Analysis of kernel density estimation of functions of random variables

In the current investigation, the problem,of estimating the probability density of a function of in in dependent identically distributed random variables, g(X-1, ..., X-m) is considered. Defining the integrated square contrast (ISC)and its mean (MISC), we study the central limit theorem of (ISO-MISC) as well as the second order approximation of both ISC and MISC. Via simulation and also using real data, we address some of the practical aspects of choosing the optimal bandwidth which minimizes the asymptotic MISC and its data based analog which minimizes ISC.