二元指數族中參數之最大概似估計與最大擬概似估計異同的研究

Abstract

當密度函數難以完整表示，例如無法求得其正規化常數，則求最大概似估計(MLE)時會有困難。因此一種替代方案就是使用擬概似函數去求得最大擬概似估計(MPLE)以取代MLE。本研究之目的在探討二元指數族中參數之MLE與MPLE的異同。文中先以常見的三個機率模型：卜瓦松-二項分配、三項分配與二元常態分配，探討模型參數的MLE與MPLE；接著推導一般二元指數族中獲得參數之MPLE的擬概似方程式；最後考慮 列聯表中，方格內參數為三種不同情況下的MLE與MPLE。當中兩種情況可以求出其精確解，而第三種則無法求出。針對第三種情況，利用Matlab程式以模擬的方式，計算出參數的MLE與MPLE，以進行分析比較，並觀察兩者之均方差如何受參數值影響。

If the density function is hard to express completely, e.g. hard to get normalizing constant, then it would be difficult to find the maximum likelihood estimate (MLE). An alternative way is to use pseudo-likelihood function to find the maximum pseudo-likelihood estimate (MPLE) instead of MLE. This research is to study the similarities and differences of the MLE and MLPE on the bivariate exponential distribution parameters. We first derive the MLE and the MPLE of parameters in Poisson-binomial distribution, trinomial distribution, and bivariate normal distribution. Then, we derive the pseudo-likelihood equation to be used for solving MPLE of parameters in bivariate exponential family. Finally, we consider three cases on the cell probabilities of the 2x2 contingency table. There are exact solutions on MLE and MPLE for the first two of these three cases. However, on the third case, there is no exact solution and we use Matlab program to do the numerical calculations for analyzing and comparing MLE and MPLE. We observe how the changes of mean square errors using MLE and those using MPLE affected by the value changes of parameters.