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標題: 多變量 t 分佈之平均數與共變異數結構之穩健聯合建模方法
A robust approach to joint modeling of mean and covariance structure using the multivariate t distribution
作者: 王鈺仁
Wang, Yun-Jen
關鍵字: Covariance structure
Maximum likelihood estimates
出版社: 應用數學系所
引用: Chi, E.M., Reinsel, G.C., 1989. Models for longitudinal data with random effects and AR(1) errors. J. Amer. Statist. Assoc. 84, 452-459. Daniels M.J., Pourahmadi, M. (2002). Bayesian analysis of covariance matrices and dynamic models for longitudinal data. Biometrika 89, 553-566. Demidenko, E. (2004). Mixed Models: Theory and Applications. New York: Wiley. Fearn, T. 1975. A Bayesian approach to growth curves. Biometrika 62, 89-100 Laird, N.M., Ware, J.H., 1982. Random effects models for longitudinal data. Biometrics 38, 963-974. Lange, K.L., Little R.J.A., Taylor J.M.G., 1989. Robust statistical modeling using the t distribution. J. Amer. Statist. Assoc. 84, 881-896. Lee, J.C., 1988. Prediction and estimation of growth curve with special covariance structures. J. Amer. Statist. Assoc. 83, 432-440. Kotz, S, Nadarajah, S. 2004. Multivariate t Distributions and Their Applications. Cambridge University Press. Healy, M.J.R. 1968. Multivariate normal plotting. Appl. Statist. 17, 157-161. Pan, J., MacKenzie, G. 2003. On modelling mean-covariance structures in longitudinal studies. Biometrika 90, 239-244. Pinheiro, J.C., Liu, C.H., Wu Y.N., 2001. Efficient algorithms for robust estimation in linear mixed-effects models using the multivariate t distribution. J. Comp. Graph. Statist. 10, 249-276. Potthoff, R.F., Roy, S.N., 1964. A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika 51, 313-326. Pourahmadi, M. (1999). Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterisation. Biometrika 86, 677-690. Pourahmadi, M. (2000). Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix. Biometrika 87, 425-435. Rao, C.R., 1987. Prediction of future observations in growth curve models (with discussion). Statist. Sci. 2, 434-471. Ryggard, K. and Spang-Thomsen, M. 1997. Quantitation and Gompertzian analysis of tumor growth data. Breast Cancer Res. Treat. 46, 303-312. Tanabe, K. and Sagae, M. 1992. An exact Cholesky decomposition and the generalized inverse of the variance-covariance matrix of the multinomial distribution, with applications. J. Roy. Statist. Soc. B 54, 211-219. Zacks S. 1971. The Theory of Statistical Inference. Wiley, New York. Zellner, A., 1976. Bayesian and non-Bayesian analysis of the regression model with multivariate student-t error terms. J. Amer. Statist. Assoc. 71, 400-405.
摘要: 本文推廣文獻Pourahmadi (1999, 2000) 的研究方法,其為考慮在多變量 t 分配架構下所提出的一個穩健性聯合平均數與共變異數結構模型。我們運用 Cholesky 分解法將其共變異數矩陣的元素分解為不受限制的自我迴歸與尺度變異參數。在這模型中,我們可以將對數概似函數表示成三種不同型式,並且探討其一些性質。對於最大概似估計,我們提出一個具效率的費雪分數演算法去計算參數估計值與標準誤。在多變量 t 所架構之模型中,我們也提出未來值的預測方法。最後,我們用兩個例子來闡述所提出的模型,並且與常態為基礎之對應模型作比較。
This thesis extends the previous work of Pourahmadi(1999, 2000) by introducing a robust version of joint mean-covariance modeling approach based on the multivariate t distribution. The modified Cholesky decomposition is adopted to factorize the variance component in terms of unconstrained autoregressive and scaling innovation parameters. We present three distinct representations of the log-likelihood function of the model and study the associated properties. An efficiently computational procedure based on the Fisher scoring algorithm is developed for carrying out maximum likelihood estimation with standard errors as a by-product. The technique for predicting future responses in this context is also investigated. The implementation of the proposed methodology is illustrated with two numerical examples and comparisons are made with those of normal counterparts.
其他識別: U0005-2506200717153400
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