Special empirical processes of independence indexed by classes of measurable functions

Abstract

When analyzing statistical data in biomedical research, insurance, demography, as well as in other areas of practical research, random variables of interest take on certain values depending on the occurrence of certain events, e.g., when testing physical systems (individuals), the values of their operating time depend on the failures of subsystems; in the insurance business, the payments of insurance companies to their customers depends on insured events. In such experimental situations, the problems of studying the dependence of random variables on the corresponding events become rather important thus entailing the necessity to study the limiting properties of empirical processes indexed by classes of functions. The modern theory of empirical processes generalizes the classical results of the laws of large numbers, central and other limit theorems uniformly over the entire class of indexing under the imposition of entropy conditions. These theorems are generalized analogs of the classical theorems of Glivenko – Cantelli and Donsker. Special empirical processes are proposed in the study to check the independence of a random variable and an event. The properties of convergence of empirical processes to the corresponding Gaussian processes are analyzed. The results obtained are used to test the validity of the random right censoring model.