Arnold and Strauss (1991) 探討2x2列聯表中的3個方格 (cell) 有相同機率θ的問題，他們比較了參數θ的最大概似估計值與最大擬概似估計值，發現參數θ的最大概似估計值與最大擬概似估計值是不相同的。在本論文中，我們將2x2列聯表中的3個方格的參數值 (機率值)，從限制為相同θ，放寬為成某種比例，並證明了在一般情況下參數θ的最大概似估計值與最大擬概似估計值也不相同。我們也提出一些使參數θ的最大概似估計值及最大擬概似估計值相同的特殊條件，諸如三個方格內的觀察值跟機率值成比例或格子內的觀察值有某些特定值。本論文也透過電腦模擬的結果，發現最大概似估計式較最大擬概似估計式來得精確，而且當參數θ在參數空間之中點附近時，最大概似估計值與最大擬概似估計值的差異為最大。

Arnold and Strauss (1991) study the cases that three of the four cells in the 2x2 contingency table have the same cell probability θ. In particular, Arnold and Strauss (1991) compare the maximum likelihood estimate (MLE) and maximum pseudolikelihood estimate (MPLE) of the parameter θ. They find that MLE and MPLE of the parameter are not the same. In this thesis, we relax the assumptions so that those three cell probabilities may not be the same and each is proportional to a parameter θ. We find that, in general, MLE’s of θ are still not the same as MPLE’s of θ. Some special cases that make MLE the same as MPLE are also given. We also find, through computer simulations, that MLE’s are accurate than MPLE’s and that the difference between MLE and MPLE is getting larger when the parameter θ is closer to the midpoint of its space.

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Arnold, B. C. and Strauss, D (1991). "Pseudolikelihood Estimation: Some Examples" Journal of the Indian Journal of Statistics 53(2): 233-243.

Strauss, D. and Ikeda. M. (1990). "Pseudolikelihood Estimation for Social Networks."Journal of the American Association 85(409):204-212

van Duijn, M. A., Gile, K. J., and Handcock , M. S. (2009). "A Framework for the Comparison of Maximum Pseudo Likelihood and Maximum Likelihood Estimation of Exponential Family Random Graph Models." Social Networks 31(1):52-62