An Efficient Single Stage Shrinkage Estimator for the Scale parameter of Inverted Gamma Distribution

Abstract The present paper agrees with estimation of scale parameter θ of the Inverted Gamma (IG) Distribution when the shape parameter α is known (α=1), bypreliminarytestsinglestage shrinkage estimators using suitable shrinkage weight factor and region. The expressions for the Bias, Mean Squared Error [MSE] for the proposed estimators are derived. Comparisons between the considered estimator with the usual estimator (MLE) and with the existing estimator are performed .The results are presented in attached tables.


Introduction
"In Reliability studies the models which are used in life testing include the Exponential, Gamma, Lognormal and Inverted Gammadistributions.If the failure is mainly due to aging or wearing out process, then its reasonable in many applicationsto choose one of the above mentioned distribution.In a sense, this distribution is unnecessary,it has the same distribution as the reciprocal of a gamma distribution .However, a catalogue of results for the inverse gamma distribution prevents having to repeatedly apply the transformation theorem in applications"; [1], [2], [3], [4].
"TheInverted Gamma distribution is prospective to use in life experiments"; [5], it has probability density function (p.d.f) with two parameters α and θ as below : Here α and θ arerespectively the shape and scale parameters.In conventional notation, wewrite X~IG (α , θ).This paper deals with the problem for estimation the unknown scale parameter (θ) of IG distribution with known shape parameter (α) when a prior estimate (θ o ) regarding the actual value (θ) is available using preliminary test single stage shrinkage estimator.
It is Well-known that, the prior knowledge regarding due reasons introduced by Thompson [9] as well as the classical estimator of ( ) and using shrinkage weight function [ψ(θ ^)] ;0 ψ (θ ^) 1 results what it isknown as "shrinkage estimator", which though perhaps biased has smaller mean squared error (MSE) than that of .
Thus "Thompson -Type" shrinkage estimator will be Now , to test the prior knowledge of weather close to actual value and to be comfortable to use this priorknowledge,the preliminary test single stage shrinkage estimator (SSSE) will be used for this mission when using the test estimator of level of significant ( ) for testing the hypotheses θO H O : θ=θOVS.H A : θ If H O correct, thenthe estimator which is defined in (2) will be used.
Conversely , if H O rejected,thendifferentshrinkage weight functionsΨ 2 ( );0 ( ) will be used and then using the followingshrinkage estimator Consequently, thecommon form of preliminary test single stage shrinkage estimator(SSSE) will be Ibn Al-Haitham J. for Pure & Appl.Sci.Vol.30 (1) 2017 ( ) ( ) =1,2 isa shrinkage weight function specifying the belief of θ ^and (1-Ψ(θ ^)) specifying the belief θ o and Ψ 1 (θ ^) may be a function of θ ^ or may be a constant (ad hoc basis ),while (R) is a pretest region for acceptance of the prior knowledge with level of significance ( ) .
The purpose of this paper is to employ the preliminary test single stage (SSSE) defined by ( 4) for estimating the scale parameter (θ) of two parameters Inverted Gamma (IG) distribution when the shape parameter (α) is known.The expressions of Bias, Mean Squared Error (MSE) and Relative Efficiency (R.Eff(.))were derived for the proposed estimator.
Numerical resultsand conclusions due mentioned expressions including some constantswereachieved and put in annexed tables.
Comparisons between the proposed estimators with the classical estimator and with existing estimator are performed.

Maximum Likelihood Estimator of θ
Let x 1 , x 2 ,---, xn be a randomsample of size n form IG (1,θ), then the natural logarithm of the Likelihood function L(1,θ) can be written as: Let ,then the maximum Likelihood estimator of θ is The distribution of is G(nα ,θ/n)

Preliminary Test Single Shrinkage Estimator (PTSSSE).
Using the form (4), we proposed the preliminary test single stage shrinkage estimator for estimator the scale parameter θ of Inverted Gamma distribution when a prior knowledge θ o available about θ with known shape α =1 as below:-
The expression for the bias of the estimator is as follow:- Where is the complement region of R in real space and f(θ ^ ) isa (P D F) of ( ) which has the following forms.
We conclude The bias ratio B(.) of the estimator ( ) is defined below And the expression for mean square error (MSE) of is given asbelow:

Conclusions and Numerical Results
The computations of Relative Efficiency[R.Ef f(.)] and Bias Ratio [B(.)] for the equation( 14) and (17 ) were used for the estimator .Thesecomputations (using Math.CAD program) were performed for =0.01, 0.05, 0.1,ζ =0.25(0.25)2and n = 4, 6, 8, 10, 12.These computation are given in attached tables No.( 1)and(2) for some samples of these constants.The observation mentioned in the tables leads to the following results.

Table ( 1) Showed Bias Ratio [B(.)]of
Relative Efficiency has the highest value at ζ =1 (θ o =θ) and decreases otherwise.6.The proportional estimator is better than the classical estimator in the sense of Mean Squared Error.