半參數分配對於描述資料的特徵相當具有彈性，其能有效地配適資料之偏態與峰數。半參數分配是由常態分配的機率密度函數與有限制之多項式係數所組成，其並不屬於指數族，在參數的估計上較為困難，因此本研究利用貝氏方法估計半參數分配之參數。首先定義參數空間並解釋為何無法使用牛頓法估計參數之最大概似估計量，接著說明如何利用 Metropolis-Hastings 演算法生成後驗分配樣本，並藉由後驗樣本估計半參數分配之參數。一般而言，為了要使均方差最小，經常會以後驗樣本平均數做為參數估計值，但使用後驗分配平均做為半參數分配之參數估計值時，會違反半參數分配參數之限制。有鑑於此，本論文採用後驗樣本與定義特定方向之平均夾角概念估計參數。此外，半參數分配中含有常態分配，本研究也提出利用此分配檢定資料常態性的方法。

Semi-nonparametric distribution is a flexible distribution to capture characteristics of data, such as skewness and multimodality. The density function of the semi-nonparametric distribution is the product of a normal density and a polynomial with certain constraints on the coefficients of the polynomial and unfortunately this distribution does not belong to the exponential family. In this thesis, we propose a Bayesian procedure for parameter estimation. We first clearly define the parameter space and explain why the popular Newton method is inapplicable to find the maximum likelihood estimate in this case. Next, we proposed a Metropolis-Hastings algorithm to generate posterior samples. To minimize mean squared error, the posterior mean is usually used. However, the posterior mean is irrelevant for the parameter estimation due to the restriction of the parameter space. Instead of averaging posterior samples, we take advantage of averaging angles between the posterior samples and the reference direction, defined in the text. Except for estimation, we also consider a Bayesian semi-nonparametric test for normality since the family of the semi-nonparametric distribution contains the normal distribution.

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