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.
Highlights
The modern asymptotic theory of empirical processes indexed by a class of measurable functions is actively developing and the current results are detailed in monographs [5, 6, 8, 9, 12,13,14], in articles [4, 10, 15]
The main results of this theory allow us to establish uniform versions of the laws of large numbers and central limit theorems for empirical measures under the imposition of entropy conditions for a class of measurable functions
To generalize Glivenko-Cantelli theorem for a certain class of sets Vapnik and Chervonenkis in 70-s years of the last century made a significant contribution to the development of statistical learning theory, which justifies the principle of minimizing empirical risk
Summary
The modern asymptotic theory of empirical processes indexed by a class of measurable functions is actively developing and the current results are detailed in monographs [5, 6, 8, 9, 12,13,14], in articles [4, 10, 15]. The main results of this theory allow us to establish uniform versions of the laws of large numbers and central limit theorems for empirical measures under the imposition of entropy conditions for a class of measurable functions. These results are essentially generalized analogues of the classical theorems of Glivenko-Cantelli and Donsker. In papers [1,2,3, 7] it was established the uniform (by corresponding indexing classes F and D ) variants of strong laws of large numbers (Glivenko-Cantelli type) and central limit theorems (Donsker type), respectively, for processes (3) and (4). We will study another variant of the processes of the form (2) and for it uniform variants of the above limit theorems will be proved
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