Master of Science / Department of Statistics / Weixing Song / The relationship between a random variable and a random vector is often investigated through the regression modeling. Because of its relative simplicity and ease of interpretation, a particular parametric form is often assumed for the regression function. If the pre-specified function form truly reflects the truth, then the resulting estimators and inference procedures would be reliable and efficient. But if the regression function does not represent the true relationship between the response and the predictors, then the inference results might be very misleading. Therefore, lack-of-fit test should be an indispensable part in regression modeling. This report compares the finite sample performance of several classical lack-of-fit tests in regression models via simulation studies. It has three chapters. The conception of the lack-of-fit test, together with its basic setup, is briefly introduced in Chapter 1; then several classical lack-of-fit test procedures are discussed in Chapter 2; finally, thorough simulation studies are conducted in Chapter 3 to assess the finite sample performance of each procedure introduced in Chapter 2. Some conclusions are also summarized in Chapter 3. A list of MATLAB codes that are used for the simulation studies is given in the appendix.

Doctor of Philosophy / Department of Statistics / James W. Neill / Checking the adequacy of a proposed parametric nonlinear regression model is important in order to obtain useful predictions and reliable parameter inferences. Lack of fit is said to exist when the regression function does not adequately describe the mean of the response vector. This dissertation considers asymptotics, implementation and a comparative performance for the likelihood ratio tests suggested by Neill and Miller (2003). These tests use constructed alternative models determined by decomposing the lack of fit space according to clusterings of the observations. Clusterings are selected by a maximum power strategy and a sequence of statistical experiments is developed in the sense of Le Cam. L2 differentiability of the parametric array of probability measures associated with the sequence of experiments is established in this dissertation, leading to local asymptotic normality. Utilizing contiguity, the limit noncentral chi-square distribution under local parameter alternatives is then derived. For implementation purposes, standard linear model projection algorithms are used to approximate the likelihood ratio tests, after using the convexity of a class of fuzzy clusterings to form a smooth alternative model which is necessarily used to approximate the corresponding maximum optimal statistical experiment. It is demonstrated empirically that good power can result by allowing cluster selection to vary according to different points along the expectation surface of the proposed nonlinear regression model. However, in some cases, a single maximum clustering suffices, leading to the development of a Bonferroni adjusted multiple testing procedure. In addition, the maximin clustering based likelihood ratio tests were observed to possess markedly better simulated power than the generalized likelihood ratio test with semiparametric alternative model presented by Ciprian and Ruppert (2004).

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