Master of Science / Department of Statistics / Weixing Song / A nonparametric lack-of-fit test is proposed to check the adequacy of the presumed parametric form for the regression function in Tobit regression models by applying Zheng's device with weighted residuals. It is shown that testing the null hypothesis for the standard Tobit regression models is equivalent to test a new null hypothesis of the classic regression models. An optimal weight function is identified to maximize the local power of the test. The test statistic proposed is shown to be asymptotically normal under null hypothesis, consistent against some fixed alternatives, and has nontrivial power for some local nonparametric power for some local nonparametric alternatives. The finite sample performance of the proposed test is assessed by Monte-Carlo simulations. An empirical study is conducted based on the data of University of Michigan Panel Study of Income Dynamics for the year 1975.

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.