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.

Definition

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.

Example

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))".

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.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Empirical process (process control model) — There are two major approaches to controlling any process: * The defined process control model. * The empirical process control model.The empirical model of process control provides and exercises control through frequent inspection and adaptation …   Wikipedia

  • Empirical measure — In probability theory, an empirical measure is a random measure arising from a particular realization of a (usually finite) sequence of random variables. The precise definition is found below. Empirical measures are relevant to mathematical… …   Wikipedia

  • Empirical distribution function — In statistics, an empirical distribution function is a cumulative probability distribution function that concentrates probability 1/ n at each of the n numbers in a sample.Let X 1,ldots,X n be iid random variables in mathbb{R} with the cdf F ( x… …   Wikipedia

  • Empirical method — is generally taken to mean the collection of data on which to base a theory or derive a conclusion in science. It is part of the scientific method, but is often mistakenly assumed to be synonymous with the experimental method.The empirical method …   Wikipedia

  • Empirical formula — This article is about analytical chemistry. For observation rather than theory, see Empirical relationship. In chemistry, the empirical formula of a chemical compound is the simplest whole number ratio of atoms of each element present in a… …   Wikipedia

  • Empirical Duration — The calculation of a bond s duration based on historical data. Empirical duration is estimated statistically using historical market based bond prices and historical market based Treasury yields. When the historical yields change, the historical… …   Investment dictionary

  • Empirical orthogonal functions — In statistics and signal processing, the method of empirical orthogonal function (EOF) analysis is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data. It is the same as performing a… …   Wikipedia

  • Defined process — There are two schools of thought about what a defined process is. Contents 1 School of thought 1 2 School of thought 2 3 References 4 Books …   Wikipedia

  • Business process orientation — The concept of business process orientation (BPO) is based upon the work of Deming (Walton, 1996), Porter (1985), Davenport and Short (1990), Hammer (1993, 1996 and 1999), Grover et al (1995), and Coombs and Hull (1996). This body of work… …   Wikipedia

  • Is logic empirical? — is the title of two articles that discuss the idea that the algebraic properties of logic may, or should, be empirically determined; in particular, they deal with the question of whether empirical facts about quantum phenomena may provide grounds …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.