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Volume 9, Issue 4, July 2020, Page: 121-126
The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence
Abdushukurov Abdurahim Ahmedovich, Department of Applied Mathematics and Informatics, Faculty of Applied Mathematics and Informatics, Tashkent Branch of Moscow State University Named After M. V. Lomonosov, Tashkent, Uzbekistan
Kakadjanova Leyla Reshitovna, Department of Probability Theory and Mathematical Statistics, Faculty of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan
Received: Feb. 15, 2020;       Accepted: May 22, 2020;       Published: Jun. 4, 2020
Abstract
In the analysis of statistical data in biomedical treatments, engineering, insurance, demography, and also in other areas of practical researches, the random variables of interest take their possible values depending on the implementation of certain events. So in tests of physical systems (or individuals) on duration of uptime values of operating systems depend on subsystems failures, in insurance business insurance company payments to its customers depend on insurance claims. In such experimental situations, naturally become problems of studying the dependence of random variables on the corresponding events. The main task of statistics of such incomplete observations is estimating the distribution function or what is the same, the survival function of the tested objects. To date, there are numerous estimates of these characteristics or their functionals in various models of incomplete observations. In this paper investigated the asymptotic properties of sequential processes of independence of the integral structure and uniform versions of the strong law of large numbers and the central limit theorem for integral processes of independence by indexed classes are established. The obtained results can be used to construct statistics of criteria for testing a hypothesis of independence of random variables on the corresponding events.
Keywords
Empirical Processes, Metric Entropy, Glivenko-Cantelli and Donsker Theorems
Abdushukurov Abdurahim Ahmedovich, Kakadjanova Leyla Reshitovna, The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence, American Journal of Theoretical and Applied Statistics. Vol. 9, No. 4, 2020, pp. 121-126. doi: 10.11648/j.ajtas.20200904.15
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