The Empirical Bayes Estimation For Parameters

Empirical Bayes was first developed by Robbins, H. (1951,1955). Generally speaking, empirical Bayes is a method that use the data to estimate some properties of he unknown parameter's prior distribution. It mostly contains parametric empirical Bayes(PEB) and nonparametric empirical Bayes(NPEB). The former assumes the prior distribution belongs to a distribution family with super parameter, the latter just assumesθi are i.i.d. In recent years, more and more attention are paid to empirical Bayes and discussion and research both in the applied area and theoretic are rising.Empirical Bayes is a successful combination of Bayes and friquentist. It inherits Bayes'thought that consider the parameter as a random variable, takes the prior information into account, and use frequentist's methods to obtain the information about the prior distribution, no matter. For example, NPEB just use the assumed prior distribution to write the Bayes rule directly but PEB obtain the estimation of the prior distribution through estimating the super parameter. This kind of processes help empirical Bayes reach to the results with better properties.In this thesis, a nonparametric empirical Bayes estimator of the parameter is constructed under square loss function for the linear exponential model by using kernel estimation method. It is also proved that the proposed estimator is an asymptotical EB estimator, and the convergence rate is established. Also, the empirical Gibbs sampling arithmetic for the estimation of the parameters of linear model is given. This work are not only meaningful in the applied field but also new tries of the empirical Bayes method.