A comparison between maximun likelihood estimation and maximun pseudo-likelihood estimation using three bivariate continuous distributions

給定一些條件分配，若其相容，我們可以試著找出對應的聯合分配，並由概似函數求其參數的最大概似估計。但當聯合密度函數不易求出或過於複雜時，我們可以利用擬概似函數去估計參數。本文透過三個分配：（1）聯合分配為Gumbel二維指數分配；（2）聯合分配為二維常態分配；（3）聯合分配為Marshall及Olkin 二維指數分配，對最大概似估計(MLE)與最大擬概似估計(MPLE)作比較，並進行探討是否可以MPLE取代MLE。我們發現在（1）、（2）情形下，MPLE與MLE一致；但在（3）情形時，MPLE與MLE不一致。在（3）情形下，透過數值模擬的實驗，發現MPLE與MLE的差異似乎有隨著相關係數變大而變大的趨勢。因此在給定一些條件分配時，雖然擬概似函數容易建立以估計參數，但MPLE相對於MLE的誤差有可能會比較大。另外，就如二維常態下的例子所示，即使MPLE與MLE一致，相對於MLE而言，MPLE的推導與計算通常較為複雜。因此仍應盡可能尋找對應的聯合密度函數，以計算最大概似估計。If the given conditional distributions are compatible, then their corresponding joint distribution exists. In such case, we may be able to find its joint p.d.f. and to find maximum likelihood estimators of the parameters. However, when it is not easy to find the joint p.d.f. or the expression of the joint p.d.f. is too complicated, we may use the maximum pseudo-likelihood estimators to estimate the unknown parameters. In this thesis, using three different bivariate joint distributions, we study the difference between their maximum likelihood estimator (MLE) and maximum pseudo-likelihood estimator (MPLE) to find out if MPLE may replace MLE. These three distributions are Gumbel’s bivariate exponential distribution, bivariate normal distribution, and Marshall and Olkin’s bivariate exponential distribution. We find that MPLE’s and MLE’s are the same under Gumbel’s bivariate exponential distribution and bivariate normal distribution. However, it’s not possible that MPLE’s and MLE’s could be the same under Marshall and Olkin’s bivariate exponential distribution. In addition, through computer simulation study on Marshall and Olkin’s bivariate exponential distribution, we find that the difference between MPLE and MLE seems getting larger if the correlation coefficient is becoming larger. Finally, the derivation and/or computation of the MPLE for some distributions may be too complicated, even their MPLE’s and MLE’s are the same. Hence, it may not be worth of using MPLE, like the bivariate normal case. Therefore, we suggest finding out the joint p.d.f. first to estimate the parameters through MLE if it is possible, instead of using MPLE