Doctor of Philosophy / Department of Statistics / Weixing Song / The regression model has been given a considerable amount of attention and played a significant role in data analysis. The usual assumption in regression analysis is that the variances of the error terms are constant across the data. Occasionally, this assumption of homoscedasticity on the variance is violated; and the data generated from real world applications exhibit heteroscedasticity. The practical importance of detecting heteroscedasticity in regression analysis is widely recognized in many applications because efficient inference for the regression function requires unequal variance to be taken into account. The goal of this thesis is to propose new testing procedures to assess the adequacy of fitting parametric variance function in heteroscedastic regression models. The proposed tests are established in Chapter 2 using certain minimized L[subscript]2 distance between a nonparametric and a parametric variance function estimators. The asymptotic distribution of the test statistics corresponding to the minimum distance estimator under the fixed model and that of the corresponding minimum distance estimators are shown to be normal. These estimators turn out to be [sqrt]n consistent. The asymptotic power of the proposed test against some local nonparametric alternatives is also investigated. Numerical simulation studies are employed to evaluate the nite sample performance of the test in one dimensional and two dimensional cases. The minimum distance method in Chapter 2 requires the calculation of the integrals in the test statistics. These integrals usually do not have a tractable form. Therefore, some numerical integration methods are needed to approximate the integrations. Chapter 3 discusses a nonparametric empirical smoothing lack-of-fit test for the functional form of the variance in regression models that do not involve evaluation of integrals. empirical smoothing lack-of-fit test can be treated as a nontrivial modification of Zheng (1996)'s nonparametric smoothing test and Koul and Ni (2004)'s minimum distance test for the mean function in the classic regression models. The asymptotic normality of the proposed test under the null hypothesis is established. Consistency at some fixed alternatives and asymptotic power under some local alternatives are also discussed. Simulation studies are conducted to assess the nite sample performance of the test. The simulation studies show that the proposed empirical smoothing test is more powerful and computationally more efficient than the minimum distance test and Wang and Zhou (2006)'s test.

In this study we develop a new method for testing the null hypothesis that the predictor function in a canonical link regression model has a prescribed linear form. The class of models, which we will refer to as canonical link regression models, constitutes arguably the most important subclass of generalized linear models and includes several of the most popular generalized linear models. In addition to the primary contribution of this study, we will revisit several other tests in the existing literature. The common feature among the proposed test, as well as the existing tests, is that they are all based on orthogonal series estimators and used to detect departures from a null model. Our proposal for a new lack-of-fit test is inspired by the recent contribution of Hart and is based on a Laplace approximation to the posterior probability of the null hypothesis. Despite having a Bayesian construction, the resulting statistic is implemented in a frequentist fashion. The formulation of the statistic is based on characterizing departures from the predictor function in terms of Fourier coefficients, and subsequent testing that all of these coefficients are 0. The resulting test statistic can be characterized as a weighted sum of exponentiated squared Fourier coefficient estimators, whereas the weights depend on user-specified prior probabilities. The prior probabilities provide the investigator the flexibility to examine specific departures from the prescribed model. Alternatively, the use of noninformative priors produces a new omnibus lack-of-fit statistic. We present a thorough numerical study of the proposed test and the various existing orthogonal series-based tests in the context of the logistic regression model. Simulation studies demonstrate that the test statistics under consideration possess desirable power properties against alternatives that have been identified in the existing literature as being important.

Master of Science / Department of Statistics / James W. Neill / Cluster based lack of fit tests for linear regression models are generally effective in detecting model inadequacy due to between- or within-cluster lack of fit. Typically, lack of fit exists as a combination of these two pure types, and can be extremely difficult to detect depending on the nature of the mixture. Su and Yang (2006) and Miller and Neill (2007) have proposed lack of fit tests which address this problem. Based on a simulation comparison of the two testing procedures, it is concluded that the Su and Yang test can be expected to be effective when the true model is locally well approximated within each specified cluster and the lack of fit is not due to an unspecified predictor variable. The Miller and Neill test accommodates a broader alternative, which can thus result in comparatively lower but effective power. However, the latter test demonstrated the ability to detect model inadequacy when the lack of fit was a function of an unspecified predictor variable and does not require a specified clustering for implementation.

Sigmoid curves have found broad applicability in biological sciences and biopharmaceutical research during the last decades. A well planned experiment design is essential to accurately estimate the parameters of the model. In contrast to a large literature and extensive results on optimal designs for linear models, research on the design for nonlinear, including sigmoid curve, models has not kept pace. Furthermore, most of the work in the optimal design literature for nonlinear models concerns the characterization of minimally supported designs. These minimal, optimal designs are frequently criticized for their inability to check goodness of fit, as there are no additional degrees of freedom for the testing. This design issue can be a serious problem, since checking the model adequacy is of particular importance when the model is selected without complete certainty. To assess for lack of fit, we must add at least one extra distinct design point to the experiment. The goal of this dissertation is to identify optimal or highly efficient designs capable of checking the fit for sigmoid curve models. In this dissertation, we consider some commonly used sigmoid curves, including logistic, probit and Gompertz models with two, three, or four parameters. We use D-optimality as our design criterion. We first consider adding one extra point to the design, and consider five alternative designs and discuss their suitability to test for lack of fit. Then we extend the results to include one more additional point to better understand the compromise among the need of detecting lack of fit, maintaining high efficiency and the practical convenience for the practitioners. We then focus on the two-parameter Gompertz model, which is widely used in fitting growth curves yet less studied in literature, and explore three-point designs for testing lack of fit under various error variance structures. One reason that nonlinear design problems are so challenging is that, with nonlinear models, information matrices and optimal designs depend on the unknown model parameters. We propose a strategy to bypass the obstacle of parameter dependence for the theoretical derivation. This dissertation also successfully characterizes many commonly studied sigmoid curves in a generalized way by imposing unified parameterization conditions, which can be generalized and applied in the studies of other sigmoid curves. We also discuss Gompertz model with different error structures in finding an extra point for testing lack of fit. / Statistics