Jeffery Prior Distribution in Modified Double Stage Shrinkage-Bayesian Estimator for Exponential Mean

This paper is concerned with Modified Double Stage Shrinkage Bayesian (DSSB) Estimator for lowering the mean squared error of classical estimator θ̂ for the scale parameter (θ) of an Exponential Distribution in suitable region (R) around available prior knowledge (θ0) about the actual value (θ) as initial estimate as well as to reduce the cost of observation. In situation where the observations are time consuming or very costly, a “Double Stage procedure “can be used to reduce the Expected Sample Size needed to obtain the estimator. This estimator has been showing a smaller Mean Squared Error for certain choice of the shrinkage weight factor ψ(⋅) and for acceptance region R. Expressions for Bias, Mean Square Error (MSE), Expected sample size [E(n/θ,R)], Expected sample size proportion [E(n/θ,R)/n], probability for avoiding the second sample 1 ˆ [p( R)] θ ∈ and percentage of overall sample saved 2 1 n ˆ [ p( R) 100] n θ ∈ ∗ for the proposed estimator are derived. Numerical results and discussions are established when the consider estimator (DSSB) are testimator of level of significance α. Comparisons with the classical estimator as well as with some existing studies were made to shown the usefulness of the proposed estimator.


Introduction 1.1 The Model:
Exponential distribution is one of the most useful and widely exploited model, Epstein [1] remarks that the exponential distribution plays as important a role in life experiments as the part played by the normal distribution in agricultural experiments.It is applied in a very wide variety of statistical procedures.Among the most prominent applications are those in the field of life testing and reliability theory.The scale parameter (θ) is known as mean life time.The maximum likelihood estimator (MLE; θ ) is the sample mean which is the minimum variance unbiased estimator.The one parameter exponential distribution has the following probability density function (p.d.f.) 1 t exp( ) , t 0, 0 f (t; ) 0 ,o.w.
where θ is the average or the mean life or mean time to failure (MTTF) and it is also acts as scale parameter, see [1].Furthermore, the Reliability function R(t) is defined as: R(t) = exp(-t/θ), t > 0, θ >0.
Note that the maximum likelihood estimator θ of the scale parameter θ of the mentioned distribution is (n 1) ˆ( ) E ( ) (n 1)
The aim of this paper is to employ Bayesian estimator which is defined in (5) in the form of double stage shrinkage estimator (DSSE) which is defined in (8) for estimate the scale parameter (θ) of Exponential Distribution.

Modified Double Stage Shrinkage-Bayesian Estimator
This section is concern with pooling approach between shrinkage estimation that uses a prior information about unknown parameter as initial value and Bayesian estimation that uses a prior information about unknown parameter as a prior distribution for the scale parameter (θ) of exponential distribution using special shrinkage weight factors as well suitable region R when a prior information about (θ) is available as initial value (θ 0 ).
In the present work we establish modified Double Stage Shrinkage-Bayesian Estimator (DSSBE) which has the following form:- represent to Bayes estimator for θ on n i(i=1,2) observation, R is suitable region (say pretest region) and 1 ˆ) ψ(θ ≤ 1 is shrinkage weight factor which may be a function of 1 θ or constant, see for example : [3], [4], [7] and [10]. where R is the complement region of R in real space and i f ( ; ) θ θ {for i=1,2} is a p.d.f. of i θ which has the following form: We conclude: The Bias ratio B(⋅) of DS θ is defined as below See [6], [7] and [9].The expression of mean square error [MSE] of DS θ is as follows …(18) where E(n α,R) is the Expected sample size, which is defined as: See for example [3], [6], [7], [9] and [10].
As well as, the Expected sample size proportion { E(n α,R)/n } equal to See [6], [7]and [9].Also, it is necessary to define the percentage of the overall sample saved (P.O.S.S) of DS θ as: See [6] , [7]and [9].And, finally, represent the probability of a voiding the second sample.
x. the Probability of avoiding second sample is very suitable especially when θ ≈ θ 0, see table (7) , (12) .xi.The considered estimator DS θ is better than the classical estimator θ especially when θ ≈ θ 0 , this will given the Effective of DS θ Relative to θ and also given an important weight of prior knowledge, and the augmentation of efficiency may be reach to tens times.see table (1) , (2).xii.The considered estimator DS θ is more efficient than the estimators introduced by [6] and [7] in the sense of higher efficiency.
, obtain x 2i ; i = 1, 2, …, n 2 , an additional sample of size n 2 and use a pooled estimator p α of α based on combined sample of size n = n 1 + n 2 , i.e.;