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UBC Theses and Dissertations

Maximum weighted likelihood estimation Wang, Steven Xiaogang

Abstract

A maximum weighted likelihood method is proposed to combine all the relevant data from different sources to improve the quality of statistical inference especially when the sample sizes are moderate or small. The linear weighted likelihood estimator (WLE), is studied in depth. The weak consistency, strong consistency and the asymptotic normality of the WLE are proved. The asymptotic properties of the WLE using adaptive weights are also established. A procedure for adaptively choosing the weights by using cross-validation is proposed in the thesis. The analytical forms of the "adaptive weights" are derived when the WLE is a linear combination of the MLE's. The weak consistency and asymptotic normality of the WLE with weights chosen by cross-validation criterion are established. The connection between WLE and theoretical information theory is discovered. The derivation of the weighted likelihood by using the maximum entropy principle is presented. The approximations of the distributions of the WLE by using saddlepoint approximation for small sample sizes are derived. The results of the application to the disease mapping are shown in the last chapter of this thesis.

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