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Asymptotic Statistics by A.W. van der Vaart

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19

Empirical Processes

           The empirical distribution of a random sample is the uniform discrete measure on the observations. In this chapter, we study the convergence of this measure and in particular the convergence of the corresponding distribution function. This leads to laws of large numbers and central limit theorems that are uniform in classes of functions. We also discuss a number of applications of these results.

19.1 Empirical Distribution Functions

Let X1,..., Xn be a random sample from a distribution function F on the real line. The empirical distribution function is defined as

image

It is the natural estimator for the underlying ...

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