On a Class of Nonparametric Density and Regression Estimators

Abstract

A class of maximum penalized likelihood estimators (MPLE) of the density function $f$ is constructed, through the use of a rather general roughness-penalty functional. This class contains all the density estimates in the literature that arise as solutions to MPLE problems with penalties on $f^{1/2}$. In addition, the flexibility of the penalty functional permits the construction of new spline estimates with improved performance at the peaks and valleys of the density curves. The consistency of the estimators in probability and a.s., in the $L_p(\mathbb{R}) -$ norms, $p = 1, 2, \infty$, in the Hellinger metric and Sobolev norms is established in a unified manner. A class of penalty functionals is identified which leads to estimators which approach the optimal rates of convergence predicted in Farrell (1972). Based on the above estimates, a class of MPLE regression estimators is introduced which has the appealing property of reducing to the classical nonparametric regression estimates when a smoothing parameter goes to zero. Finally, a theoretically justifiable and numerically efficient method for a data based choice of the smoothing parameter is proposed for further study. A number of numerical examples and graphs are presented.