Shrinkage Estimators for the Exponential Scale Parameter under Multiply Type II Censoring

Abstract: We consider the problem of estimating the scale parameter of an exponential distribution under multiply type II censoring when a prior point guess of the parameter value is available. Shrinkage estimators are obtained from the approximate maximum likelihood estimators proposed in Singh et al. (2004) and in Balasubramanian and Balakrishnan (1992). These estimators are then compared by their simulated mean squared errors.


Introduction
In life testing experiments a fixed number of items, say n, is often put on test simultaneously.But the experimenter may not always be in a position to observe the life times of all these items because of time limitations or other restrictions on the data collection process.Let us suppose that out of the n items only the first l life-times have been observed and the life-times of the other (n − l) components remain unobserved and are missing.This type of censoring is known as right type II censoring.Another way to get censored data is to observe only the largest m life times.In this case the life times of the first (n−m) components are missing.Such censoring is known as a left type II censoring scheme (see Leemis and Shih, 1989).Moreover, if left and right censoring happen together, this is known as doubly type II censoring (see Sarhan and Greenberg, 1957).A reverse situation to doubly type II censoring is mid censoring, where the data on two extremes are available but some of the middle observations are censored (see Sarhan and Greenberg, 1962).If mid censoring arises amongst doubly censored observations, the scheme is known as a multiply type II censoring scheme.Balakrishnan (1990) has discussed a more general version of such a multiply type II censoring, where only the r 1 th, r 2 th, . .., r k th (1 ≤ r 1 < • • • < r k ≤ n) failure times are available.Under this multiply type II censoring scheme even the likelihood estimate for the one parameter exponential distribution is difficult to obtain directly from the likelihood equation.Balasubramanian and Balakrishnan (1992) and Singh et al. (2004) proposed some approximate maximum likelihood estimators, which are denoted as θBL and θUA , respectively.
In real world situations, particularly in life testing problems, the experimenter may have evidence that the value of the parameter under study, say θ, is in the neighborhood of θ 0 .We call θ 0 the experimenter's prior point guess.For example, for a patient suffering from cancer the doctor may believe that the patient will survive two more months.In this case θ 0 can be taken to be equal to two months.Similarly, a bulb producer may know that the average life time of his product may be close to 900 hours.Here we may take θ 0 = 900.Now the following questions arise: "Should we use θ 0 in the estimation procedure, which may be a close guess of θ but may not be its true value?", or "Should we base our estimator on sample information only?"Furthermore, if one wishes to incorporate the additional information θ 0 in the estimation of θ the question may be "How to use it?" The purpose of this paper is to study the procedures, which answer the above questions in order to estimate the scale parameter of an exponential distribution under a multiply type II censoring scheme.It may be recalled that Thompson (1968) was the first who proposed a procedure popularly known as shrinkage procedure, which suggests the use of a prior point guess of the parameter for improving the performance of the existing estimator θ.If a prior point guess θ 0 is available with known confidence α, 0 < α < 1, the shrinkage estimator for θ is defined as (1) Using Thompson's technique, the respective shrinkage estimators based on the approximate maximum likelihood estimators θUA and θBL can easily be defined.Studies of such types of other estimators reveal that these perform better than the original estimators provided the true value of θ is close to θ 0 and α is taken to be large.It is also noted that the performance of these estimators strongly depends on the choice of α.If α is not set in accordance with the reality (i.e., large α when θ is close to θ 0 , and small α when θ is away from θ 0 ), it may happen that either there is no significant gain in the performance of T SH or there is actually a significant loss.In general, the true value of the parameter is unknown and, hence, a proper choice of α can not be guaranteed.Therefore, in the situations when the experimenter is either not able to provide a fixed value of α or it is feared that the value of α may not be in accordance with the real situation, it may be proposed to consider (1) as a class of estimators and select the best by choosing α such that the mean squared error (MSE) of T SH is at its minimum.It is easy to verify that the optimum value of α for which MSE(T SH ) is minimized, is It is clear that α opt depends on θ.It is therefore suggested to replace θ in (2) by its estimate, giving αopt .Needless to mention that due to the use of θ in α opt , the performance of the shrinkage estimator is expected to be adversely affected.
Comparisons of the performance of shrinkage estimators with the usual estimators are quite common in the existing literature.But the present paper discusses for the first time the effect of the use of different estimators on the corresponding shrinkage estimators.Comparing MSEs, it will be seen that shrinkage estimators based on the approximate likelihood estimator proposed by Singh et al. (2004) perform better than the one based on results in Balasubramanian and Balakrishnan (1992).Singh et al. (2004) proposed a procedure to obtain an approximate maximum likelihood estimator as an alternative to the one given in Balasubramanian and Balakrishnan (1992).The present paper aims to develop the shrinkage estimators from these approximate maximum likelihood estimators and compare their performances.
In the next section we obtain the shrinkage estimators for θ using a prior point guess.In Section 3 the proposed estimators are computed for the data given in Balasubramanian and Balakrishnan (1992) in order to illustrate the procedure discussed here.The MSEs of all estimators are then compared in Section 5. Finally, a brief conclusion is given.

Shrinkage Estimation
Consider a one parameter exponential distribution with pdf Suppose that n items, whose life-times follow model ( 3), are placed on test and that the r 1 th, r 2 th, . .., r k th failure times are recorded as x 1 , . . ., x k , respectively.The likelihood function for such a multiply type II censored sample is The approximate likelihood estimator of θ proposed by Singh et al. (2004) is whereas the one proposed by Balasubramanian and Balakrishnan (1992) is

Specified Confidence
As discussed earlier, the shrinkage estimators θUA(α) or θBL(α) can be defined by replacing θ in (1) by θUA or θBL .Their MSEs can be easily obtained from after replacing θ by θUA and θBL , respectively.Expressions for bias( θUA ), bias( θBL ), MSE( θUA ), and MSE( θBL ) can also be easily obtained by using where The terms δ i and γ i , i = 1, . . ., k have been already defined.

Unspecified Confidence
As suggested in Section 1, the shrinkage estimator based on θUA when point guess θ 0 is available with unspecified confidence can be obtained as where α U A , as defined in (2), can be rewritten as It may be noted from (5) that MSE( θUA )/θ 2 and bias( θUA )/θ are independent of θ but α U A still depends on θ due to the term θ 0 /θ, which can be estimated by θ 0 / θUA in (5) giving its estimated value αUA .Substituting αUA in place of α U A , we get the shrinkage estimator based on θUA when a point guess is given with unspecified confidence.This may be written as, Similarly, the shrinkage estimator based on θBL may be obtained from ( 6), after replacing θUA and αUA by θBL and αBL , respectively, where θBL is given in (4) and αBL can be obtained from (5) after replacing θ and θUA by θBL .It may be noted here that the expressions for the MSEs of θUA(α) and θBL(α) can not be obtained and, therefore, one has no option except to go for a simulation study to compare their MSEs.

Illustrative Example
For illustration we take the example from Balasubramanian and Balakrishnan (1992), where n = 30 items were placed on a life-test experiment and their failure times (in hours) were recorded.The data reported is This is a simulated data set from model ( 3) with θ = 20, where some middle observations were not recorded and the experiment is supposed to be terminated as soon as the 26th item failed.Based on this multiply-II censored sample the estimates are calculated and given in Tables 1 to 3. We see that θBL(α) or θUA(α) only provide improvements compared to θBL or θUA , if the guess θ 0 equals the true value, i.e. θ 0 = 20.For guesses less than 20, the estimates move away from the true value as α increases.Since θBL and θUA both underestimate the true parameter, an improvement can be seen if θ 0 is larger than the true value.On the other hand, if no point guess is available and we use θUA(α) and θBL(α) , the estimates only improve if θ 0 is quite close the truth.For other values, the estimates move away from the true value but the magnitude of deviation is smaller as compared to the case when θ 0 is given with specified confidence.However, it may be remarked here that we should not infer about the performance of the estimator on the basis of a single sample.To study the performance of all the estimators we should study the behavior of their MSEs in order to draw fair conclusions.

Specified Confidence
We now compare the performance of the shrinkage estimators θUA(α) and θBL(α) with that of the corresponding approximate maximum likelihood estimators when a point guess is available with specified confidence.Notice that MSE( θUA(α) ) and MSE( θBL(α) ) are both functions of θ, θ 0 , n, α, and r i , i = 1, . . ., k.The MSEs of these estimators have been calculated for various values of θ, n, α, and r i .A number of values have been assigned to θ 0 so that the relative variation φ = (θ − θ 0 )/θ takes values in (−0.60(0.20)0.60).This was done to provide a wide variation in the values of θ 0 around the truth.
It is noted that as sample size n increases the MSE of the estimators θUA(α) and θBL(α) decreases generally, provided the sampling fraction and the type of sample observations do not change too much.It was further noted that as θ increases the MSE increase without affecting the relative performances of the estimators.Therefore, only for n = 10 and θ = 5 the MSEs of the estimators have been shown here in Figure 1.
Moreover, if φ is close to zero, i.e. if θ 0 is close to θ, the shrinkage estimator θBL(α) has smaller MSE than θBL for all choices of α.However, a greater reduction is obtained for large values of α.It may be further noted that for moderate values of φ, i.e. for |φ| ≤ 0.5, θBL(α) has always smaller MSE than θBL for all α.But if |φ| ≥ 0.5, the MSE of θBL(α) may be larger than that of θBL for large values of α.The range of φ for which θBL(α) has smaller MSE than θBL can be increased by taking α small, though the magnitude of reduction in MSE also decreases.It is also interesting to note that if the sample contains higher order observations, the greater reduction in MSE is seen for positive values of φ, i.e., when θ 0 is smaller than the true value.The situation is reversed when the observed sample contains lower order values.A similar trend can be observed for the MSE of θUA(α) , which is generally smaller than that of θBL(α) for small values of α.As α increases, MSE( θUA(α) ) remains smaller than MSE( θBL(α) ) for small values of φ, but for large value of φ the trend is reversed.For α = 0.9 both shrinkage estimators have approximately equal MSE.Except for large values of φ, MSE( θUA(α) ) is smaller than MSE( θBL(α) ).

Unspecified Confidence
As already mentioned, although the shrinkage estimators are obtained in closed forms, analytically closed form expressions for their MSE are not available.Therefore, a comparison of their MSEs will be based on results of a simulation study.For this purpose, a Monte Carlo study of 1000 samples each of size 10 was conducted for various values of θ, φ, n, k and r i .The parameter values considered here are the same as in Section 4.1.Notice that a change in the r i 's, for k fixed, results in a change of the magnitude of the MSE.In general, for a fixed number of observations (i.e., k fixed), if the higher order observations are taken, the MSE decreases slightly for almost all estimators (see Figure 2).The amount of decrease, however, differs from estimator to estimator.Further, on the basis of a thorough study of the results, it was noted that the MSEs of all the proposed estimators increase as θ increases but the trend remains more or less the same.
Austrian Journal of Statistics, Vol. 34 (2005) As shown in Figure 2, if φ = 0 then θUA(α) has the smallest MSE.The MSE of θBL(α) is also smaller than the ones of θUA and θBL .As φ increases the MSEs of θBL(α) and θUA(α) also increase and become larger than those of θUA and θBL beyond certain limits of φ, say, φ ∈ (φ 1 , φ 2 ) with φ 1 < −0.6 and φ 2 ≈ 0.4.Thus, the shrinkage estimator provides an improvement only in a subspace around the true parameter value.Generally, MSE( θUA(α) ) is also smaller than MSE( θBL(α) ) in this subspace.For values of φ outside this range, the MSE of θUA is smaller than the MSEs of all other estimates.

Conclusion
From the above results we may conclude that if a prior point guess is close to the truth, we can safely use the shrinkage estimator θUA(α) together with a large value of α, because it provides the smallest MSE.On the other hand, if the point guess is expected to be in the immediate neighborhood, one can still use θUA(α) .However, if it is suspected that the true value of θ may be far away from the guessed value θ 0 , one should never use a shrinkage estimator.In such situations the best one can do is to use θUA , i.e., the approximate maximum likelihood estimator proposed by Singh et al. (2004).

Table 1 :
Estimates θBL(α) based on θBL when the guess θ 0 is given with confidence α

Table 2 :
Estimates θUA(α) based on θUA when the guess θ 0 is given with confidence α

Table 3 :
Shrinkage estimates when a guess θ 0 is given with unspecified confidence