BAYES ESTIMATORS OF EXPONENTIAL PARAMETERS FROM A CENSORED SAMPLE USING A GUESSED ESTIMATE

This paper provides the Bayes estimators of the failure rate and reliability function for a one-parameter, exponential distribution by utilizing a point guess estimate of the parameter. For deriving the Bayes estimators, the prior distributions are chosen such that they are centered at the known prior values of parameters. The validity of proposed estimators is examined with respect to their maximum likelihood estimators (MLE) and Thompson's Shrinkage estimator on the basis of Monte Carlo simulations of 1000 samples.

the mass k according to some distribution ( ) g θ .Such a prior has a point of discontinuity at 0 θ .Moreover, a sudden change in the confidence for a value close to 0 θ seems to be unrealistic.Another Bayesian estimation procedure as an alternative to the shrinkage estimation procedure that utilizes the point guess 0 θ has been suggested by Pandey and Srivastava (1985) and Pandey and Upadhyay (1988).Their technique chooses a subfamily from a family of priors such that the mean of the prior distribution is equal to 0 θ .Equalization of the mean to 0 θ , however, does not seem appropriate unless the prior guess 0 θ is specified as the average value of θ .As mentioned above, prior to the sample selection, the experimenter believes that the point guess 0 θ is a possible value of the parameter.Thus, it seems more appropriate to interpret 0 θ as the most probable value rather than interpreting it as the average value of θ .Therefore, we propose that the choice of subfamily of the prior be made by equalizing the mode of the prior distribution to 0 θ .As an illustration of our proposition, we have considered the problem of estimation of the failure rate and reliability for a one-parameter exponential distribution.
The exponential distribution is a widely used model in a variety of statistical procedures.Among its most prominent applications are those in the field of life testing and reliability problems.
Data Science Journal, Volume 7, 20 October 2008 The probability density function (pdf) of one parameter exponential distribution is ( ) where λ is the failure rate.The reliability function R(t) which is probability of survival for at least time t is given by . (2) The Bayesian and classical inferences regarding parameter λ and R(t) have been considered by many authors (Martz & Waller, 1982;Sinha, 1985) ).In the next section, we obtain the Bayes estimator of λ and R(t) under a quadratic loss function when prior distribution is chosen as mentioned above.In Section 3, performance of the proposed estimators is compared with the corresponding MLE and shrinkage estimators on the basis of a Monte Carlo simulation study.

ESTIMATION OF EXPONENTIAL PARAMETERS
We assume that n items are tested and the experiment is continued until the first r failures are observed (type II censoring).If denotes the first r ordered observations, then the likelihood function is given by Where and is interpreted as 'total observed failure times.'

Estimation of λ
It is well known that the MLE of is Suppose that prior to the sample information X , some a priori information including a point guess regarding is also available.In such a situation, one may be willing to use a shrinkage estimator which is defined below: where (1-k) is the confidence in as prespecified by experimenter.The estimator performs better than if is close to , but in other situations it be may worse than .The shrinkage estimator utilizes only a single piece of prior information in the form of a point guess, which may always be questionable.Perhaps a better and more easily justifiable way of utilizing the prior information would be by the Bayes method of selecting a family of priors that can describe a variety of information.One such family of priors is the natural conjugate prior (see Raiffa & Schlair, 1961).The natural conjugate prior for is c, a>o, λ>o The mode of this distribution is at a/(c-1).In the presence of point guess about , a subfamily of (6) may be obtained by choosing a and c such that , i.e., 0 λ λ The posterior distribution of λ can easily be obtained by combining (3) and ( 7) by the Bayes rule as Hence the Bayes estimator of λ , denoted by under a quadratic loss function, which is the mean of the

Estimation of R(t)
In this subsection, we obtain the Bayes estimator of the reliability function ( ) -λt R t =e , which is the probability that a unit will survive until a specified time t.If we make the transformation where R=R(t), the density and likelihood function, given in ( 1) and ( 3) respectively, can be written in terms of R as A flexible and rich prior (in the sense of being capable of describing a variety of information) is the Beta Prior (see Martz & Waller, 1985;Canfield, 1970).If the point guess regarding the reliability at time t is R 0 , and equating it to the mode of Beta Prior, we get a subfamily of priors as ( ) Data Science Journal, Volume 7, 20 October 2008 The posterior distribution of R given X can be obtain from the following Substituting ( L X/R)and g(R) from ( 11) and ( 12) in ( 13) and simplifying under the assumption that 0 Under the quadratic loss function, the Bayes estimator of R, R P , that is the mean of ( 14) is The The shrinkage estimator of R, , with confidence (1-k) in R R S 0 can be defined as

COMPARISON OF ESTIMATORS
In the previous section, Bayes estimators of λ and R(t) have been obtained as alternative estimators to shrinkage estimators, utilizing the knowledge of a point guess.Now the question of comparison of the proposed estimators with the corresponding shrinkage and MLE estimators arises.For comparison, we propose the risk criterion, which is well accepted by non Bayesians.It is well known that the risk of an estimator T 1 of parameter is defined as ( ) ( ) where the expectation is taken over the whole sample space.Naturally, T 1 will be superior to another estimator 2 ) > R( , T θ 1 ), or in other words, we can say the risk efficiency RE(T 1 , T 2 ) > 1, where Unfortunately, in the present case, we cannot find RE , RE , RE , or RE in a nice closed form, making the analytical comparison impossible.Therefore, we obtain these risk efficiencies on the basis of simulated data using a Monte Carlo technique.

Comparison of estimators of R(t)
The proposed estimator of R(t) and corresponding MLE and shrinkage estimator are given in ( 15), ( 16), and (17) respectively.To compare with and , the risk efficiencies RE and are obtained on the basis of simulated samples of size = 10 for different values of R from ( 10).
The values of R have been chosen such that R/R ⎞ ⎟ 0 = 0.50(0.50)3.00where R 0 was prefixed at 0.18.Similarly the values of r were taken such that r/n=0.20(0.20)0.80.The variation in the prior constant was taken as =3.00 (3.00)12.00with k=0.25(0.25)0.75.The results are summarized in Moreover, the range of R/R 0 , for which risk efficiency is greater than one, increases as k and increase except in a few cases when k=0.50.It may also be seen that an increase in the value of k increases RE if .00.The initial increase in the value of R/R o results in an increase in the risk efficiency, but at some values of R/R o, which is small for large values of , a further increase causes reduction in risk efficiency.
Increases in r/n do not have a uniform effect on risk efficiency.However, it may be seen that for large k and β 0 β 0 namely, k=0.75 and β 0 =12, an increase in r/n decreases the risk efficiency.

CONCLUSION
We have seen that the proposed estimator ^P λ performs better than MLE over a wide range of parameter space.
Thus, ^p λ may be recommended as an alternative estimator against MLE with large c ( )

≥
, provided λ is not very small as compared to λ 0 .It has also been observed that ^P λ is superior to the shrinkage estimator ^s λ for k=0.75 for large values of c.For small k, i.e., k=0.25 and k=0.50, however, the proposed estimator ^p λ may still be preferred to ^s λ if the true value of λ is greater than the guessed value λ 0 .In brief we may conclude that the proposed estimator can be safely used with large values of c when the guessed estimate is an underestimate of true value.It has also been seen that the proposed estimator of reliability is always more efficient than the corresponding MLE if estimate R 0 is less than the true value of parameter R=R(t). On For generation of data, different values of λ have been considered such that λ / λ 0 =0.50, 1.00, 1.50, 3.00 and 4.00 where λ 0 =2.00.The values of r were chosen in each case such that the sample fraction r/n=0.20(0.20)0.80.The various values of c and k considered here are c=1.50,3.00,6.00 and k=0.25, 0.50, 0.75.The results thus obtained are compiled in Table1.
(1).It may be noted that risk efficiency depends on λ, λ 0 , c, k, and r besides n, which has fixed to 10 in the present study./ λ 0 .On the other hand, increases in the value of r/n decrease the risk efficiencies for small values of λ / λ 0 and c.

Table 2 .
generally reduce as r/n increases.It may also be seen that as R/R 0 increases from 0.50, risk efficiency increases in the beginning but then starts decreasing with further increases in R/R 0 .For an increase in the value of the A comparison of the proposed estimator with the shrinkage estimator shows that RE

Table 1 .
Risk efficiency of

Table 2 .
Risk efficiency of the other hand, comparing the estimator ^p R with shrinkage estimator ^s R , it is noticed that if R 0 is an underestimate of its true value, with the risk of ^p R generally less than that of ^s R for k=0.75, the region of gain increases for large values of β 0 .However, if k=0.25 and 0.75, RE if R/R R may be recommended for use with large only if R 0 β 0 is expected to be smaller than its true value.