| The opinion of Bayes'school regard the unknown-parameterθas a random variable, and according to the prior information of parameterθdetermine prior distributionπ(θ).Therefore, to choose appropriate prior distribution is first role of Bayes'school. 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 the unknown parameter prior distribution. It mostly contains parametric empirical Bayes and nonparametric empirical Bayes. 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 areas and theoretic are rising.In discussion parameter Bayes estimated and empirical Bayes estimate in the question, the domestic and foreign scholars done the foundations work. They selected some representative density functions, to estimate the parameters that are contained in these functions. Then they selected the suitable loss functions to discuss the asymptotic optimality and the convergence rate of the estimate. For example, many works has done about scale exponential family. The distribution ofΓ(α,β) is a special scale exponential function. This paper researched the distribution ofΓ(θ, 1/2) which is a specificΓ(α,β) function. It is very significant to study the distribution.In this thesis, a empirical Bayes estimator of the parameter of the distributionΓ(θ, 1/2) is constructed under square loss function by using kernel estimation method. Then to proved the asymptotic optimality of the estimator and gained the convergence rate. Linex loss function is inducted. Bayes estimate, empirical estimate and maximum likelihood estimate for the parameterθof the positive exponential family were gained under squared loss function and Linex loss function. The three estimates were compared with each others.
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