A Study on LINEX Loss Function with Different Estimating Methods

Abstract

LINEX means linear exponential loss function which used in the analysis of statistical estimation and prediction problem which rises exponentially on one side of zero and almost linearly on the other side of zero. It is used in both overestimation and underestimation problems. Ali Shadrokh and Hassan Pazira [1] presented Shrinkage estimator in Gamma Type-II Censored Data under LINEX loss function. In that paper, they have explained how the LINEX loss function works however no practical or detail explanations were given in terms of changing the shape parameter and the error function. In this study we have explained how the LINEX loss function works through practical or detail explanations in terms of changing the shape parameter and the error function, also see how the loss function works with the data generated from gamma distribution through resampling methods to compare the performance of LINEX loss function considering the relative estimation error and usual estimation error through generating random numbers from gamma distribution like randomization method and by using bootstrapping samples. The very intention is to find out which resampling method performs well in using the LINEX loss function. Using Monte Carlo Simulations these estimators are compared. It is doing draw random number from the gamma distribution and finds the maximum likelihood estimate of θ is and using this estimator to explain the LINEX loss function ; , or , where c is the shape parameter and is any estimate of the parameter . The shape of this loss function is determined by the value of c. In the analysis we use the values of shape parameter c = -0.25, -0.50, -0.75, -1 and c = 0.25, 0.50, 0.75, 1. The same procedure is done by using bootstrapping method, and finally compared between this two methods. The relative estimation error should be used instead of the estimation error where the LINEX loss function works better in both of the cases. Between the two estimators, bootstrap method is better work because although the characteristics are same, bootstrap method is more dispersed than others.