On Nonparametric Estimation of the Mathematical Expectation of a Function of Random Variables with Identical Distributions

Abstract

We consider the classical problem of nonparametric estimation of the mathematical expectation of a function of independent random variables. In contrast to the traditional formulation, it is assumed that some random variables have identical distributions. For estimation, there are the samples whose number coincides with the number of unknown distributions. Traditional nonparametric estimation uses empirical distribution functions, which leads to biased estimates. The resampling approach proposes the following procedure. For each random function’s argument, an element from the corresponding sample is drawn at random without replacement, and it is taken as the argument’s value in the given realization. Then the function’s value is calculated and stored. After that all drawn elements are returned to their samples and the procedure is repeated many times. The estimator is the arithmetic mean of the obtained function’s values. This estimator is unbiased. This paper describes the process of calculation of the resampling estimator variance and the bias of the traditional nonparametric estimator.