Improved Estimation Strategies in Normal and Poisson Models

Abstract

In this chapter, we consider the estimation of the mean parameter from two commonly used statistical models in practice. In a classical approach, we estimate the parameter based on the available sample information at hand only. On the other hand, in Bayesian framework, we assume prior distribution on the parameter of interest to obtain an improved estimation. In semi-classical approach, we assume that an initial value of the parameter is available from past investigation or any other sources whatsoever. The main focus in this chapter is to combine sample information and nonsample information to obtain an improved estimator of the mean parameter of normal and Poisson models, respectively. To improve the estimation accuracy linear shrinkage and pretest estimation strategies are suggested. The performance of the suggested pretest estimator is appraised by using the mean squared error criterion. The relative efficiency of the suggested estimators with respect to a classical estimator is investigated both analytically and numerically. Not surprisingly, the linear shrinkage estimator outperforms its competitors when the nonsample information is nearly correctly specified. The pretest estimator is relatively more efficient than the classical estimator in the most interesting part of the parameter space. The suggested shrinkage pretest estimation strategy is easy to implement and does not require any tuning or hyperparameter. We strongly recommend using the shrinkage pretest estimation method for practical problems, since it does not drastically suffer postestimation bias or any other implications, unlike other methods, which fail to report the magnitude of the bias whether negligible or not negligible. The shrinkage pretest strategy precisely reports its strength and weakness.