| We address some inferential issues for unknown regression curves and surfaces under a nonparametric regression framework.;In the first portion, we examine the properties of three nonparametric tests, which are available in the literature, for testing the equality of two regression curves. All of these tests involve user-defined smoothing parameters, which have a heavy influence on the performance of each test. We propose two different approaches to select data-based smoothing parameters and show that the proposed smoothing parameters optimize the power performance in each test as sample sizes get large. We also propose a new test using Gateaux derivative of the likelihood function and study its asymptotic properties. Simulations have been performed to evaluate each procedure.;In the second portion, we investigate some estimation properties of the Generalized Additive Models (GAM), which are a special case of multiple covariate regression models. We first justify the validity of the derivation of normal equations for a data setting, which have been used to find component estimators of a GAM. We then examine the existence of component solutions in a two dimensional case providing some necessary and sufficient conditions.;In the third portion, we consider the test of equality of two regression surfaces with two covariates. We develop a novel testing procedure by reducing the problem of testing two dimensional regression surfaces to a problem of testing one dimensional regression curves. The asymptotic properties for the proposed method have been obtained.;In addition, we also establish some results on uniform convergence rates of the kernel estimators of density and regression functions, strengthening the results in the literature. |
