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 R^k; 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

$G\left(t\right)\hspace{0.17em}=\hspace{0.17em}P(h\left(X_1,\mathrm{...},\hspace{0.17em}{X}_m\right)\hspace{0.17em}\le \hspace{0.17em}t)$

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:

$Gn\left(t\right)\hspace{0.17em}\text{=}\hspace{0.17em}{\left(\underset{m}{\overset{n}{}}\right)}^{-1}{\displaystyle \sum _{\left(n,m\right)}1}\left(h\left(X_{i1},X_{i2},\mathrm{...},X_{im}\right)\le t\right),$

where 1(A) denotes the indicator of A and ${\displaystyle \sum _{\left(n,m\right)}}$ denotes sum over all possible $\left(\begin{array}{c}n\\ m\end{array}\right)$ combinations. Applications and asymptotic properties of the alternative estimator are investigated.