Limiting Distributions of Deviations of Kernel-Type Density Estimators

Abstract

1. In the preceding chapter we investigated the problem of strong consistency in various functional metrics of kernel-type density estimators. The theorems proved therein were basically of a qualitative nature. The problem of accuracy and reliability of approximations for large samples was not addressed in these theorems. In this section the limiting behavior of the distribution of the maximal deviation of an empirical density f_n(x) of the type (1.1.1) from the unknown density f(x) (the so-called N. V. Smirnov problem) is investigated. The theorems we are going to prove permit us to carry out statistical estimation of the degree of approximation and to construct for an unknown theoretical density a confidence region with a preassigned confidence coefficient. As it is known, the groundbreaking work in this direction is N. V. Smirnov’s paper [2] in which the limiting distribution of the maximal deviation for the case when f_n(x) is a ‘histogram’ was obtained.