of X r, conditioned on R ρ, is PX r x X ρ τ < X ρ+ r ρ! and the conditional density is Ḡτ x τ log Ḡτ r ρ gv dv Ḡv f X r X ρ τ<x ρ+ x log Ḡx r ρ gx r ρ! Ḡτ Ḡτ, x > τ, which is incidentally the same as the density of the r ρ th minimal repair time based on G truncated on the left at τ > 0 5

8 If ρ r, then by the Markovian property of records [see Arnold et al 998 and Kamps 995, p 32], we have Gx Gy Gτ Gy Py Pτ < X ρ+ x X ρ y Gy Gy and consequently PX r x, X r τ < X r Thus, we obtain τ PX r x X r τ < X r Gx Gτ, Gy Gx Gτ log[ Gy] r 2 gydy r 2! Gy r! [Gx Gτ][ logḡτ ] r Gx Gτ, x > τ, Gτ which is same as the distribution function of Y Y based on G truncated on the left at τ Hence, the theorem Based on the cumulative exposure model and underlying exponential distributions, the conditional distribution of X r, given R r, turns out to be a power function distribution, which does not depend on the model parameters and Lemma 23 Let the minimal repair times X, X 2, be based on G in 2 Then, we have x µ j PX j x X j τ < X j+, j N, x τ τ µ Proof For x τ, upon using the expression of the joint density of two records in Arnold et al 998, p or Kamps 995, p 68, we have PX j x X j τ < X j+ PR j PX j x, X j+ > τ PR j x µ τ gu j! Ḡu x µ j logḡuj gv dvdu PR j j! θ j Then, the assertion follows readily by applying Theorem 2 3 One-sample case exp τ µ } We now suppose that we have a sample of observations of minimal repair times X,, X r based on distribution function G in 2 Given a number of ρ observations before time τ, ie, given R ρ, ρ < r, the likelihood function becomes L, exp x r τ τ µ }, µ x < < x ρ τ < x ρ+ < < x r θ ρ θr+ρ 2 The above likelihood function yields the conditional MLEs as given in the following theorem Theorem 3 For the exponential cumulative exposure model in 2, the conditional MLEs, conditioned on R ρ for some ρ < r, of and are given by 3 ˆ τ µ ρ and ˆθ2 x r τ r ρ 6

9 Obviously, ˆ depends only on the number of minimal repairs up to time τ; hence, the distribution of τ µ R, conditioned on R ρ, is discrete Further, as in the case of a sample of records based on an exponential distribution, the MLE of only depends on the largest observation x r cf Arnold et al 998, pp Remark 32 i Given R r, ie, all observations are smaller than τ, the likelihood function is independent of Hence, a MLE of does not exist The MLE of in this case is given by ˆ xr µ r ; ii Given R 0, ie, all observations are larger than τ, the likelihood function is a monotonic increasing function of, and so MLE of does not exist The MLE of in this case is given by ˆ xr τ r Now, we shall derive the distributions of ˆ and ˆ, conditioned on R,, r }, which simply means conditioned on the event that the MLEs ˆ and ˆ both exist For this purpose, let us first denote 32 p ρ PR ρ ρ! Ḡτ logḡτρ τ µ ρ exp τ µ } ρ! Then, P R r r ρ p ρ p, say Remark 33 Let x ρ τ µ ρ, ρ r Then, Pˆ x ρ R r p r Pˆ x ρ, R ρ p ρ p, ρ r ρ The conditional distribution of ˆ turns out to be a mixture of gamma distributions as presented below Theorem 34 The conditional distribution of ˆ in 3 is given by r Pˆ x R r ρ p r ρ p FΓρ, ρ x, x > 0, with p and p j, j r, as defined in 32, where F Γn,θ denotes the distribution function of a gamma distribution with parameter n and θ, ie, and the corresponding density function is Proof Since Pˆ x R r p Theorem 22: n F Γn,θ x e x/θ x/θ j, x > 0, j! θ n j0 f Γn,θ x n! xn e x/θ, x > 0 r Pˆ x R ρ PR ρ, we have upon applying ρ r Pˆ x R r PY p r ρ x PR ρ, ρ 7

10 where x xr ρ+τ, and the minimal repair times Y,, Y r are based on the distribution function Hz Gz Gτ Gτ, z τ, and corresponding density function hz For z τ, we find Hz Ḡz Ḡτ exp z τ } Since the density of Y r ρ τ r ρ is given by log Hx r ρ hx r ρ r ρ! r ρ r ρ x r ρ exp r ρ! which is a gamma density with parameters r ρ and r ρ, and PR ρ p ρ, we derive r Pˆ x R r which establishes the required result p ρ p FΓr ρ, ρ r r ρ x ρ p r ρ p F Γρ, ρ x, r ρ x It is worth noting that in the case of the usual step-stress experiment under cumulative exposure model and exponential lifetimes, the conditional distribution of the MLE of is also a mixture of gamma distributions, but with different mixing coefficients; see Balakrishnan et al 2007, p 38 From the conditional distributions of ˆ and ˆ presented above, conditional moments can be readily found In particular, ˆ turns out to be conditionally unbiased Remark 35 We have: i E ˆ R r τ µ p exp τ µ } r τ µ ρ ρ! ρ ρ, ii E ˆ R r, E ˆ 2 R r r θ2 2 p r iii V ar ˆ R r θ2 2 p 4 Multi-sample case ρ p r ρ ρ ρ ρ+ ρ p r ρ, The motivation for considering the multi-sample situation is two-fold On the one hand, we may have data from two or more minimal repair experiments with possibly different numbers of observations and different change points, in which case the MLEs based on a larger number of observations is in the sense of a meta-analysis On the other hand, in the planning phase of a life-testing experiment with minimal repair schemes, one may intend to control or minimize the non-existence probabilities of the MLEs since if there is no observation at one of the stress levels, the corresponding MLEs of or do not exist see Theorem 3 and Remark 32 In this multi-sample set-up, let X k i i N denote the minimal repair times in the k th sample k s, where the samples are assumed to be independent; further, in the k th sample, the stress level changes at time τ k and that these change points may be different for different samples; Consider Type-II censored samples with r k observations in the k th sample, and let the corresponding observations be denoted by x k i, for k s; i r k In each sample, analogous to the one-sample situation, the minimal repair times are based on the cumulative exposure distribution G in 2 with τ replaced by τ k for the k th sample }, 8

11 Moreover, let R k denote the number of minimal repairs before time τ k in the k th sample for k s, and let indicators I k and J k be defined by I k Rk } and J k Rk <r k } A realization of R k is denoted by r k Obviously, the implication I k 0 J k holds true, and that J k 0 implies R k r k, and I k 0 implies R k 0 Theorem 4 In the multi-sample simple step-stress model as described above, with the minimal repair times in each sample being based on G in 2, with τ replaced by τ k in the k th sample, the unique MLEs of and are given by ˆ R k J k X k τ k τ k + τ k µ} if there exists a k,, s} with R k, and ˆ r k R k J k X k τ k τ k if there exists a k 2,, s} with R k2 r k2 Consequently, in the multi-sample case, the MLEs of both and exist iff there is at least one observation under the first stress level ie, one observation before one of the τ k s and at least one observation under the second stress level ie, one observation after one of the τ k s Proof Let L k, Lk,2 and Lk 2 denote the likelihood functions given R k r k, R k < r k and R k 0, respectively In case the arguments of I k and J k are deterministic, we use the notations i k and j k, respectively Then, the joint likelihood function is given by L, ; x k r k, k s θ θ [ s j k +i k j k θ r k θ r k θ r k r k 2 P s P r k s r k r k θ2 [ s j k exp P s P r k s r k r k θ s [ ] j k L k + i k j k L k,2 + i k L k 2 exp xk r k µ } } exp xk r k τ k τ k µ + i k 2 exp xk r k µ + i k j k + i k exp [ j k xk r k µ + j k θ r k 2 xk r k x k r k exp xk r k }] τ k τ k µ ] τ k τ k µ }] τ k + τ k µ upon using the fact that i k j k + i k j k Hence, the log-likelihood function is given by l, ; x k r k, k s r k log r k r k log j k xk r k µ + j k x k r k } τ k + τ k µ Equating the partial derivatives of l with respect to and to zero readily yields the possible MLEs of and to be 9

12 ˆ sp r k } j k x k r k τ k + τ k µ and ˆθ2 sp j k x r k k τ k sp, r k r k where ˆθ exists iff s r k > 0, ie, iff k : r k, and ˆθ 2 exists iff s r k r k > 0, ie, iff k : r k < r k Inspecting the second derivatives, it can be shown that ˆ, ˆ is the global maximum if k : r k and if j : r j < r j, and so the MLEs become unique in this case We shall now derive the conditional moment generating functions of the MLEs ˆ and ˆ For simplicity in notation, we shall use R s R k to denote the total number of observations under the first stress level, and r s r k for the total number of observations in the s samples altogether Since the random variables R,, R s are independent and, according to Theorem 2, distributed as Poisson, the distribution of R is R Poisson log Ḡτ k Since Ḡτ k F τ k exp τ k µ }, we readily have R Poisson τ with τ Hence, PR l τ l l! e τ π l, say, l N 0 Moreover, let π r l π l τ k µ The conditional moment generating functions of ˆ and ˆ, under the condition R r ensuring the existence of both ˆ and ˆ, are as given in the following theorem Theorem 42 The conditional moment generating functions of the MLEs ˆ and ˆ presented in Theorem 4 are as follows for t 0: i Ee tˆ R r r r r s s PR k r k π l r 0 r s 0 }} t exp l P s r kl } s rk! exp t l τ k µ τ jk k µ} τ k µ rk t A j k l r k k, where PR k r k λr k k r k! e λ k, λ k τ k µ t and A k exp l τ k µ r k j0 } t l τ k µ j j! ; ii E e tˆ R r r l π l π h lt, where h l t t r l r l θ is the moment generating function of Γ r l, 2 r l, the gamma distribution with parameters r l and r l Proof i Recalling that ˆθ } R J k X k r k τ k + τ k µ, we have for l,, r } Ee tˆ R l r r s s } t PR k r k exp τ k µ π l l r 0 r s 0 }} P s r kl 0

13 s E exp where the latter expected value can be expressed as s t E exp l J kx k r k k} τ Rk r k Since and E j k t exp l Xk E e t l Xk r k Rk r k e tµ l r k τ k µ r k r k k} τ Rk r k τ k µ µ r k! t/l r k e t x µ l x r k τ k µ τ k µ 0 t l τ k µ j t l J kx k r k k} τ R r,, R s r s, rk s e tτ k E j k t exp l l jk E j k X k e t l Xk r k tµ r k dx e l τ k µ τ k µ r k r k τ k } Rk r k Rk r k τ k µ t/l r k r k! xrk e t l x r k! e tµ l dx τ k µ rk t A k, l r k with A k e t l τ k µ r k j0 j!, we immediately find Ee tˆθ r r s s R l P R k r k PR l r 0 r s 0 }} t exp l P s r kl Thus the assertion follows by noting that ii Next, we have Ee tˆ t R l E exp r l 0 e t l x x r k dx } s rk! exp t l τ k µ τ jk k µ τ k µ rk t A j k l r k k r Ee tˆ π l R r π E etˆ R l π l r r s r 0 r s 0 }} P s r kl t E exp r l l R J k X k r k k} τ l s PR k r k J k Now, the latter expected value equals s t Rk E exp r l J kx k r k k} τ r k X k r k τ k } R r,, R s r s s [ t } ] jk E exp X k r l r k τ Rk k r k

14 s exp tτ } jk k E j k t Rk exp r l r l Xk r k } r k Upon applying Theorem 22, this conditional expected value equals Eexp t r l Y r k r k }, wherein the minimal repair time Y rk r k is based on H k x Gx Gτ k Gτ k, x τ k Hence, by the definition of G in 2, we have H k x exp τ k µ exp x τ k exp τ k µ τ k µ exp x τ k, x τ k Thus, for 0 t <, the moment generating function of Y r based on H is given by Ee ty r r! θ r 2 τ k e tx x τ k r e Using the fact that j k r k r k r k r k, we arrive at P s r kl x τ k e tτ k dx t r Ee tˆ R l r r s s PR k r k t π l r l r, 0 r s 0 }} r l Setting t 0 and observing that the latter factor does not depend on the summation variables any more, we obtain r E e tˆ π l R r π E r e tˆ π l R l t π r l l l r l Remark 43 The conditional distribution of ˆ is discrete if j k 0 for all k,, s}; otherwise, it is a continuous distribution As in the one-sample situation, the distribution of ˆ is a mixture of gamma distributions with π l /π, l r, as the mixing proportions From the form of the conditional moment generating function of ˆ, we immediately obtain its moments as given below Remark 44 E ˆθ2 R r, E ˆθ2 2 R r θ2 2 r π l π + r l, l V arˆθ2 R r θ2 2 r π l Finally, we compare the conditional variances of the MLEs of in the one-sample case viz, ˆθ 2 with r s s r k observations and change point τ and in the multi-sample case viz, ˆ with s independent samples having r,, r s observations and change points τ,, τ s Note that the comparison is proper in this case since the same number of observations are present in both sampling situations Lemma 45 With the above notation, we have l r l π V arˆθ s 2 R r V arˆ R r τ k µ τ µ 2

15 Proof With λ τ µ, we have from Remark 35 that V ar while from Remark 44 that Hence, where V ar r 2 R r θ2 2 ˆθ ˆθs i r 2 R r θ2 2 i r i ˆθ ˆθs V ar 2 R r V ar 2 R r θ2 2 r r A λi τ j τ i λ j r r r i i! j! i j i j with a ij i! r ij! r j a ji We can then express r A from which the assertion follows i 5 Concluding remarks r ji+ r i λ i i! r λ j j! j ˆτ i i! r ˆτ j j! j, r λ j r τ j j! j! j j A, i j i!j! r i r j λi τ r r j a ij i jλ i τ j i j }} a ij i j λ i τ i τ j i λ j i, In this paper, we have discussed a special form of accelerated life-testing, viz, the simple step-stress scheme in order to analyze data from minimal repair processes based on exponential distributions In the one-sample as well as multi-sample cases, we have derived explicit expressions for the unique maximum likelihood estimators of the model parameters and have further derived their exact distributions as well From these distributional results, further inferential procedures such as confidence intervals and tests of hypotheses may be developed Acknowledgements We express our sincere thanks to the referees and the editor for making some constructive comments and suggestions which led to an improvement in the presentation and writeup of this article References Arnold, BC, Balakrishnan, N, Nagaraja, HN, 998 Records Wiley, New York Ascher, HE, 968 Evaluation of repairable system reliability using the bad-as-old concept IEEE Trans Reliab R-7, 03 0 i j Ascher, HE, Feingold, H, 984 Repairable Systems Reliability Dekker, New York 3

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