Performance of Efron and Tibshirani's semiparametric density estimator

TL;DR

This paper evaluates the performance of Efron and Tibshirani's semiparametric density estimator by deriving bias and variance, comparing it with other methods, and demonstrating its strengths and limitations through empirical analysis.

Contribution

It provides explicit formulas for bias and variance of the estimator, enabling performance comparison with other recent semiparametric density estimators.

Findings

The estimator often outperforms kernel methods near the normal distribution.

It is generally less effective than the Hjort and Glad estimator in most test cases.

Performance varies depending on the choice of polynomial order in the exponential correction.

Abstract

Recently, Efron and Tibshirani (Annals of Statistics, 1996) proposed a semiparametric density estimator, which works by multiplying an initial kernel type estimate with a parametric exponential type correction factor, chosen so as to match certain empirical moments. While Efron and Tibshirani investigate and illustrate many aspects of their method, the basic questions of performance, and comparison with other density estimators, were not directly addressed in their article. The purpose of the present paper is to provide formulae for bias and variance and hence mean squared error for the estimator. This additional insight into the method makes it easy to compare its performance with that of other recently proposed semiparametric constructions. A brief comparison study is carried out here. It indicates that the new method, used with lower order polynomials in the exponential correction term, is often better than the kernel estimator, in a reasonable neighbourhood around the normal distribution, but that its performance as a density estimator is more than equalled by other methods. In particular, the recently developed Hjort and Glad estimator (Annals of Statistics, 1995), using a parametric start times a nonparametric correction, wins in eight out of nine test cases, from the list of such suggested by Wand and Jones (Annals of Statistics, 1992).