# Empirical process

The study of

**empirical processes**is a branch ofmathematical statistics and a sub-area ofprobability theory . It is a generalization of thecentral limit theorem forempirical measure s.**Definition**It is known that under certain conditions

empirical measure s $P\_n$ uniformly converge to theprobability measure "P" (seeGlivenko-Cantelli theorem ). The theory of "Empirical processes" provides the rate of this convergence.A centered and scaled version of the empirical measure is the signed measure:$G\_n(A)=sqrt\{n\}(P\_n(A)-P(A))$It induces map on measurable functions "f" given by

:$fmapsto\; G\_n\; f=sqrt\{n\}(P\_n-P)f=sqrt\{n\}left(frac\{1\}\{n\}sum\_\{i=1\}^n\; f(X\_i)-mathbb\{E\}f\; ight)$

By the

central limit theorem , $G\_n(A)$converges in distribution to a normal random variable "N(0,P(A)(1-P(A)))" for fixed measurable set "A". Similarly, for a fixed function "f", $G\_nf$ converges in distribution to a normal random variable $N(0,mathbb\{E\}(f-mathbb\{E\}f)^2)$, provided that $mathbb\{E\}f$ and $mathbb\{E\}f^2$ exist.**Definition**:$igl(G\_n(c)igr)\_\{cinmathcal\{C$ is called an "empirical process" indexed by $mathcal\{C\}$, a collection of measurable subsets of "S".:$igl(G\_nfigr)\_\{finmathcal\{F$ is called an "empirical process" indexed by $mathcal\{F\}$, a collection of measurable functions from "S" to $mathbb\{R\}$.A significant result in the area of empirical processes is

Donsker's theorem . It has led to a study of the "Donsker classes" such that empirical processes indexed by these classes converge weakly to a certainGaussian process . It can be shown that the Donsker classes are Glivenko-Cantelli, the converse is not true in general.**Example**As an example, consider

empirical distribution function s. For real-valuediid random variables $X\_1,X\_n,...$ they are given by:$F\_n(x)=P\_n((-infty,x]\; )=P\_nI\_\{(-infty,x]\; \}.$

In this case, empirical processes are indexed by a class $mathcal\{C\}=\{(-infty,x]\; :xinmathbb\{R\}\}.$ It has been shown that $mathcal\{C\}$ is a Donsker class, in particular,:$sqrt\{n\}(F\_n(x)-F(x))$ converges weakly in $ell^infty(mathbb\{R\})$ to a

Brownian bridge "B(F(x))".**References*** P. Billingsley,

*Probability and Measure,*John Wiley and Sons, New York, third edition, 1995.

* M.D. Donsker, Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems,*Annals of Mathematical Statistics*, 23:277-281, 1952.

* R.M. Dudley, Central limit theorems for empirical measures,*Annals of Probability,*6(6): 899-929, 1978.

* R.M. Dudley,*Uniform Central Limit Theorems,*Cambridge Studies in Advanced Mathematics, 63, Cambridge University Press, Cambridge, UK, 1999.

* M.R. Kosorok,*Inroduction to Empirical Processes and Semiparametric Inference,**Springer, New York, 2008.*

* Aad W. van der Vaart and Jon A. Wellner,*Weak Convergence and Empirical Processes: With Applications to Statistics,*2nd ed., Springer, 2000. ISBN 978-0387946405

* J. Wolfowitz, Generalization of the theorem of Glivenko-Cantelli.*Annals of Mathematical Statistics,***25**, 131-138, 1954.**External links**** [**http://www.stat.yale.edu/~pollard/Iowa/ Empirical Processes: Theory and Applications*] , by David Pollard, a textbook available online.

* [*http://www.bios.unc.edu/~kosorok/current.pdf Introduction to Empirical Processes and Semiparametric Inference*] , by Michael Kosorok, another textbook available online.

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