Quadratic Bayesian Estimation For Two Classes One-parameter Distributions

Pareto distribution and uniform distribution attract great attention from the researchers due to their wide application.They propose a variety of parameter estimation methods,such as:the maximum likelihood estimation,the unbiased estimation and the Bayes estimation etc.In the case of large samples,these methods above can obtain accurate results.In the case of small samples,we usually use the Bayes estimation.But it is often difficult to calculate the Bayes estimation due to the complex integrals.Using the method of Gibbs sampling will always make troubles for parameter estimation.So in this paper we propose a new parameter estimation which is called the quadratic Bayes estimation.In this paper,we introduce the statistic T2 based on statistic T to construct the quadratic Bayes estimation.This estimation not only adopt the prior information,but also avoid the complicated calculation of the posterior expectation.Firstly,we obtain the quadratic estimation expressions of these two distributions by calculation.Then,we prove that the quadratic estimation is better than the maximum likelihood estimation and the uniformly minimum variance unbiased estimation under the mean square error criterion.Numerical simulations show that the quadratic estimation is very similar to the Bayes estimation regardless of the choice of priors and the change of the parameters of the priors.The quadratic estimation is approaching to the Bayes estimation as the sample size increases.Meanwhile,the quadratic estimation gets closer to the Bayes estimation as the concentrating of prior information.In summary,the quadratic estimation for Pareto distribution and uniform distribution are both effective.