Abstract:
In this thesis, we are concerned with the uniform in bandwidth consistency of kernel-type estimators of the regression function derived by modern empirical process theory, under weaker conditions on the kernel than previously used in the literature. Our theorems allow data-driven local bandwidths for these statistics. We extend existing uniform bounds on kernel type-estimator and making it adaptive to the intrinsic dimension of the underlying distribution, which will be characterising by the so-called intrinsic dimension. The thesis is divided in three main parts, we describe as follows. The first part is devoted to general empirical processes indexed by classes of functions. The results are obtained for uniformly bounded classes of functions or unbounded with envelope functions satisfying some moment conditions. The purpose of the second part is the statistical applications to illustrate the usefullness of the main contribution. Applications include the uniform in bandwidth consistency of the kernel type estimators for density, regression, the conditional distribution, multivariate mode, Shannon’s entropy, derivatives of density and regression functions. The third part is devoted to the uniform in bandwidth consistency for non-parametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship. These new results are applied for the non-parametric conditional density and conditional distribution functions.
