Sometimes Pooled Testimation in the Inverse Gaussian Model for Measure of Dispersion

This paper suggests sometimes pool estimators for the measure of dispersion in the inverse Gaussian distribution and their properties are studied in terms of the relative bias and relative efficiency under two different loss functions.


Introduction
The inverse Gaussian distribution has useful applications in a wide variety of fields such as Biology, Economics and Medicine [1][2][3].It plays an important role in reliability theory and life testing problems.Let x be the inverse Gaussian variate with the parameters μ (measure of location) and σ (measure of dispersion), having probability density function ( ) Here, stands as the shape parameter.Let ( In many real life situations the overestimation and underestimation are not of equal consequences.For such situations a symmetric loss function like as a square error loss (SELF) function is not appropriate [4][5][6].A useful asymmetric loss function was introduced by [4], named as the LINEX loss function.The invariant version of the LINEX loss function (ILLF) [7] for the parameter θ is given as ( ) a approximately linearly on the other side.Here, is an estimate of the parameter The sign and magnitude of ' represent the direction and degree of asymmetry respectively.The positive (negative) value of is used when overestimation is more (less) serious than underestimation.For close to zero, the LINEX loss is approximately squared error and therefore almost symmetric.Recently, some shrinkage testimators for the inverse dispersion of the inverse Gaussian distribution under the ILLF has been studied elsewhere [7].
' a ' ' a ' ' a Han and Bancroft [8] have studied the sometimes pool estimator for the mean of a Normal distribution.They have considered the situation when two independent random samples are available from two Normal distributions with means μ 1 and μ 2 and the common variance.The problem of pooling in different situations has also been considered by other workers [9][10][11][12].Rai [13] has estimated the mean life of Exponential distribution.Sometimes pool estimator for shape parameter of the Pareto distribution under the SELF has been proposed by [14].
In the present article, we have studied the performances of the sometimes pool estimators for σ under the SELF and ILLF.

The Proposed Class of Estimators
We consider a class of estimators for σ of the model (1) as The risk of Y under the SELF and ILLF are obtained respectively as ) The suffix S and L stand respectively for the risk under the SELF and ILLF criteria.The values of C which minimize the risks R (S) (Y) and R (L) (Y) respectively are given as The minimum risk estimators of σ in the class Y with their respective risks are given as Estimator Risks ( )

The Proposed Pool Class of Estimators
We consider a class of the pooled estimator for σ of the distribution (1) as The expressions of the risks for under the SELF and ILLF, respectively are given as The values of for which the risks The improved classes of pooled estimator for σ with their respective risks are given as Estimator Risks ( )

The Proposed Sometimes Pool Estimators and their Properties
Our interest is to estimate the parameter σ 1 when it is suspect but not known for certain that σ 1 = σ 2 .Before pooling the two sample estimates for the estimation of the parameter σ, the test of hypothesis H 0 : σ 1 = σ 2 may be performed at some pre-assigned level of significance α.The test statistic for H 0 is given as The proposed sometimes pooled estimator is given as (11) where i = 1,2 and f 1 , f 2 are the lower and upper 100α/2% points of the F distribution with (n 1 -1) and (n 2 -1) degrees of freedom.The hypothesis H 0 is rejected when and it is never pool estimator.Otherwise it is sometimes pool estimator.The relative biases for the estimators are obtained as ( ) ( ) The expressions of the risks under the SELF and ILLF for are obtained The relative efficiencies for the pooled estimator The expressions of the relative biases and relative efficiencies are the functions of n the relative biases and relative efficiencies have been calculated.The 16-point Gauss-Legendre quardature formula is used to solve the integrations involved in relative biases and relative efficiencies.The relative biases are not presented here and the relative efficiencies have been presented in the Tables 1 to 4 for α = 0.01 and 0.05.

Recommendation
The absolute relative bias (ARB) of first decreases and then increases steadily as n 1 increases for all considered values of δ for fixed of size n 1 and n 2 drawn from two inverse Gaussian distributions.The maximum likelihood estimates of and under the SELF and ILLF criteria are defined as 2

1 ˆ ST σ α and n 2 . 2 ˆδ
The bias is almost negligible near δ = 100.The values of ARB of for all considered values of δ.The ARB of ST σ decreases asincreases for all δ when other parametric values are fixed but it increases as increases for all δ (except δ near 1.00).The opposite trend has been seen decreasing trend also has been seen as increases for δ ≥ 0.80 when other values are fixed.≥ 0.80 when other values are fixed.In addition, the gain in efficiency decreases with increase of α when δ ≥ 0.80 for all considered values of the parametric space.
≤ δ ≤ 1.6 (when other parametric values are fixed).The gain in efficiency decreases

Table 1 .
Relative efficiency between and under SELF.