Journal of Kufa for Mathematics and Computer Vol.1, No.6, December , 2012, pp.—   On Jeffery Prior Distribution in Modified Double Stage Shrinkage-Bayesian Estimator for Exponential Mean Abbas Najim Salman, Assel Hussein Ali , Maha A. Mohammed Department of Mathematics-Ibn Al-Haitham College of Education - University of Baghdad   Computer Center - University of Baghdad Abstract This paper is concerned with Modified Double Stage Shrinkage Bayesian (DSSB) Estimator for lowering the mean squared error of classical estimator ˆ θ for the scale parameter (θ) of an Exponential Distribution in suitable region (R) around available prior knowledge (θ 0 ) 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. 1. 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; ˆ 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. -  ≥ θ>  θ= θ θ    …(1) 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 ˆ θ of the scale parameter θ of the mentioned distribution is n i i 1 t t n = =  . 1.2 Jeffery Prior distribution (Bayesian Estimator): Consider the one parameter Exponential Distribution which is define in (1).