Bias reduction for kernel estimates of density functionals

Professor Martin Hazelton
Institute of Information Sciences and Technology, Massey University

There are a number of important statistical functions that can be expressed as simple functionals of probability densities. These include the relative risk function (a ratio of typically bivariate densities used in geographical epidemiology and elsewhere) and the binary regression function. In many cases parametric models are insufficiently flexible to describe these functionals and a nonparametric approach is to be preferred.

Nonparametric estimation of such functionals can be achieved by substituting kernel estimates in place of the unknown densities. Moreover, in principle we can obtain improved performance in the functional estimates by applying a range of bias reduction techniques developed for density estimation per se. However, in practice this approach tends to lead to poor results.

In this talk I will describe a new methodology which combines local bias reduction techniques borrowed from the density estimation literature with global smoothing optimized for the particular functional to be estimated. The results are encouraging.

The methodology is illustrated through examples on binary regression for low birth weight data, and on geographical variation in the relative risk of cancer of the larynx.

Session 1a, Statistical Methodology: 11:50 — 12:10, Room 446

Presentation Program