Nonparametric density estimation with a parametric start

TL;DR

This paper introduces a semiparametric density estimation method that improves upon kernel estimators by starting with a parametric estimate and applying a correction, especially effective in higher dimensions.

Contribution

It develops a novel class of semiparametric estimators combining parametric starts with kernel corrections, outperforming traditional kernel methods in many scenarios.

Findings

The new method often outperforms kernel density estimators, especially when the true density is close to the parametric start.

Extensive comparisons show advantages in normal mixture cases and higher-dimensional settings.

Procedures for selecting smoothing parameters are also proposed.

Abstract

The traditional kernel density estimator of an unknown density is by construction completely nonparametric, in the sense that it has no preferences and will work reasonably well for all shapes. The present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities, for example the normal, while not losing much in precision when the true density is far from the parametric class. The idea is to multiply an initial parametric density estimate with a kernel type estimate of the necessary correction factor. This works well in cases where the correction factor function is less rough than the original density itself. Extensive comparisons with the kernel estimator are carried out, including exact analysis for the class of all normal mixtures. The new method, with a normal start, wins quite often, even in many cases where the true density is far from normal. Procedures for choosing the smoothing parameter of the estimator are also discussed. The new estimator should be particularly useful in higher dimensions, where the usual nonparametric methods have problems. The idea is also spelled out for nonparametric regression.