Integral Estimate from the Square of the Probability Density for a One-Dimensional Random Variable
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INTEGRAL ESTIMATE FROM THE SQUARE OF THE PROBABILITY DENSITY FOR A ONE-DIMENSIONAL RANDOM VARIABLE A. V. Lapko1,2 and V. A. Lapko1,2
UDC 519.24
The problems of the direct estimate of a functional under study are examined with the use of nuclear statistics. In selecting an optimal number of sampling intervals of the range of values of a one-dimensional random variable, it has been established that the functional of the square of the probability density is a constant, the value of which does not depend on the probability density parameters. The functional dependences of the constant being studied on the coefficient of antikurtosis of the distribution law of the random variable is determined. The established dependences for sets of log-normal probability density, the Student distribution laws, and sets of probability densities with Gaussian distribution are analyzed. From the results obtained, a generalized model between the studied constant and the coefficient of antikurtosis that does not depend on the form of the probability density is generated, and is determined by an estimate of the specified coefficient. A methodology for estimating the integral from the square of the probability density was developed. The effectiveness of the proposed methodology is confirmed by the results of computational tests. The conditions of the computational tests differ significantly from the information applied in synthesis of models of dependence between the studied constant and the coefficient of antikurtosis. Conditions for the adequacy of the methodology for estimating the integral of the square of a probability density from the coefficient of antikurtosis were established using the proposed models of the dependence of the studied constant on the conditions of computational tests. Keywords: functional of the square of probability density, one-dimensional random variable, coefficient of antikurtosis, mixture of probability densities, log-normal distribution, Student distribution law.
Introduction. The traditional approach for calculating functionals from the probability density of p(x) under conditions of uncertainty consists in the use of nuclear estimates of p(x) [1–11]. The implementation of this approach is accompanied by significant computational expenditures and the necessity for studying the asymptotic properties of nonparametric estimates of the considered functionals from the probability density. In the analysis in [12] of the asymptotic properties of the change of the nonparametric estimate of probability density, it has been well-established that the functional W2
p2 (x) dx
(1)
is a constant for distribution laws with finite limits α and β of changes of values of random variables. Here Δ = (β – α) is the length of the interval of modification of their values. In order to compare the regularities inherent in random variables with finite and infinite intervals of values, we will introduce a definition. For infinite α and β, the value of Δ will be understood to be the minimum length D of the basic part of Δ for whi
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