| This thesis studies some asymptotics of some error density and distribution function estimators in nonparametric regression models. First, the histogram type density estimator based on nonparametric regression residuals obtained from the full sample is shown to be uniformly weakly and strongly consistent with some rates. Uniform weak consistency with a rate is also obtained for the empirical distribution function of these residuals. We also show the weak and strong uniform consistency of the kernel type error density estimator.; Furthermore, if one uses a part of the sample to estimate the regression function and the other part to estimate the error density, then the asymptotic distribution of the maximum of a suitably normalized deviation of the density estimator from the true error density function is the same as in the case of the one sample setup. Similarly, a suitably standardized nonparametric residual empirical process based on the second part of the sample is shown to weakly converge to a time transformed Brownian bridge. These asymptotic distribution results can be used to test the goodness-of-fit hypothesis, pertaining to the error density and distribution functions, thereby enhancing the domain of their applications. |
