Estimation of the moments of a quantized random variable

A significant part of the research on the problems of quantization of random variables is devoted to practical aspects of optimal quantization in the sense of filling in information. For these purposes, certain quantitative characteristics of quantized random variables are used, such as: mathematical expectation, variance and mean square deviation. At the same time, to determine the quantitative characteristics of quantized random variables, as a rule, well-known parametric distributions are used: uniform, exponential, normal and others. In real situations, it is not possible to identify the initial parametric distribution based on the available statistical information. In this paper, a nonparametric model is proposed for determining such numerical characteristics of a quantized random variable as the highest initial moments. The mathematical formalization of the problem of estimating the higher initial moments of a quantized random variable in the conditions of incomplete data represented by small samples of a quantized random variable is performed in the form of an optimization model of a certain integral of a piecewise continuous function satisfying certain conditions. The final estimates of the highest initial moments of the quantized random variable are found as extreme (lower and upper) estimates of a certain integral on a set of distribution functions with given moments equal to the sample moments of the quantized random variable. A model of the higher initial moments of a quantized random variable is presented in the form of a definite integral of a piecewise continuous function; in the general case, the problem of finding extreme (lower and upper) estimates of the higher initial moments of a quantized random variable on a set of distribution functions with given moments is solved. Examples of finding higher initial moments and optimal quantization of a random variable are given. The obtained results can be used by specialists in evaluating and optimizing the quantization of various information presented by random signals.