J. Koronacki

Abstract

Let X (1) ; : : : ; X (m) be m independent, real-valued random variables with unknown densities and let Y = \Psi(X (1) ; : : : ; X (m) ) be their known functional. The aim is to estimate the density of the random variable Y , given n independent realizations of the vector (X (1) ; : : : ; X (m) ). In the paper, three natural ways of solving the estimation problem, all based on the Rosenblatt-Parzen estimator, are proposed and compared by means of a simulation study. Applicability of each of the three methods is thus assessed. Simulations suggest that the "direct" method based on the observations of Y only, is "almost always" outperformed by the other two methods. Recommendations for choosing between the methods are given. Key words: density estimation, kernel estimators, estimating density of a functional. Subject classification: Primary 62G05. ) The paper was completed while on leave of this author to the Department of Statistics, Rice University, Houston, Texas. Partia..