Estimation of the Multi-Dimensional Probability Density Function.
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TL;DR
The author examines the rate of convergence of the empirical densities to f, and considers the situation when there is 'noise' in the observations (X sub k).
Abstract
Suppose X1,X2,... are independent, identically distributed (i.i.d.) (R sup d) valued random variables with common probability density function f. A problem of considerable practical importance and also of theoretical interest is the estimation of f through some statistic based on the observed sequence (X sub k). Such statistics are called empirical density functions, and the author examines the rate of convergence of the empirical densities to f. The author also considers the situation when there is 'noise' in the observations (X sub k).