Empirical Bayes Estimation of Reliability

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1 Empirical Bayes Estimation of Reliability Introduction Assessment of the reliability of various types of equipment relies on statistical inference about characteristics of reliability such as reliability function, mean lifetime of the devices, or failure rate. General techniques of statistical inference estimation and hypotheses testing) are reviewed in Estimation; Least-Squares Estimation; Maximum Likelihood; Nonparametric Tests; Hypothesis Testing; andsignificance Level, respectively. Consider the following model. Let T = T 1,T 2,...,T m ), T i T, be independent identically distributed i.i.d.) observations with the joint probability density function pdf) ft θ), t T m, depending on an unknown parameter θ. If we are interested in estimating θ or a known function uθ) of θ, we can, for example, use the maximumlikelihood estimation MLE) of θ described in Maximum Likelihood. This technique is, however, of very little help if we want to accommodate some prior information about θ which may be available from previous experiments, expert opinions, or other sources of knowledge. To take advantage of this information, we assume that parameter θ is also a random variable or vector, and the particular value of θ associated with our sample comes from a pool of possible values of θ, which have a prior pdf gθ). Using Bayes formula one can obtain the updated posterior pdf of θ given the sample T as [ ] ft θ)gθ) gθ T)= [ ] 1) ft θ)gθ)dθ The estimator ût ) = uθ)gθ T)dθ 2) is called the Bayes estimator of uθ) and it minimizes the quadratic risk Rg) = EûT ) uθ)) 2 = ût ) uθ)) 2 T m ft θ)gθ)dt dθ 3) Here E is the expectation with respect to T and θ the concept of expectation is introduced in Expectation). The prior density gθ) is usually chosen from some familiar distribution family, or by some other considerations to minimize the information introduced by a prior distribution or to make integration in equation 2) easy, etc.). Extensive introduction to Bayesian methods can be found in [1]. In practice, however, none of the choices for the prior may offer sufficient motivation. Moreover, we may have some prior measurements that are indirectly related to the recent data of interest. In particular, suppose that n past data sets are available, T i = T i1,t i2,...,t imi ), T ij T, i = 1,...,n 4) where, conditional on θ i, vectors T i have pdfs f i t θ i ), and we also have a present data set T = t 1,...,t m ), which carries information about the parameter of interest τ. Now, if we assume that θ 1,θ 2,...,θ n and τ are not just fixed constants unrelated to each other but realizations of a random variable with the unknown pdf gθ), we can use past data T i, i = 1,...,n,for assessment of the value of τ or some function uτ) of τ. Decision theoretic procedures that utilize past data as a means for bypassing the necessity of identifying a completely unknown prior distribution are called empirical Bayes EB) decision procedures [2, p. 65]. Empirical Bayes Estimation Estimator and its Risk The EB estimation was introduced by Robbins [3] and became increasingly popular in various areas of statistics see, e.g., [4] and [5] for a general review of EB methods). To simplify our introduction, we consider the case when all conditional densities f i t θ) are identical and equal to ft θ) the case of different conditional densities is treated in, e.g., [6]). Let one observe random vectors T 1,θ 1 ),..., T n,θ n ), where each θ i is distributed according to some unknown prior distribution G with the pdf gθ) and, given θ i, T i has a known conditional density function ft θ i ), i = 1,...,n. In each pair the first

2 2 Empirical Bayes Estimation of Reliability component is observable, but the second is not. For the observation T with the conditional pdf ft τ), the goal is to estimate τ or a function uτ). Here T, τ) can be a new pair of observations T T n+1, τ = θ n+1, or one of the old ones T T k, τ = θ k, k = 1,...,n. If the prior pdf gθ) were known, the best estimator for uτ) would be the Bayes estimator where ût ) = T )/pt ) 5) t) = uθ)f t θ)gθ) dθ, pt) = ft θ)gθ) dθ 6) Here, pt) is the marginal pdf of T at the point t. As the prior density gθ) is unknown, we construct an EB estimator û n T ) û n T ; T 1,...,T n ) as the estimator of the right-hand side of formula 5) based on observations T 1, T 2,...,T n. Denote by E T and E T the expectations with respect to T and T 1,...,T n, respectively. An EB estimator û n T ) can be characterized by its quadratic risk see [5]) Rû n ; g) = E T E T û n T ) uτ)) 2 7) which can be partitioned into two components. The first component Rg) = E T ût ) uτ)) 2 = inf E T vt ) uτ)) 2 8) v is the risk of the Bayes estimator, while the second component ˆRû n ; g) = E T E T û n T ) ût )) 2 9) is the error of estimating ût ). Robbins [3] called an EB estimator asymptotically optimal if lim n ˆRû n ; g) =. All techniques for EB estimation can be roughly divided into two classes: parametric and nonparametric. Parametric methods are based on the assumption that gθ) has a known parametric form with one or several unknown parameters. In nonparametric methods, gθ) is assumed to be completely unknown and unspecified. Parametric EB Estimation Let gθ) have a known form with a vector or scale parameter σ, which is unknown: gθ) gθ σ).in this case, t) and pt) in formula 6) also have known forms t) t σ) = uθ)f t θ)gθ σ) dθ, pt) pt σ) = ft θ)gθ σ) dθ 1) Hence, the Bayes estimator ût ) is a known function ût σ) of σ and T, and the problem reduces to estimation of a parametric function ût σ) on the basis of observations T 1,...,T n with the pdf pt σ). This is a standard statistical problem. We can, for example, construct the MLE ˆσ of σ on the basis of observations T 1,...,T n see Maximum Likelihood) and then estimate ût ) by û n T ) =ût ˆσ). In addition, we can calculate the posterior pdf of θ [ ] ft θ)gθ σ) gθ T,σ) = 11) [ ft θ)gθ σ) d θ] and estimate it by ĝ n θ T)= gθ T, ˆσ). We can use ĝ n θ T) for construction of the 1 γ) credibility interval ˆτ l, ˆτ r ) for τ by choosing ˆτ l and ˆτ r such that ˆτr gθ T, ˆσ) = 1 γ 12) ˆτ l see Confidence Intervals for the review of confidence and credibility intervals). The credibility interval for uτ) is of the form û l, û r ),whereuτ) û l, û r ) whenever τ ˆτ l, ˆτ r ). Parametric EB estimation is reviewed in [7] and [8] as well as in [4] and [5]. Nonparametric EB Estimation If gθ) is completely unknown, then we can construct a nonparametric estimator of ût ). One possible way of doing this is to estimate gθ) by ĝθ) on the basis of the observations T 1,...,T n and then plug ĝθ) into formulae 5) and 6). The first attempt in this direction was to estimate the unobserved parameters θ i by ˆθ i on the basis of T i, i = 1,...,n,andthen use observations ˆθ i for estimation of gθ) see, e.g., [2]). The difficulty, though, is that in order for

3 Empirical Bayes Estimation of Reliability 3 ˆθ i to be consistent estimators of θ i, the past data should contain series of observations, that is, m i s in 4) should be relatively large. This drawback can be avoided by constructing an estimator of gθ) directly on the basis of T 1,...,T n see, e.g., Maritz and Lwin [5] or Walter [9]). However, the shortcoming of any approach that is based on estimating prior density gθ) is that estimation of gθ) is a much harder task than estimation of ût ) itself. Consequently, the risk ˆRû n ; g) of the resulting estimator will converge to zero very slowly as n. In order to avoid estimation of gθ), one can estimate uτ) directly. For example, if uτ) = τ and ft θ) belongs to a one-parameter exponential family ft θ) = wt)cθ) expθt) with ut) > for t>a 13) Singh [1] observed that ˆτ = p T ) 14) pt ) where pt) is the marginal density of T defined in formula 6). Note that the gamma and the normal pdfs belong to the family 13). Estimation of the pdf pt) and its derivative p t) at a point t is a very common problem in statistics and is addressed in a variety of papers and monographs see, e.g., [11] for a review). One of the possibilities is to construct kernel estimators for pt) and p t) ˆp n t) = 1 nh ˆp n t) = 1 nh 2 ) t Ti K, h ) t Ti K 1 h 15) where h is small and the kernels K j, j =, 1, satisfy the conditions t i K j t) dt = 1ifi = j and t i K j t) dt = ifi = j, i =, 1,...,r. Here r 2 is a positive integer. The approach of Singh [1] can be generalized as follows see [6]). Note that in order to construct an EB estimator û n T ) one needs to estimate the numerator and the denominator in formula 5), and then estimate the ratio. The denominator pt ) can be estimated using, for example, the first formula in 15). To estimate the numerator T ) = uθ)f T θ)gθ) dθ one needs to find a sequence of functions ϕ ε t) such that ϕε 2 t) dt < for every ε> and for every s and θ ft θ)ϕ ε s, t) dt uθ) as ε 16) Then T ) can be estimated by ˆ n T ) = n 1 ϕ ε T, T i ) 17) with small ε = ε n. Finding the sequence ϕ ε s, t) may be tricky; however, luckily, in many particular situations, for example, when θ is the location or a scale parameter of the family ft θ), ft θ)ϕ ε s, t) dt = uθ) 18) T has an exact solution ϕs,t) see, e.g., [12, 13]). The last step is estimation of the ratio ût ) = T )/pt ) by û n T ) = Hˆ n T )/ ˆp n T ), δ) 19) where δ> is a small parameter. Here, function Ha,δ) is chosen to ensure that the overall mean squared error tends to zero as n grows: Ha,) = a and Ha,δ) is bounded for any δ>. Singh [1] proposed the use of Ha,δ) = ai a δ 1) + δ 1 Iaδ > 1) δ 1 Iaδ < 1) 2) where IA) is the indicator function of the set A. Pensky [6, 12] suggested a smooth version Ha,δ) = a1 + δ a 2τ ) s, τ = 1, 2,...; s>; 2τs 1 21) EB Estimation of the Reliability Function and the Mean Lifetime Let T 1,...,T n be observations on the lifetimes of the devices of a certain type corresponding to unobserved parameters θ 1,...,θ n and let T be a measurement with the pdf ft τ). The general ideas reviewed

4 4 Empirical Bayes Estimation of Reliability above can be used for EB estimation of the mean lifetime mτ) = tf t τ)dt 22) and the reliability function at a point t Rτ, t ) = PT t ) = = t ft τ)dt It t )f t τ)dt 23) For this purpose, one should carry out EB estimation of u 1 τ) = mτ) and u 2 τ) = Rτ, t ) where t is treated as a fixed known constant. Consider, for example, estimation of the mean lifetime and the reliability function in the case when ft θ) belongs to the family of generalized gamma distributions ft θ) = αtαν 1 θ ν Ɣν) exp t α ) It > ), θ θ 24) where α and ν are known parameters, αν 1, and the parameter space is a subset of, ), the positive real line. The family of pdfs 24) includes most of the pdfs used in reliability, for example, the exponential α = ν = 1), the gamma α = 1), and the Weibull ν = 1) pdfs. It is easy to calculate that for the family of pdfs 24) we have mτ) = τ 1/α [Ɣν)] 1 Ɣν + α 1 ) 25) Rτ, t ) = [Ɣν)] 1 γν,τ 1 t α ) 26) where γa,x)is the incomplete gamma function see [14, p. 26]). Parametric EB Consider an example of parametric EB estimation of mτ) and Rτ, t ) when gθ) is the inverted gamma prior of the form gθ) gθ σ) = [Ɣβ)] 1 σ β θ β+1) exp σ/θ)iθ > ) 27) This prior was considered by many authors, for example, Dey and Kuo [15] and Li [16]. Here, for simplicity, we assume that parameter σ is unknown while parameter β is known. In this case, the marginal pdf of T is pt ) pt σ) = α[bν, β)] 1 σ β T αν 1 T α + σ) ν+β) IT > ) 28) Hence, the MLE ˆσ of σ is the value minimizing the likelihood function n Lσ ; T 1,...,T n ) = pt i σ) 29) see Maximum Likelihood) and ˆσ is the root of the equation 1 Ti α n T α i +ˆσ = ν 3) ν + β Therefore, Bayes estimators of mτ) and Rτ, t ) are given by formulae 5) and 6) with u 1 θ) = mθ) and u 2 θ) = Rθ, t ) see formulae 25), 26), and 28)): ˆmT ) = [Ɣν)Ɣν + β)] 1 Ɣν + 1/α) ˆRT, t ) = Ɣν + β 1/α) T α + σ) 1/α 31) t αν Ɣ2ν + β) Ɣν + 1)Ɣν + β) T α + σ) ν F 2ν + β, ν; ν + 1; t α ) 32) T α + σ where Fa,b; c; z) is the hypergeometric function see [14, p. 556]). The EB estimators ˆm n T ) and ˆR n T, t ) are obtained by replacing σ with ˆσ in the expressions 31) and 32). By formula 11), the posterior pdf of τ given T is the inverted gamma gτ T, ˆσ) = T α +ˆσ) ν+β τ ν+β+1 Ɣν + β) exp T α ) +ˆσ) τ and the confidence intervals for τ form 12). Nonparametric EB 33) are of the Now, let us consider nonparametric EB estimation of mτ) and Rτ, t ). In this subsection, following [17],

5 Empirical Bayes Estimation of Reliability 5 we carry out EB estimation under the assumption that θ belongs to some interval: = [a 1,a 2 ], <a 1 < a 2 < where a 1 and a 2 are unknown. The problem reduces to estimation of ˆmT ) = T )/pt ) and ˆRT, t ) = T )/pt) where T ) and T ) are of the form 6) with u 1 θ) = mθ) and u 2 θ) = Rθ, t ) given by formulae 25) and 26), respectively. Denote EB Estimation of the Failure Intensity Function Consider a repairable system and let T = t 1,...,t m ) be its failure times. The number of failures Nt) during time interval [; t] is commonly described by the Poisson distribution Qz, ε) = [Ɣ1/α)] 1 1/α 1 z εz) 1/α 1)/2 J 1/α 1 2 ) z/ε 34) PNt) = k) = [t)] k exp t))/k!, k =, 1, 2,... 39) where J b ) is the Bessel function of integer order see [14, p. 36]). Estimate T ) and T ), respectively, by ˆ n T ) = n 1 ˆ n T, t ) = n 1 ϕ ε T, T j ), j=1 ψt,t j,t ) 35) j=1 with α αν T αν 1 j Tj α αt T α ) 1/α 1 Ɣ1/α) ϕ ε T, T j ) = IT j T), if <α<4/3 36) T αν 1 T α αν j QTj α T α,ε), if α 4/3 and ψt,t j,t ) = α αν αtj T αν 1 Tj α T α ) 2ν 1 Ɣ 2 ν) ) B T α j t α T α ; ν, ν It α T α j T α ) 37) Here BA; ν, µ) is the incomplete beta function see [14, p. 263]), ε = 2ln n ln ln n) 1. Estimate pt ) by ˆp n T ) see formula 15)) with K x) = x) 1/2 J 1 2 ) x) Ix < ) 38) with h = 2ln n ln ln n) 1. Finally, choose Ha,δ) = ai <aδ 1) + δ 1 Iaδ > 1) and estimate ˆmT ) and ˆRT, t ) by ˆm n T ) = Hˆ n T )/ ˆp n T ), δ) and ˆR n T, t ) = Hˆ n T )/ ˆp n T ), δ), respectively. Here, t) is the average number of failures during time interval [; t], and λt) = t) is the failure intensity function. It is known see, e.g., [18]) that in this situation the joint pdf of t 1,...,t m is of the form m ) ft 1,...,t m ) = λt i ) exp { t m )} 4) The main concern of a practitioner is the behavior of the failure intensity λt): whether it is increasing, decreasing, or staying constant. It is common in reliability to work with the power law processes with the intensity function of the form λt) = θβt β 1, t > 41) Here, β is the growth parameter, so that λt) is increasing if β>1, decreasing if β<1, and equal to a constant failure rate λt) λ if β = 1. The value of β can be estimated on the basis of the sample T but the estimator is not reliable when m is small. Now, consider the situation when one observes failure times of n repairable systems T i = T i1,t i2,...,t imi ), i = 1,...,n, where the ith system has an intensity function of the form equation 41) with parameters θ i, and a common parameter β. UsingEB approach reviewed above, one can use these past data for estimation of the intensity function λt) = τβt β 1 corresponding to data T = t 1,...,t m ). Following [18], we assume that θ 1,...,θ n,τ have the gamma prior pdf pθ a, b) = [Ɣa)] 1 b a θ a 1 exp bθ), θ> 42) where hyperparameters a and b are considered fixed but unknown. Hence, the marginal pdf of

6 6 Empirical Bayes Estimation of Reliability T = T 1,T 2,...,T n ) is of the form see [18], p. 223) ban pt 1,...,T n a, b, β) = β M [Ɣa)] n n n m i T ij j=1 Ɣn i + a) T β im i + b) m i+a β 1 43) where M = n m i. Parameters a, b, andβ can be found by the MLE procedure. In the general situation, MLEs of a, b, andβ cannot be expressed explicitly; however, if for all n systems the values of T imi are identical and equal to s, then the MLE â of a is the solution of the equation m i j=1 1 â + j 1 = n ln 1 + M ) nâ 44) while the MLEs ˆβ and ˆb of β and b have, respectively, the following forms 1 m i ˆβ = M lns/t ij ), ˆb = nâs ˆβ /M j=1 The posterior pdf of τ is then of the form 45) pτ T)= t m + β) m+â τ m+â 1 exp τˆb + t ˆβ Ɣm +â) m )) 46) and can be used for point or interval estimation of τ. Discussion EB estimation of reliability was carried out by a number of researchers in the last 4 years. Martz and Waller [2], Chapter 13) and Attia [19] offer a review of the techniques on the EB approach to reliability which complements this article. Parametric EB estimation of the mean lifetime and the reliability function was considered in [15, 16, 2 23] among others. Dey and Kuo [15] investigated EB estimation of the mean lifetime for Weibull distribution with type II censored data, while Li [16] treated the case when fx θ) belongs to the exponential family. Both Dey and Kuo [15] and Li [16] employed the inverted) gamma prior. Abu-Salih et al. [2] and Chiou [22] studied EB estimation of the reliability function in the case of the exponential distribution, Ahsanullah and Ahmed [21] for a special class of distribution families, Jaheen [23] in the case of the Burr type X distribution, and Padgett [24] in the case of the lognormal model. Various authors were involved in nonparametric estimation of reliability function and mean lifetime, see, for example, [17, 25 28] among others. For instance, Nakao and Liu [27], Papadopoulos [28], and Sarhan [29] estimated θ i from previous data T i = T i1,t i2,...,t imi ) and then constructed a nonparametric estimator ĝθ) of gθ) using estimators ˆθ i as observations. Lahiri and Park [25] and Liang and Padgett [26] carried out nonparametric EB estimation of the mean lifetime and the reliability function, respectively, based on Dirichlet process prior. The concept of nonparametric Bayes and EB estimators based on Dirichlet process prior is reviewed systematically in, for example, [3]; however, it is too mathematically involved to be treated in this small article. Finally, Pensky and Singh [17] used the approach based on estimating the numerator and the denominator of the Bayes estimator reviewed above. In addition, Liang and Singh [31] carried out EB testing for the value of the mean lifetime [32] and the reliability function [31]. EB estimation of the failure rate was studied in [33 36], and [18]. Gupta and Liang [33] used nonparametric EB approach for selecting the most reliable Poisson population, that is, the population with the smallest failure rate. Li [34] studied admissibility of EB rules for different types of priors. Mazzuchi and Soyer [35] estimated software failure rate when the life length of software on each stage of testing is described by exponential distribution and failure rates λ i have the gamma pdfs. Finally, Rigdon and Basu provided extensive treatment of the EB estimation of the failure intensity function in their book [18]. Acknowledgment This research was supported in part by the NSF grant DMS References [1] Berger, J.O. 1993). Statistical Decision Theory and Bayesian Analysis, 2nd Edition, Springer-Verlag, New York. [2] Martz, H.F. & Waller, R.A. 1982). Bayesian Reliability Analysis, John Wiley & Sons, Chichester.

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