一些條件分佈族的先驗分佈估計及經驗貝氏估計量

Abstract

在貝氏理論中，分佈函數中的參數是作為隨機變量考慮的，這種變量的分 佈叫做事前機率，當參數固定為某值時，隨機變數的分佈叫做條件分佈， 當隨機變數為一觀測給定值時，參數的分佈就稱為事後分佈，條件分佈對 事前機率作積分後，所得不含參數的隨機變數的分佈叫做邊際分佈。本論 文在連續參數型空間上探討三種不同的條件分佈集，即變形位置族 (variational location family)，位置族(location family)及位置比例 族(location-scale family)，其事前分佈的估計。我們利用隨機樣本， 得到此逼近估計量。在平方差損失函數下，得到參數的經驗貝氏估計量。 本文第二章討論條件分佈是變形位置族時，我們利用正交展開級數及拉普 拉轉換，得到事前分佈的逼近估計量，及參數的經驗貝氏估計量。並利用 估計的事前分佈，推展出貝氏選擇過程，選擇最佳母體。第三章討論條件 分佈是位置族時，我們利用正交開級數及傅利葉轉換得到事前分佈的逼近 估計量，及參數的貝氏估計量。第四章討論條件分佈是位置比例族時，我 們利用正交展開級數及傅利葉—拉普拉斯轉換，得到事前分佈的逼近估 驗貝氏估計量。第二，三及第四章的貝氏估計量，用蒙地卡羅法作模擬， 和真正的貝氏估計量作了比較。 The Bayes principle involves notion of a distribution on the parameter space Θ which is so-called prior distribution. When a density of random variable Χ involves a parameter θ, it is usually called a conditional density of Χ, denoted by f(x|θ). Observing a realization of ｘ, a density of the random parameterθ is called a posterior density, denoted by g(θ|x). The unconditional distribution function of random variable Χ is then called the marginal distribution. In this thesis, we consider three different types of conditional distributions for continuous parameter space, they are, respectively, variations of location-family, location- family and location-scale family. Through orthogonal expansion method, we estimate the prior density based on observations from marginal distribution. And under square loss function, we obtain empirical Bayes estimator for unknown parameter θ. In Chapter 2, we use orthogonal expansion and Laplace transform to obtain approximated prior distributions for variational location family. Empirical Bayes estimators of θ or (θ(1),θ(2)) under square loss are also obtained. We then derive the Bayes selection procedure with respect to the estimated prior distribution. In Chapter 3, we use orthogonal expansion and Fourier transform to obtain prior distributions for location family. Empirical Bayes estimators of θ are also obtained. In Chapter 4, we use orthogonal expansion and Fourier-Laplace type transform to obtain estimated prior distributions for location-scale family. Empirical Bayes estimators of θ are also obtained. Some Monte-Carlo methods are applied to study the performances of the proposed estimators both for prior distributions and the empirical Bayes estimators for medium and small sample size.