CHAPTER 8 - ESTIMATION OF DISTRIBUTIONS

Abstract

This chapter discusses the nonparametric estimation methods for distribution of random variables. The estimation methods for the numerical characteristics of random variables and distribution parameters allows to find from the results of trials the distributions of random variables, which represent known functions, depending on a finite number of unknown scalar parameters, that is, on an unknown finite-dimensional parameter. Such an estimation of distributions is called a parametric estimation. The direct estimation of distributions without supposition that they are known functions depending on a finite-dimensional unknown parameter is called a nonparametric estimation. To obtain an estimator of a distribution function F(x), it is natural to replace the probabilities of the events X <x for various x by their frequencies. The problem of finding a confidence region for a distribution function is very complex and is not solved yet for vector random variables. For scalar random variables, this problem is rather simple because of the fact that for any continuous distribution function F(x), the distribution of the random variable S = max | F^(x)—F(x) | is independent of F(x).