On Nonparametric Estimation of the Density of a Non-Negative Function of Observations

Abstract

Let { X1, ..., X n} be a random sample from a continuous distri- bution F defined on the k−dimensional Euclidean space Rk; for some k≥1. In many statistical applications we are interested in statistical properties of a function h( X1, ..., X m) of m ≥ 1 observations. Frees (1994, J. Amer. Stat. Assoc.) considered estimating the density function g associated with the distribution function [Formula: see text] using the kernel method. In many applications, though, the functions of interest are non-negative where the usual symmetric kernels applied in the kernel density estimation are not appropriate. This paper adapts the alter- native density estimator developed in Chaubey and Sen (1996, Statistics and Decisions) by smoothing the so called empirical kernel distribution function: [Formula: see text] where 1( A) denotes the indicator of A and [Formula: see text] denotes sum over all possible [Formula: see text] combinations. Applications and asymptotic properties of the alternative estimator are investigated.