Empirical process

The study of empirical processes is a branch of mathematical statistics and a sub-area of probability theory. It is a generalization of the central limit theorem for empirical measures.


It is known that under certain conditions empirical measures P_n uniformly converge to the probability measure "P" (see Glivenko-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 certain Gaussian process. It can be shown that the Donsker classes are Glivenko-Cantelli, the converse is not true in general.


As an example, consider empirical distribution functions. For real-valued iid 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))".


* 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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