Bayes Estimator as a Function of Some Classical Estimator

Abstract

Maximum likelihood estimation method, uniformly minimum variance unbiased estimation method and minimum mean square error estimation, as classical estimation procedures, are frequently used for parameter estimation in statistics, which assuming the parameter is constant , while Bayes method assuming the parameter is random variable and hence the Bayes estimator is an estimator which minimize the Bayes risk for each value the random observable and for square error lose function the Bayes estimator is the posterior mean. It is well known that the Bayesian estimation is hardly used as a parameter estimation technique due to some difficulties to finding a prior distribution.
 The interest of this paper is that whether above classical estimators of the parameter for a particular probability distribution can be obtained from Bayes estimator is determined. In this analysis one-parameter Pareto distribution is used to examine the relationship between Bayesian and classical estimators. Considering improper prior distribution for shape parameter of the Pareto distribution of the first kind with known scale parameter which equals one, we have tried to show how the classical estimators can be obtain from Bayes estimator for various choices of hyper parameters of the prior function.