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 Title: 以三個二維連續分配對最大概似估計與最大擬概似估計作比較A comparison between maximun likelihood estimation and maximun pseudo-likelihood estimation using three bivariate continuous distributions Authors: 張嘉福 Contributors: 宋傳欽姜志銘張嘉福 Keywords: 相容條件分配最大概似估計最大擬概似估計compatibilityconditional distributionmaximum likelihood estimatormaximum pseudo-likelihood estimator Date: 2012 Issue Date: 2013-09-02 16:46:31 (UTC+8) Abstract: 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. Reference: [1] Besag, J. E. (1975) Statistical Analysis of Non-Lattice Data. The Statistician, Vol. 24, No. 3, pp. 179-195[2] Arnold, B. C. and Strauss D. (1991) Pseudolikelihood Estimation: Some Examples. The Indian Journal of Statistics Vol.53, Series B, Pt. 2, pp. 233-243[3] 蕭惠玲（2010）：二維聯合分配下條件常態分配相容性之探討。國立政治大學應用數學系教學碩士在職專班碩士論文。[4] Kotz, S., Balakrishnan, N. and Johnson N. L. (2000) Continuous Multivariate Distributions, Vol. 1. John Wiley, New York.[5] 羅純、王築娟（2002）：Gumbel分佈參數估計及在水位資料分析中應用。上海應用技術學院數理教學部。[6] Nadarajah, S. and Kotz, S. (2005) Reliability for some bivariate exponential distributions. Mathematical Problems in Engineering, Vol.2006, Article ID 41652, pp. 1-14[7] 李國安（2000）：多元Marshall~Olkin型指數分布的特徵及參數估計。工程數學學報，第22卷第6期，1055-1062。[8] 彭江艷、何平（2004）。多維指數分布模型。數學的實踐與認識，第34卷第7期，102-106。 Description: 碩士國立政治大學應用數學系數學教學碩士在職專班99972010101 Source URI: http://thesis.lib.nccu.edu.tw/record/#G0099972010 Data Type: thesis Appears in Collections: [應用數學系] 學位論文

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