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Bayes Estimators of Reliability Function of Exponential Distribution Under Different Prior Distributions

Dev Singh1 and J. S. Chaudhary2

1Aryabhatt College of Engineering and Technology, Daula, Baghpat 2Ganjdundwara P G College, Ganjdundwara

ABSTRACT: Bayesian analysis of the exponential model, based on life tests that are terminated at pre assigned time points or after pre assigned no. of failures, has been developed. For the prior distribution of the parameter, uniform, inverted gamma and exponential densities have been considered. The role of prior quasi-densities when a life tester has no prior information has been illustrated and it has been observed that the reliability estimate for a diffuse prior which is uniform over the entire positive real line closely resembles the classical MVU estimate obtained. Bayesian estimation of parameters of a double exponential distribution under different priors has also been developed in this paper.

KEYWORDS: exponential distribution; reliability; Bayesian estimates; Prior-quasi density; Posterior density; Posterior estimates

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Introduction

In the theory of reliability and life testing analysis the exponential distribution has a great importance. The main features of the utility of the distribution are its positive ranges its property of forgetfulness and the graphic shape of the distribution. It has a mirror shape J when plotted on the range of random variable x , at Χ-axis with respect to its probability density function at Υ-axis. The shape of the distribution supports the idea of life that is in the beginning when the product is recently created the probability of its life is highest and as the time (the random variable x ) increase the probability of its life decreases.

Exponential distribution is second commonly used distribution in the theory of Statistics after the Normal distribution but it is first in the applications of reliability and life testing analysis, it is used more than the normal distribution.

A modification of the well-known exponential life testing model:

Formula

 

when the parameter Θ is itself a random variable distributed uniformly over a finite range has recently been considered by Bhattacharya and Holla (1965). An interpretation of this uniform distribution in terms of a prior density for the parameter will be of direct interest for a Bayesian analysis. In the sequel, the term Bayesian refers to any use or user of prior densities on a parameter space with the associated application of Bayes’s theorem in the analysis of a statistical problem. Such an analysis rests on the belief that in most practical situations, the mathematician will possess some subjective a priori information concerning the probable values of the parameter.

This information may often be reasonably summarized and made objective by the choice of a suitable prior distribution on the parameter space. Another reason that is usually advanced in support of the use of prior densities is the ‘admissibility’ of the Bayes estimates as discussed by Karlin and Rubin (1956) under the general set up of densities with monotone likelihood ratio, of which (1) is, of course, a member. The object of the present paper is to exploit the use of suitable prior information in the framework of life testing techniques that are currently used in practice. A systematic development of the subject of life testing originated from the work of Epstein and Sobel (1953), and the subsequent progress made in the field can be gauged from the two workers, Mendenhall (1958) and Govindarajulu (1964). To date, the most commonly employed procedures are those in which a life test is terminated either after a pre assigned number of failures have taken place or at a pre assigned time point, and the statistical inferences are based on a statistic representing the accumulated life.

Prior Distributions

These situations have been examined in detail here in the light of prior information about the parameter Θ, that is available to the life tester from his past experience. It frequently happens that the experimenter knows in advance that the probable values of the parameter lie over a finite range  [α,β] but he does not have any strong prior opinions about any subject of values over this range. In such a case, a uniform distribution of  over the range mentioned above my serve as a nice approximation to the prior density representing the experimenter’s beliefs. More generally, we shall consider a prior density:

Formula

 

For the case α=0, equation (2) reduces to a uniform density on [α,β] . The reasons for considering the general form (2) will be apparent in due course. The problem will be examined for two other prior densities, namely the exponential density:

Formula

 

and the inverted gamma density of Ralffa and Schalfer [16]

Formula

 

By suitably adjusting the parameters involved, one may be able to reasonably express the decision-makers prior judgments in terms of these prior densities. An excellent account of the choice of prior densities on the parameter space that are capable of summarizing experimenter’s prior beliefs has been given by Raiffa and Schlaifer(1961) and may be consulted for the details. The prior density (3) for the parameter involved in (1) gives rise to a Bessel-function model in life testing and has recently been discussed by Bhattacharya (1967). The reasons for considering the inverted gamma prior density (4) are that it is flexible enough to capture almost any kind of prior experience, and it also possesses the attractive property that the posterior distribution of the parameter after the sample has been observed is also of the inverted gamma type. A family of prior densities which gives rise to posteriors belonging to the same family is very useful in as much as the mathematical tractability is maintained, and this ‘nice’ property has been termed ‘closure under sampling’ by Wetherill(1961). For densities which admit sufficient statistics of fixed dimensionality, Raiffa and Schlaifer(1961) have considered a method of generating prior densities on the parameter space that possess this desirable property. A family of such densities has been called by them a ‘natural conjugate family’, and for the exponential density (1), the inverted gamma prior forms such a family. Under the assumption of the prior densities mentioned above, reliability estimation will also be considered by using the Bayesian approach and a situation in which the times to failure of the test items are not available Ehrenfeld (1962), Bhattacharya (1962) will be discussed. A difficulty arises when the prior information available to the experimenter is rather ‘vague’, or worse still, there is no prior information whatever. What does a Bayesian life tester do in such a situation? Can he objectively assign prior densities on the parameter space in such a state of ‘ignorance’? This has led to the consideration of what are known as ‘prior quasi-densities’ which assign infinite measures on the parameter space. The use and justification of such densities in the framework of life testing problem will be discussed in some detail, and it will be noticed that a prior quasi-density of the form 1/θ2  leads to an estimate which identifies with the classical minimum variance un–biased (MVU) estimate of Θ, Epstein and sobel(1953) based on the first  ordered observations. The classical result of Pugh(1963) for the MVU estimate of the reliability function will be shown to closely resemble the Bayes estimate obtained under the assumption of a uniform prior Θ for  over the entire positive real line.

Bayesian Estimation of Reliability Function When Uniform Distribution is Taken as a Prior Distribution

In this section we shall consider the case when the prior density of  involved in (1) is restricted to a finite range [α,β]  as specified by the family (2). We assume that  items from the population (1) are placed on a life test and the experiment is continued till the first  r failures are observed  x1,x2,x3…..,xr If  denote the first r ordered observations, then the sample likelihood conditional on Θ is given by:

Formula

 

where

Formula

 

Under the assumption of the prior density (2), the posterior distribution of  after observed the first  ordered values in a sample of size , can be obtained by using Bayes’s theorem as;:

Formula

 

which is a truncated ‘inverted gamma density’.

The Bayes estimate of the mean life is, therefore, provided by the posterior mean:

Formula

where

Formula

 

defines an incomplete gamma function for n>0, the condition being tacitly assumed in the sequel. The variance of the estimate obtained at (8) works out to be:

Formula

 

Where, for brevity, we write

Formula

 

For the case in which the complete sample (x1,x2,x3…..xn). is available, the results follow from the preceding ones by a mere substitution r=n , and  Tr then reduces to the sample total nx . If, however, a life test is terminated at a pre-assigned time point t , the number of failures during that period would be a random variable. If r be the observed value in a sample and if the corresponding life-times be x1,x2,x3…..xr , then the conditional likelihood (for a given Θ ) of such a sample has a kernel

Formula

 

where,

Formula

 

Under the assumption of a prior density (2), the posterior distribution of Θ turns out to be the same as (7) except that Tr has now to be replaced by T’r. Thus, the results for a life test terminated at a pre-assigned time point follow from the preceding ones and in the sequel therefore, we shall restrict our–selves to the case of first  ordered observations only. Note that in either case the statistic (Tr  or T’r )  denotes the accumulated life at the termination of the experiment.

A mere substitution α=0 in the expressions given above leads to corresponding results for the uniform prior density over [α,β] to which we now confine our attention, and obtain the Bayesian estimate of the reliability function

Formula

 

Reliability problems have currently drawn the attention of various research workers in the field and the classical minimum variance unbiased (MVU) estimate of the reliability function for an exponential distribution has been obtained by Pugh(1963) based on a complete sample. The Bayesian method considered here applies to a life test that is terminated after the first  r failures out of a sample of n items. Under the assumption of a uniform prior over [α,β] , the prior distribution of the reliability function at (14) is given by a probability density:

Formula

 

However, when a life test is terminated and the statistic Tr is observed, the posterior distribution of Θ computed at (7) for α=0 is used to obtain the posterior distribution of the reliability function. Thus, for values of  Z in the range stated at (15), the cdf of the posterior distribution of the reliability function is given by:

Formula

 

Now, a differentiation with respect to z yields the probability density:

Formula

 

The posterior mean of this distribution provides the Bayes’ estimate of reliability as:

Formula

 

and the variance of this estimate works out to be:

Formula

 

These results will be used later to obtain estimates for situations when the prior knowledge about Θ is slight or none whatever. Such results provide interesting comparisons with corresponding results of the classical theory.

Bayesian Estimation of Reliability Function when Inverted Gamma Distribution is Taken as a Prior Distribution

First the ‘natural conjugate family’ of prior densities has been considered for the parameter . This is an inverted gamma family as given by (4). Under this assumption, one can use the sample likelihood conditional on Θ stated at (5) to obtain the posterior distribution:

Formula

 

which, is again an inverted gamma density with parameters (μ+Tr) (r+v )and thus, the inverted gamma prior distribution is ‘closed under sampling’. The Bayes estimate of the mean life and its variance are, therefore, provided by:

Formula

 

As regards the reliability estimation, the posterior distribution of  given at (20) is used to obtain the Bayes estimate.

Formula

and the variance of this estimate works out as:

Formula

 

When the prior distribution of Θ is given by an exponential density (3), we obtain, by virtue of Bayes’s theorem applied to (5), the posterior distribution:

Formula

 

The denominator in the last expression is evaluated by using an integral representation of (z) , the modified Bessel function of the third kind of order v , Erdelyi et al. (1953) and subsequent use of the same formula in conjunction with the fact that:

yields for the Bayes estimate:

Formula

 

The Variance of this estimate is evaluated as;:

Formula

 

Based on the posterior distribution (25), The Bayes estimate of the reliability function is obtained as:

Formula

and the variance of this estimate comes out as:

 

Formula

 

Bayesian Estimates Without the Information of Prior Distribution

In this section the situation where the experimenter does not have any prior knowledge about the parameter has been considered. The problem confronting a Bayesian life tester is then the choice of suitable prior densities on the parameter space. In such a case neither can he make any prior guess about the parameter values lying over a finite subset of the infinite parameter space nor can he straightaway proceed to choose a suitable prior from the ‘natural conjugate’ family or some other family of densities. One customary way out of the dilemma is to use a constant density g(Θ)=1 over the positive real line 0<Θ< ∞. The uniform prior over the entire real line has frequently been used by Jeffrey (1961) and has also been a subject of much controversy amongst the Bayesian theoreticians for various reasons besides the one that it cannot be interpreted as a probability density in the usual fashion. Clearly, a uniform density assigns infinite measure usual fashion. Clearly, a uniform density assigns infinite measure on the parameter space. More generally, if g(Θ) is any non- negative function defined on the parameter space Θ∈ Ω  such that g(Θ)≠0 , then g(Θ)  is called a “prior quasi-density.” Here, the integral ∫g(Θ may or may not converge. Such densities have been used in recent years by Bayesian theoreticians and Wallace (1959) calls a prior quasi-density g(Θ)  “admissible” with respect to a density f( x : Θ ) defined for x ∈ X if:

Formula

 

almost everywhere on X. For every prior quasi-density  g(Θ) which is admissible with respect to f( x : Θ ), there is defined a density on g(Θ) as follows;

Formula

 

If g(θ)  or some constant multiple of g(θ) is a proper prior density, then by Bayes’s theorem, g*  in (31) is clearly a posterior density for θ give x. If, however, ∫g(ΘdΘ =∞, then g(θ)  is merely a formal substitution in the Bayes’s theorem on probability stated in (31). Strictly speaking, the theorem does not apply anymore but  is still a g* proper probability density on Ω , and thus, an enthusiastic Bayesian may proceed undeterred with his usual analysis by taking g* as a posterior density. To distinguish, however, Wallace (1959) calls such as g* a “week posterior density.” A life tester facing problems of applied nature need not bother himself about this terminology but certainly one has a right to seek some justification of the use of densities which assign infinite measures on the parameter spaces for one’s own aesthetic satisfaction. To provide one, if on a subset ,

vol-1no-1_baye_dev_for4

then we can interpret

vol-1no-1_baye_dev_for5

as a conditional density given that θ ∈ Ω1 and clearly, this density defined on  Ω1 is normalized. The special case g(θ)= 1 that is, uniform prior over the real line has earlier been interpreted in this manner by Lindley (1965). Various other justifications for the use of prior quasi-densities have also been given. If over some important subset Ω1 of parameter points where the likelihood is effective, the quasi-density  resembles the actual prior, the fact that g(θ)  is not integrable over Ω-  Ω1 , may not be important. Or if g(θ) leads to posteriors which provide Bayesian inferences that closely resemble those of classical inference, then such a density may be used as a satisfactory approximation to the unknown prior distribution. Further theoretical justifications have been provided in recent literature by Bayesian statisticians and the reader interested in theory may consult Stone (1958) and Wallace (1959) for the details. The object here is neither to focus our attention on the controversies raised by the uniform prior of Jeffreys nor to make propaganda in favor of the use of prior quasi-densities but to merely explore the consequences of using such densities in the context of life testing problems. We hope some of the results may be useful for applied workers too. Thus, for situations where the life tester has no prior information about the parameter, we may use the uniform prior or more generally, we examine the quasi-density:

Formula

 

Under the assumption of this prior quasi-density, the posterior density of Θ after having observed the sample of first r ordered observations out of a sample of size n , turns out to be an inverted gamma density

Formula

 

Thus, a comparison with (7) immediately reveals that the results for this case can be obtained from earlier results by simply letting α→o and β→∞. Under these limiting conditions the results at (8) and (10) yield for the Bayes estimate and its variance respectively:

Formula

The substitution  α=0 above yields the corresponding results for the diffuse prior that is uniform over the positive real line. Further, it is evident that only for α=2, (34) reduces to the well-known MVU estimate Tr /r, Epstein and Sobel (1953). Thus, for the family of quasi-densities considered at (32), the Bayes solution is identical with classical result only when the parameter  is assumed to have a prior quasi-density 1/Θ2  over the positive real line. As regards the estimates of the reliability function when the life has no prior information, we let α→o and β→∞ and  in (22) and (23). Thus, a uniform prior over the positive real line leads to the following expressions:

 

Formula

 

When the complete sample is available, the reliability estimate obtained here reduces to:

vol-1no-1_baye_dev_for8

That closely resembles the classical MVU estimate:

vol-1no-1_baye_dev_for9

Given by Pugh (1963) if

vol-1no-1_baye_dev_for10

For the numerical example considered by Pugh, the Bayes estimate of the reliability function and its variance work out to be  0.8387 and 0.0069  respectively, whereas Pugh obtains a value 0.89 for the reliability estimate but he has not evaluated any expression for the precision of his estimate. One may note that Pugh’s result as well as ours is valid for  only  n>1.

Strictly speaking, Pugh’s result holds for 0< t < nx̄

and for, t ≥ nx̄

vol-1no-1_baye_dev_for11

That is, the classical method does not provide any information about the reliability function for large values of the time parameter. However, this is not the case with the Bayesian estimate, and the result holds for all t > 0.

The estimating function

vol-1no-1_baye_dev_for12

is a monotonically decreasing function of t with R̂(0)=1 and  R̂(t)→0 as t →∞ . This is a desirable property.

Similar results may be arrived at under suitable limiting conditions from the expressions corresponding to the exponential and the inverted gamma prior densities. Thus, letting  μ → 0 and  v→ 1  in (21), (22), (23) and (24) we obtain estimates corresponding to the prior quasi-density 1/Θ2 . On the other hand, for the exponential prior (3) if we let , λ → ∞ the results correspond to those for a diffuse prior that is uniform over (o,∞ ). Under the limiting conditions stated above the limiting behavior of the prior densities has been thoroughly discussed by Raiffa and Schlaifer (1961) and we do not reproduce them here. The estimate of the reliability function based on attribute life tests may be obtained under the assumption of a uniform prior over the positive real line by letting α→o and β→∞   Thus,

Formula

 

The variance of this estimate is obtained by subtracting from

Formula

 

The square of the expression at (38).

That comes out as:

Formula

 

Bayes Estimate of Both Parameters in a Two Parameter Exponential Distribtuion

Exponential distribution is a widely used lifetime distribution which has appeared in the literature since the early 1800s. This distribution is one of the commonly used statistical distributions in practice. Epstein and Sobel (1953, 1954, 1955), Epstein (1954), Bartholomew (1957), Mendenhall (1958), Johnson, Kotz and Balakrishnan (1994, 1995), Lawless (2003) and others have discussed this distribution with applications.

As given in Sinha (1986), the pdf of two parameter exponential distribution is given by:

Formula

 

If y1,y2………yn be individual observations from an exponential distribution, then the likelihood function is given by

 

Formula

 

Where y1 is the first order statistic in the sample  such that Y=(Y1,Y2,…….Yn)Formula

Jeffrey (1961) and others make extensive use of improper prior pdf’s. A more general class of priors has been considered here as,

Formula

 

Posterior density for μ and θ :

According to Bayes theorem, we have

Since, Posterior density ∝ (Prior density *likelihood)

That is

Formula

From the equations (42) and (43), the posterior density of μ and θ is given by

Formula

Where K is normalizing constant and is given by

 

Þ

Formula

 

Marginal Posterior densities for μ and  θ

The marginal posterior density of μ is given by

Formula

The marginal posterior density of θ is given by

 

Formula

BAYES ESTIMATES FOR μ AND θ

The posterior estimate of μ is given by

 

Formula

 

The posterior estimates of θ is given by

Formula

 

Numerical and Graphical Results

Nineteen military personnel carriers failed in services for one reason or the other at the following mileages: 162, 200, 271, 302, 393, 508, 539, 629, 706, 777, 884, 1008, 1101, 1182, 1463, 1603, 1984, 2355 and 2880 miles. Posterior estimates μ of  and θ are given in Table-I. The graphical representation for marginal posterior densities of μ and  are θ shown in figures 1 and 2 respectively.

Table 1: Posterior estimates of μ and θ under different priors

Table 1
Click on image to enlarge

 

Figure 1

Figure 1
Click on image to enlarge

 

Figure 2

Figure 2
Click on image to enlarge

 

The posteriors of μ and θ are plotted in figures 1 & 2 respectively. The posteriors μ are quite robust for varying  in the prior P

(μ, θ) ∞ (1/θc) while the posteriors of θ are less robust.

References

  1. Bartholomew, D.J. (1957): “A problem in life testing”, JASA, Vol. 52, 350-355.
  2. Bhattacharya, S.K. and Holla, M. S. (1965): “On a life test distribution with stochastic deviations in the mean”. Ann. Stat. Math. 17, 97-104.
  3. Bhattacharya, S.K. (1962) : “On a probit analogue used in a life test based on Weibull distribution,” Aust. Jour. Stat. 4, 101-5.
  4. Bhattacharya, S.K. (1967) : “Bayesian approach to life testing and reliability estimation”, JASA, vol. 62, 48 -62.
  5. Donato, M C and E, Mellilli (2000) : “Some new results for Dirichelt priors”, The Annals of Statistics, Vol. 28, No. 5.
  6. Ehrenfeld, S.(1962) : “Some experimental design problems in life testing,” JASA, 57, 668-79.
  7. Epstein, B. and Sobel, M. (1953) : “Life testing,” JASA, 48, 486-502.
  8. Erdelyi, A. et al(1953): “Higher transcendental functions,” Vol. II, McGraw-Hill Book Company, Inc. N. Y.
  9. Govindarajulu, Z.(1964) : “A supplement to Mendenhalls bibliography on life testing and related topics,” JASA, 59, 1231-91.
  10. Jeffrey, H. (1961): “Theory of Probability”. Oxford University Press.
  11. Johnson, N.L., Kotz, S., and Balakrishnan, N. (1994) : “Continuous Univariate Distributions”, Vol. 1. John Wiley & Sons, New York.
  12. Johnson, N.L., Kotz, S., and Balakrishnan, N. (1995):” Continuous Univariate Distributions”, Vol. 2. John Wiley & Sons, New York.
  13. Karlin, S. and Rubin, H.(1956):”The theory of decision procedures for distributions with monotone likelihood ratio,” Ann. Math. Stat. 27, 272-99.
  14. Lawless, J.F. (2003) :” Statistical Models and Methods for Lifetime Data”, John Willey and Sons Inc., NY.
  15. Lindley, D. V.(1965) : “Introduction to probability and Statistics from a Bayesian view- point,” Part II, Inference, Cambridge, University Press.
  16. Mathew Stephens (2000) : “Bayesian analysis of mixture models with an unknown number of components – an alternatives to reversible jump methods”, the Annals of Statistics, Vol. 28, No.1, 40-74.
  17. Mendenhall, W. (1958): “A bibiliography on life testing and related topics”, Biometrika 45, 521-543.
  18. Pugh, E. L.(1963): “The best estimate of reliability in the exponential case” Opeartions Research (U.S.A.), 11, 57-61.
  19. Raiffa, H. and Schlaifer, R. (1961) : “Applied statistical decision theory”, Harvard University Press.
  20. Sinha, S.K.(1986) : “Reliability and life testing,” Willey Eastern Limited New Delhi.
  21. Stone, M. (1965) : “Right haar measure for convergence in probability to quasi-posterior distributions,” Ann. Math. Stat., 36, 440-53.
  22. Wallace, D. L.(1959): “Conditional confidence level properties,” Ann. Math. Stats. 30, 864-76.
  23. Wetherill, G.B.(1961) : “Bayesian sequential analysis”, Biometrika 48, 281-92.
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