Empirical Processes

Empirical Processes

Corstanje, M.A.

Jongbloed, G. (mentor)

Electrical Engineering, Mathematics and Computer Science

Delft Institute of Applied Mathematics

2016-06-27

Let us start with a random sample X1, . . . , Xn that is independent and identically distributed and has distribution function F. When we wish to test the hypothesis H0 : F = F0 for some known function F0, we are doing a goodness of fit test. To do a goodness of fit test, several test statistics can be used. Examples are the Kolmogorov-Smirnov test statistic p n supt jFn(t) 􀀀F(t)j and the Cram´er-Von Mises test statistic n R (Fn 􀀀 F)2dF where Fn denotes the empirical distribution function of X1, . . . , Xn. When looking at several goodness of fit test statistics, one might p notice that most of them can be expressed in terms of the function Un(t) = n(Fn(t) 􀀀 F(t)). This process is called the empirical process and its asymptotic behaviour tells us a lot about the asymptotic behaviour of the goodness of fit test statistics. However, various problems can be encountered when looking atUn as a function. Even the question whetherUn is a random function whenever X1, . . . , Xn are random variables has an answer that is far from trivial. In this thesis, this problem will be solved by introducing a function space equipped with a metric such that the empirical distribution function becomes a random element of this function space. In order to make sure that the empirical distribution function is indeed a random function, we will consider the uniform metric and the Skorohod metric and discuss the corresponding Borel s-fields. Furthermore, we will look into some asymptotic properties of the empirical process and prove Donskers theorem. This result will be used to estimate the distributions of some goodness of fit test statistics using R.

http://resolver.tudelft.nl/uuid:fd43d5d7-27f7-4629-bd80-7da848925504

Student theses

bachelor thesis

(c) 2017 Corstanje, M.A.