Double Stage Shrinkage Estimator of Two Parameters Generalized Exponential Distribution

This paper is concerned with double stage shrinkage estimator (DSSE) for lowering the mean squared error of classical estimator (MLE) for the shape parameter (α) of generalized Exponential (GE) distribution in a region (R) around available prior knowledge (α0) about the actual value (α) as initial estimate in case when a scale parameter (λ) is known as well as to reduce the cost of experimentations. In situation where the experimentations 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 is shown to have smaller mean squared error for certain choice of the shrinkage weight factor ψ(⋅) and for acceptance mentioned region R. Expressions for Bias, Mean square error (MSE), Expected sample size [E(n/α,R)], Expected sample size proportion [E(n/α,R)/n], probability for avoiding the second sample 1̂ [p( R)] α∈ and percentage of overall sample saved 2 1 n ˆ [ p( R) 100] n α∈ ∗ for the proposed estimator are derived. Numerical results and conclusions are established when the consider estimator (DSSE) are estimator of level of significance Δ. Comparisons with the classical estimator and with the last studies shown the usefulness of the proposed estimator


[p( R)] α ∈
and percentage of overall sample saved

Introduction
Gupta and Kundu (1999) proposed the generalized exponential (GE) distribution as an alternative to the well known Weibull or gamma distributions.It is observed that the proposed two-parameter GE distribution has several desirable properties and in many situations it may fit better than the Weibull or gamma distribution.Extensive work has been done since then to establish several properties of the generalized exponential distribution.The readers are referred to the recent review article by Gupta and Kundu (2007) for a current account of it.[1].Generalized exponential (GE) distribution has been proposed and studied quite extensively recently by Gupta and Kundu [2,3,4,5,6].The readers may be referred to some of the related literature on (GE) distribution.[7].The two-parameters (GE) distribution has the following distribution function:
In this paper we introduce the problem of estimating of the shape parameter (α) of GE distribution with known scale parameter (λ) when some prior information (α 0 ) regarding the actual value (α) available due past experiences such a prior estimate may arise for any one of an umber of reasons [8], e.g., we are estimating α and; i.We believe α 0 is close to true value of α, or ii.We fear that α 0 may be near the true value of α, i.e.; something bad happens if α = α 0 and we do not know about it.In such a situation it is natural to start with an estimator α (e.g.MLE) of α and modify it by moving it closer to α 0 , so that the resulting estimator, though perhaps biased, has smaller mean square error than that of α in some interval around α 0 .This method of constructing an estimator of α that incorporates the prior value α 0 leads to what is known as a shrinkage estimator.
It is an important aspect of estimation that one should be able to get an estimator quickly using minimum experimentation.This also economizes cost of experimentation.To achieve this, double stage shrinkage estimator were introduced.
A double stage shrinkage estimator procedure is defined as follows: Let x 1i ; i = 1, 2, …, n 1 be a random sample of n 1 from GE distribution and 1 α be a "good" estimator of α based on these n 1 observation.Construct a preliminary test region R in the parameter space based on α 0 and an appropriate criterion.  .
Thus, the double stage shrinkage estimator (DSSE) of α will be: The motivation of this study was provided by the work of [9], [10] and [11].The aim of this paper is to employ the double stage shrinkage estimator (DSSE) α % defined by (3) for estimate the shape parameter (α) of two parameters generalized Exponential (GE) distribution when the scale parameter (λ) is known.
The expression of Bias, Mean squared error (MSE), Relative Efficiency [R.Eff(⋅)], Expected sample size, Expected sample size proportion, probability for avoiding the second sample and percentage of overall sample saved are derived and obtained for the estimator α % .
Numerical results and conclusions due mentioned expressions including some constants are performed and displayed in annexed tables.
Comparisons between the proposed estimator with the classical estimator ( ) α and with some of the last studies are demonstrated.

Unbiased -Maximum Likelihood Estimator of α
In this section, we consider the maximum likelihood estimator (MLE) of GE distribution with shape and scale parameter α and λ respectively i.e.GE (α,λ).Assume x 11 , x 12 , …, x 1 1 n be a random sample of size n 1 from GE(α,λ) then the log-likelihood function L (α,λ) can be written as: L( , ) In this paper we take λ = 1 (λ is known). So, Then, the MLE ofα, say , see [4]. i.e.; Using ( 6), an unbiased estimator 1 α of α can be easily obtained as:
Where R is a pretest region for testing the hypothesis H 0 : α = α 0 vs H A : α ≠ α 0 with level of significance (Δ) using test statistic function Where are the lower and upper 100(Δ/2) percentile point of chi-square distribution with degree of freedom (2n 1 ) respectively.
The expression for Bias of DSSE ( α % ) is defined as below

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[ ] Where R is the complement region of R in real space and We conclude, The Bias ratio [B (⋅)] of DSSE ( α % ) is defined as: Bias( / , R) B( ) The expression of Mean squared error [MSE (⋅)] of α % derived as below: - And by simple computations, one can get: Now, the Efficiency of α % relative to α denote by R.Eff ( α % /α, R) is defined by: MSE Where E(n/α,R) is the Expected sample size, which is defined as: See for example [10] and [11].
As well as the Expected sample size proportion E (n/α, R)/n equal to

Conclusions and Numerical Results
The computations of Relative Efficiency value with increasing value of u (u = n 2 / n 1 ) andζ.ix.R.Eff( % α ) is an increasing function with respect to u.This property shown the effective of proposed estimator using small n 1 relative to n 2 (or large n 2 ) which given higher efficiency and reduce the observation cost.x.The considered estimator % α is better than the classical estimator especially when α≈α 0 , this will given the effective of % α relative to α and also given an important weight of prior knowledge, and the augmentation of efficiency may be reach to tens times.xi.The considered estimator % α is more efficient than the estimators introduced by [8], in the sense of higher efficiency.
for the proposed estimator are derived.
have to define the percentage of the overall sample saved (p.o.s.s.) of α % as: the probability of a voiding the second sample (stage).

Table ( 5) Shown Probability of a Voiding Second Sample w.r.t Δ, u, n 1 and ζ
Shown Percentage of Overall Sample Saved w.r.t Δ, u, n 1 and ζ