Asymptotic theory of simultaneous estimation of Poisson means

The Poisson distribution is often a good approximation to the underlying sampling distribution and is central to the study of categorical data. In this paper, we propose a new unified approach to an investigation of point properties of simultaneous estimations of Poisson population parameters with general quadratic loss functions. The main accent is made on the shrinkage estimation. We build a series of estimators that could be represented as a convex combination of linear statistics such as maximum likelihood estimator (benchmark estimator), restricted estimator, composite estimator, preliminary test estimator, shrinkage estimator, positive rule shrinkage estimator (James–Stein type estimator). All these estimators are represented in a general integrated estimation approach, which allows us to unify our investigation and order them with respect to the risk. A simulation study with numerical and graphical results is conducted to illustrate the properties of the investigated estimators.