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Journal of Automation and Information Sciences
SJR: 0.275 SNIP: 0.59 CiteScore™: 0.8

ISSN Imprimir: 1064-2315
ISSN En Línea: 2163-9337

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Journal of Automation and Information Sciences

DOI: 10.1615/JAutomatInfScien.v51.i12.20
pages 7-17

Estimation of Experimental Distribution Function on the Basis of Finite Samples of Random Variable

Andrey B. Lozynsky
G.V. Karpenko Physico-Mechanical Institute of National Academy of Sciences of Ukraine, Lvov
Igor M. Romanyshyn
G.V. Karpenko Physico-Mechanical Institute of National Academy of Sciences of Ukraine, Lvov
Bogdan P. Rusyn
G.V. Karpenko Physico-Mechanical Institute of National Academy of Sciences of Ukraine, Lvov

SINOPSIS

A new approach to estimating the distribution function of a random variable based on finite (including small) samples is proposed. The approach is based on determining the estimates of positions of points of the distribution function. The theoretical substantiation of the approach and the algorithm for estimating the distribution function are presented. The equations (the integral) for determining the points on the required distribution function are exact. Therefore, with the knowledge of the distribution of any order statistics the points of the required distribution function are determined exactly. In the simplest case the approach can be reduced to determining the median of statistics. Usually its exact value is unknown. Therefore the sample median is used as the median of the statistics distribution. The consistency, bias and effectiveness of estimates of the points position on the required distribution function are considered. It is shown that the proposed estimates are consistent but biased, and the bias depends on the form of the required distribution and decreases with the increasing size of the subsample. Numerical experiments for the most important practical cases indicate the increased effectiveness of the proposed approach compared to the classical one. The simplified algorithms for the approximate determination of distribution parameters are proposed and the error estimates are given which indicate the acceptability of the proposed simplified algorithms. The paper presents the results of numerical modeling which illustrate the correspondence of the constructed distribution function with the model one and the advantages of this approach. It is noted that the estimation of the distribution function based on the proposed approach in the form of the interpolated curve allows one to perform the numerical differentiation, filtering and other signal processing operations. At the same time the classical distribution function in the form of the step curve has a number of known limitations at the application of these processing operations. The method for filtering a signal with uncorrelated pulse interference based on the proposed approach is described. The Appendix gives the determination of abscissas and ordinates of the required distribution function of a random variable by solving the integral equations in the analytical form for the case of uniform distribution based on subsamples of two elements. The obtained solution is fully consistent with simplified algorithms.

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