Theoretical Statistics and Mathematics Unit, ISI Delhi

January 16, 2013 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Y. P. Chaubey,
Concordia University, Montreal, Canada

Title:
On Nonparametric Estimators of the Density of a Non-negative Function of Observations

Abstract of Talk

Let $\{X_1,...,X_n\}$ be a random sample from a continuous distribution
$F$ defined on the $k-$dimensional Euclidean space $\mathbf{R}^k,$ for some $k\ge 1.$ In many statistical applications we are interested in statistical properties of a function $h(X_1,...,X_m)$ of $m\ge 1$ observations. Frees (1994, {\it J. Amer. Stat. Assoc.}) considered estimating the density function $g$ associated with the distribution function
$$ G(t)=P\bigl(
h(X_1,...,X_m)\le t \bigr)$$
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 alternative density estimators developed in Chaubey and Sen (1996, {\it Statistics and Decisions}) and Chaubey {\it et al.} (2012, {\it J. Ind. Statist. Assoc.}) by smoothing the so called {\it empirical kernel distribution function}:
$$ G_n(t)={n\choose m}^{-1}\sum_{(n,m)}
\mb{1}\bigl(h(X_{i_1},X_{i_2},...,X_{i_m})\le t\bigr), $$ where
$\mb{1}(A)$ denotes the indicator of $A$ and $ \displaystyle{\sum_{(n,m)}}$ denotes sum over all possible ${n\choose m}$ combinations. Applications and asymptotic properties of the alternative estimators are investigated.