The Annals of Statistics

On a Class of Nonparametric Density and Regression Estimators

V. K. Klonias

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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.

Article information

Ann. Statist., Volume 12, Number 4 (1984), 1263-1284.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G10: Hypothesis testing 60F15: Strong theorems 41A25: Rate of convergence, degree of approximation 60F25: $L^p$-limit theorems 40A30: Convergence and divergence of series and sequences of functions 62E10: Characterization and structure theory

Nonparametric density estimation nonparametric regression estimation maximum penalized likelihood method strong consistency rates of convergence likelihood functional spline function data based smoothing parameters


Klonias, V. K. On a Class of Nonparametric Density and Regression Estimators. Ann. Statist. 12 (1984), no. 4, 1263--1284. doi:10.1214/aos/1176346791.

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