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Reliability Estimation in a Generalized Life-Model With Application to the Burr-XII Ahmed A. Soliman
predictive pdf implies the Bayes estimator MLE : a real-valued, increasing function of ) ( GLM vector parameter: range space for vector of prior parameters posterior -expectation of prior pdfs posterior pdf
Abstract—Estimators of the reliability function in a GLM (generalized life model) are considered. The class of the GLM includes (among others) the Weibull, Pareto, Beta, Gompertz, and Rayleigh distribution. A proper general prior density and the predictive function for general class of distribution proposed by Al-Hussaini [1] are used to obtain the exact estimate. Also, the Bayes estimates relative to symmetric loss function (quadratic loss), and asymmetric loss function (LINEX loss, and GE loss), are obtained. Comparisons are made between those estimators and the MLE applying to the Burr-XII model using the Bayes approximation due to Lindley. Monte Carlo simulation was used. Index Terms—Bayes prediction, Burr-XII model, lifetime distributions, Lindley procedure, symmetric and asymmetric loss functions, type-2 and completely censored samples.
ACRONYMS1 BS, BL, BG BP Cdf GLM GE MLE LINEX r.h.s. pdf SE Sf
(subscripts) implies: Bayes estimates relative to: SE loss, LINEX loss, and GE loss (subscript) implies: Bayes estimate using predictive distribution cumulative distribution function generalized life model general entropy maximum likelihood estimator linear-exponential; see (18) right-hand side probability density function squared error survivor function.
shape parameter of the loss function (21)
. I. INTRODUCTION
NOTATION [Cdf, pdf] of reliability function (Sf) likelihood function : log-likelihood function number of items on test number of failed items, failure-time of failure during a test failure-time of a future failure number of future observations pdf of a future observation , Manuscript received May 13, 2000; revised April 13, 2001 and June 9, 2001. Responsible Editor: P. S. F. Yip. The author is with Makkah, Kingdom of Saudi Arabia (e-mail:
[email protected]). Publisher Item Identifier 10.1109/TR.2002.801855. 1The
singular and plural of an acronym are always spelled the same.
E
STIMATION of the reliability function of some equipment is one of the main problems of reliability theory. In most practical applications and life-test experiments, the distributions with positive domain, e.g., Weibull, Burr-XII, Pareto, Beta, and Rayleigh, are quite appropriate models. There have been many papers on estimating the reliability function of these distributions in non-Bayes as well as Bayes contexts, e.g., [12], [15], [22], [28]. It is well-known that, for Bayes estimators, the performance depends on the form of the prior distribution and the loss function assumed. So, this paper considers the choice of optimal estimators of the reliability function in a GLM, this GLM includes among others, the Weibull, exponential, Burr-XII, Pareto, Beta, Gompertz, and compound Gompertz. A proper general prior pdf and the predictive Cdf of the general class of distributions [1] are used to obtain the exact estimate of reliability function. The Bayes estimate relative to the symmetric loss function (SE loss function), and the Bayes estimates relative to the asymmetric
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loss function (LINEX loss and GE loss) are obtained. Approximate Bayes estimators using the well known Bayes approximation procedure [11] is considered. An example illustrates the proposed estimators for the Burr-XII model. Based on a Monte Carlo simulation study, those estimators are compared with the corresponding MLE by applying the Burr-XII model.
and
are given in (8). The corresponding predictive Sf
is: (11)
II. ESTIMATOR USING PREDICTIVE DISTRIBUTION The GLM Cdf, Sf, pdf respectively are [1]: (12)
(1) (2) (3) , as , can be a parameter vector. , the corresponding Sf can be written in the For any Cdf . form (2), by putting In problem such as life-testing, ordered observations are common. In that case, time and cost can be saved by stopping the experiment after ordered observations have occurred, rather than waiting for all failures. items from the GLM are placed on test Assumption #1: and the experiment is continued until failures are observed. The likelihood function for this censored sample is of the form (4)
For more details, see [1]. Under an SE loss, the Bayes estimator mission time is the posterior mean:
, of
, at a (13)
Using (7) and (9),
(14) is the reliability function of the Thus the Bayes estimator of predictive distribution [18]. For simplicity, this paper uses the predictive pdf of a first observation , which can be obtained in (9). Then, from (11), (13) and (14): by setting (15)
(5) Let
be a proper prior pdf with the form:
(16) III. BAYES ESTIMATORS
(6) Multiply (4) by (6), then (7)
Under the SE loss (quadratic) function, the usual Bayes estimator of a given function of the parameters is the posterior mean , of this function [18]. Thus, the Bayes estimator of under the SE loss, is: (17)
(8) Assumption #2: The 2 samples are -independent, and each of their corresponding random samples is obtained from a population with Cdf of (1). The predictive pdf of is [1]:
(9) the normalized constant (10)
posterior expectation with respect to the posterior function defined in (7). A. Asymmetric Loss Functions Most of Bayes-inference procedures have been developed under the usual SE loss function, which is symmetrical and gives equal importance to the losses due to overestimation and underestimation of equal magnitude. However, such a restriction might be impractical; for example, in estimating reliability and failure rate functions, an overestimate is usually much more serious than an underestimate, in this case the use of symmetrical loss function might be inappropriate [3], [5]. Reference [9] states that in the disaster of a space shuttle, the management overestimated the average life or reliability of solid fuel rocket booster. This is an example where an asymmetrical loss function might be more appropriate.
SOLIMAN: RELIABILITY ESTIMATION IN A GENERALIZED LIFE-MODEL WITH APPLICATION TO THE BURR-XII
A useful asymmetric loss (LINEX loss function) was introduced in [24], and was widely in several papers, e.g., [4], [13], [14], [21], [29]. This function rises approximately exponentially on one side of zero and approximately linearly on the other side. , the Under the assumption that the minimal loss occurs at can be expressed as: LINEX loss function for (18)
is an estimate of . The sign and magnitude of represent the direction and dethe overestimation is gree of symmetry respectively. (For more serious than underestimation, and vice-versa.) For close to 0, the LINEX loss is approximately the SE loss and therefore almost symmetric. The posterior -expectation of the LINEX loss function (18) is:
(19) posterior -expectation with respect to the posterior of under the LINEX loss pdf . The Bayes estimator function is the value which minimizes (19):
[11] via an asymptotic expansion of the ratio of 2 nontractable integrals, was chosen.2 Lindley Procedure: Reference [11] shows that the posterior -expectation of an arbitrary function : (24) can be asymptotically estimated by:
(25) ; is the prior distribution of ; ; is the logarithmic likelihood function; . . RefAll functions of the r.h.s. of (25) are to be evaluated at , the Bayes estierence [19] shows that up to the order of mate using the linear approximation (25) is more efficient than the MLE. Reference [10] states that (25) is a “very good and operational approximation for the ratio of multi-dimensional integrals.”3
(20) exists and is finite [4]. provided that Another useful asymmetric loss function is the GE loss [4]: (21) . This loss function is a generwhose minimum occurs at , alization of the entropy-loss used in several papers where , a positive error ( ) causes more e.g., [7], [8]. When serious consequences than a negative error. The Bayes estimate of under the GE loss (21) is:
B. Estimators Relative to Asymmetric Loss
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IV. APPLICATION EXAMPLE The Burr-XII model was chosen to compare the estimators of the reliability function of the GLM obtained in Section III. It has a nonmonotone hazard function which can accommodate many hazard shapes. It has been used in quality control and acceptance sampling in situations where the Gaussian distribution is not appropriate [27]; [6] used the Burr-XII model to obtain better fits to a uranium survey data-set. Reference [20] shows that the Burr-XII model has excellent properties for modeling the size distribution of incomes; [26] used the Burr-XII model to fit the life data in a medical study; [30] discusses the statistical and probabilistic properties of the Burr-XII distribution, and its relationship to other distributions used in reliability analyses. The pdf, Cdf, Sf of the 2-parameter Burr-XII distribution are:
, and is given by (2), then the Bayes If in (20), estimate of the reliability function (2) relative to the LINEX loss function (18) is:
(26) (27) (28)
(22) Combining the GE loss (21) with the posterior distribution (7), of the reliability function (2) is the value of that the : minimizes the posterior -expectation of
so that (29) The logarithm of the likelihood function (4) in this case is:
(23) provided that
exists and is finite.
C. Bayes Approximation In a failure-time distribution, such as Weibull, the Bayes estimates obtained in (15), (17), (22), (23) do not lead to a simple form for the estimators, but require numerical integration, programming, and computer time. Thus a simple approximation
(30) 2For
an English reference, see [16, pp. 73-75]
3Other approximation forms, e.g., Laplace approximation and approximation
form in [23], can be used to evaluate the ratio of integrals of the form (24); see [23].
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Differentiate (30) with respect to and , and equate each result , of , can be obtained iteratively by to 0. The MLE solving the 2 resulting equations (see, e.g., [25]). For a given , can be obtained by replacing by , and the MLE by in (28).
B. Approximate Bayes Estimates , and (25)
For the Burr-XII model, reduces to:
A. Estimate Using Predictive Distribution . Let and be unknown, and let Reference [2] suggests using the prior pdf:
(37)
is the
conjugate prior, when is known, and
for for
is the gamma Multiply by
pdf. to obtain the bivariate prior pdf:
(31)
All functions in (37) are evaluated at the MLEs of and . and SE Loss: Using the posterior distribution (7) with as in (35), then from (17) and (29), for under an SE loss is:
(32)
(38)
, are positive real numbers. Compare (6) and (31); then
in (37), then the approximate After setting: Bayes estimate in (38) becomes:
is the vector prior parameter. Also, from (5) and (29): (33)
(39) (34)
evaluated at (
), where
The corresponding posterior pdf is of the form in (7) with:
(35) From the results in Section II, the use of (35) and (29) in (15), and integrating the result over , yields the Bayes estimator of for a given time : (36)
the and can be obtained by setting (37), respectively; the details are in [2].
and
in
SOLIMAN: RELIABILITY ESTIMATION IN A GENERALIZED LIFE-MODEL WITH APPLICATION TO THE BURR-XII
LINEX Loss: Use the result in (22); substitute from (28) and (29) into (22); then for a given , obtain the Bayes estimate of relative to LINEX loss function (18):
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TABLE I ESTIMATORS FOR THE RELIABILITY FUNCTION
(40) provided that the -expectation is finite. The -expectation in (40) is evaluated as follows: , in (37); ; then, Set after some manipulations the result is:
TABLE II R
(t) AND
R
(t) WITH CORRESPONDING VALUES OF FOR t = 0:5
a
AND
q
(41)
(42) , . All functions in the r.h.s. of (41) are to be evaluated at General Entropy Loss: Use the result in (23), substitute from (28) and (29) into (23), then for a given , obtain the Bayes estimate of the reliability function relative to GE loss (21) as: (43) provided that the -expectation is finite. , in (37) where Set
; then
soring; 30 components were involved in the life test, which was censored after 20 failures. The failure times (in months) are: 0.1, 0.1, 0.2, 0.3, 0.4, 0.5, 0.5, 0.6, 0.7, 0.8, 0.9, 0.9, 1.2, 1.6, 1.8, 2.3, 2.5, 2.6, 2.9, 3.1. , and for , the estimaFor : tors of , , , with , and with ; are computed and presented in Table I. The MLEs of and used in computing are: , . D. Numerical Example #2 To study the sensitivity of the assumed values of the shape parameters and of the LINEX loss functions (18), and GE loss function (22), respectively, the same random sample of nuand merical example #1 was used. Table II summarizes at with the corresponding values of and . V. SIMULATION STUDY AND COMPARISONS
(44) evaluated at
; and
(45) C. Numerical Example #1 This example uses an example from [26] based on the failure times of certain electronic components and using type-II cen-
The expressions of the estimators show that an analytic comparison of these estimators is not possible. Therefore, a Monte Carlo simulation study was used. To compare the different estimators of the reliability function of a Burr-XII distribution obtained in this paper, random samples of various sizes are generated as follows: ), #A. For a given vector ( , from the prior pdf (31), generate as explained in [2]. (The generated values of and can be used as the true values.) from #A, generate random samples of #B. Using the , 50, 100, by observing that if is uniform sizes is Burr-XII over (0, 1), then . , , , , are then #C. computed, at some (chosen here as 0.7 and 0.9); the with and are true values of , and .
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TABLE III MEANS AND RMSES OF VARIOUS ESTIMATORS OF R(t)
TABLE IV MEANS AND RMSES OF VARIOUS ESTIMATORS OF R(t)
3) For a large sample, the performance of the Bayes estimators depends on the loss-function that is believed. For the LINEX loss function ( , designed to minimize RMSE) the Bayes estimator performed better than the others in the sense of comparing the RMSE of the estimates. As gets very large, the Bayes estimates tend to take the values of the MLE. and 4) Table II shows that the Bayes estimators relative to asymmetric loss functions (LINEX and General Entropy), respectively, are sensitive to the values of the shape parameters , . The problem of choosing the value of the shape parameter of the selected loss-function are discussed in [4]. 5) As anticipated, the difference in the RMSE for the 2 values of decreases with increasing sample (censoring) size. In most situations involving reliability estimation, the asymmetric loss functions are more appropriate than the quadratic loss functions. ACKNOWLEDGMENT The author would like to thank the associate Editor, Dr. P. S. F. Yip, and the referees for their constructive comments and suggestions. REFERENCES
#D.
Steps #A–#C are repeated 500 times. Define a Monte Carlo estimator of a function by:
true value of
The computation results for various estimators are displayed in Tables III and IV. Entries within parentheses represent the corresponding RMSE. VI. REMARKS 1) The performance of each estimator depends on the value of . The RMSE for all estimators decreases with increasing . ), from tables Tables III 2) For small and ( , (estimators based on the and IV, the exact estimates predictive distribution) perform better than both the MLE, and the Bayes estimates; also as anticipated, the MLE could be better than the Bayes estimates. The parameter uncertainty is greater for smaller sample size; this might have led to the higher RMSE for the Bayes estimators corresponding to the cases of small .
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Ahmed A. Soliman (born 1957) received his M.S. in mathematics from Assiut University, Egypt, and Ph.D. in mathematical statistics from South Valley University, Egypt. His research interests are in statistical estimation and prediction.
