The stochastic approximation method for the estimation of a multivariate probability density

TL;DR

This paper explores recursive kernel estimators for multivariate probability densities using stochastic approximation, comparing their properties and performance to Rosenblatt's nonrecursive estimator, with findings favoring recursive methods for confidence intervals.

Contribution

It introduces a broad class of recursive kernel estimators via stochastic approximation and compares their effectiveness to traditional nonrecursive estimators.

Findings

Recursive estimators are less suitable for pointwise estimation than Rosenblatt's estimator.

Recursive estimators outperform Rosenblatt's for constructing confidence intervals.

Nonrecursive Rosenblatt's estimator is preferable for pointwise density estimation.

Abstract

We apply the stochastic approximation method to construct a large class of recursive kernel estimators of a probability density, including the one introduced by Hall and Patil (1994). We study the properties of these estimators and compare them with Rosenblatt's nonrecursive estimator. It turns out that, for pointwise estimation, it is preferable to use the nonrecursive Rosenblatt's kernel estimator rather than any recursive estimator. A contrario, for estimation by confidence intervals, it is better to use a recursive estimator rather than Rosenblatt's estimator.