The Kernel distribution estimator of functions of random variables

摘要

Let X_1, ... , X_m be identically and independently distributed (i.i.d.) observations on a random variable X with unknown distribution function F and density f. The kernel density estimate of the function g(X_1, ... , X_m) was proposed by [Frees, E., 1994, Estimating densities of functions of observations. Journal of the American Statistical Association, 89, 517-525]. In this paper, we propose a kernel method to estimate the distribution function of the function g(X_1, ... X_m) and we derive the asymptotic mean-square error for the estimator. In addition, we propose two data-based methods to obtain the bandwidth.