限制型多項分佈之參數估計

Abstract

這篇文章主要在討論當多項分佈每一種可能出現的結果有最大次數限制時的參數估計，首先我們會先簡介一下引發動機的問題，接著將問題參數化，並先處理較簡單的情況，推導該情況的最大概似估計量，並討論其極限分配。接著我們想知道如果雖然有最大次數限制，但在並不知情的情況下去估計會發生什麼樣的偏差，我們針對此情況做了模擬，並討論在什麼樣的參數下，不知情的估計會如何出錯。最後，我們稍微推廣了原本容易處理的問題，並比較同樣是最大概似估計量，但“不同取樣順序”對變異數的影響。

The thesis mainly focuses on the estimation of parameters for the multinomial distribution in which the maximum occurrence times of each possible outcome are limited. The inspiration is briefly introduced first. We will then parameterize the problem and deal with the easier case. We derive the maximum likelihood estimator and then move on to its asymptotic distribution. Next, we wonder the estimation bias if we estimate the parameters without knowing the existence of occurrence restriction. We do several simulations to observe how the estimation will go wrong in accordance with different parameter settings. At last, we slightly generalize the easier case mentioned before and compare how different sampling orders will affect the standard deviations of maximum likelihood estimators.