在給定樣本資料及模型假設下，經常會使用最大概似估計方法來估計模型的未知參數。但是在遇到具有複雜形式的概似函數情況下，可能無法經由常規的數學計算推導出其未知參數的最大概似估計之通式解，因此需要採取其他模式處理。在此，與其尋找最大概似估計之通式解通式解，我們將最大概似估計之問題轉化為一個全域最佳化問題進行數值求解。本論文中，我們以混合效應模型作為模型設定，探討使用粒子群最佳化演算法來尋找數值解的可行性。為提高估計的精準度，除了以概似函數為最佳化之目標函數外，亦結合概似函數與其之一次偏微分方程式所構成之新的目標函數來求取最大概似估計之數值解。除了本論文中所以之方法外，亦有與過往研究中提出之估計方法進行以數值模擬得比較。

Given sample data and model assumptions, the Maximum Likelihood Estimation (MLE) method is often used to estimate the unknown parameters of a model. However, when dealing with likelihood functions of complex forms, it may be impossible to derive a general solution for the MLE of unknown parameters using conventional mathematical methods. In such cases, alternative approaches are required. Instead of seeking a general solution for the MLE, we transform the MLE problem into a global optimization problem for numerical resolution. In this thesis, we use the mixed-effects model as the model setting to explore the feasibility of using the Particle Swarm Optimization (PSO) algorithm to find numerical solutions. To improve estimation accuracy, we consider not only the likelihood function as the objective function for optimization but also a new objective function that combines the likelihood function with its first-order partial derivatives. We then use this new objective function to obtain numerical solutions for the MLE. Furthermore, the methods proposed in this thesis are compared with those from previous studies through numerical simulations.

[1] Chung, Y., Rabe-Hesketh, S., and Choi, I.-H. (2013a). Avoiding zero between-study variance estimates in random-effects meta-analysis. Statistics in medicine, 32(23):4071– 4089.
[2] Chung, Y., Rabe-Hesketh, S., Dorie, V., Gelman, A., and Liu, J. (2013b). A non- degenerate penalized likelihood estimator for variance parameters in multilevel models. Psychometrika, 78:685–709.
[3] Cochran,W.G.(1937).Problems arising in the analysis of a series of similar experiments. Supplement to the Journal of the Royal Statistical Society, 4(1):102–118.
[4] Faouri, A. O. and Kasap, P. (2023). Maximum likelihood estimation for the nakagami distribution using particle swarm optimization algorithm with applications. Necmettin Erbakan Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 5(2):209–218.
[5] Hardy, R. J. and Thompson, S. G. (1996). A likelihood approach to meta-analysis with random effects. Statistics in medicine, 15(6):619–629.
[6] Kasap,P.andFaouri,A.O.(2024).Comparison of the meta-heuristic algorithms for max-imum likelihood estimation of the exponentially modified logistic distribution. Symmetry, 16(3):259.
[7] Kennedy, J. and Eberhart, R. (1995). Particle swarm optimization. In Proceedings of ICNN’95-international conference on neural networks, volume 4, pages 1942–1948. ieee.
[8] Myung,I.J.(2003).Tutorial on maximum likelihood estimation. Journal of mathematical Psychology, 47(1):90–100.
[9] Raudenbush, S. W. (2009). Analyzing effect sizes: Random-effects models. The hand-book of research synthesis and meta-analysis, 2:295–316.
[10] Rukhin, A. L. (2011). Maximum likelihood and restricted likelihood solutions in multiple-method studies. Journal of research of the National Institute of Standards and Technology, 116(1):539.
[11] Shi, Y. and Eberhart, R. C. (1999). Empirical study of particle swarm optimiza- tion. In Proceedings of the 1999 congress on evolutionary computation-CEC99 (Cat. No. 99TH8406), volume 3, pages 1945–1950. IEEE.
[12] Vangel,M.G.and Rukhin,A.L.(1999).Maximum likelihood analysis for heteroscedas-tic one-way random effects anova in interlaboratory studies. Biometrics, 55(1):129–136.
[13] Veroniki, A. A., Jackson, D., Viechtbauer, W., Bender, R., Bowden, J., Knapp, G., Kuss, O., Higgins, J. P., Langan, D., and Salanti, G. (2016). Methods to estimate the between-study variance and its uncertainty in meta-analysis. Research synthesis methods, 7(1):55–79.