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 the empirical measures.

Contents

Definition

Example

References

External links

Definition

It is known that under certain conditions empirical measures $P\_n$ uniformly converge to the probability measure ''P'' (see Glivenko-Cantelli theorem). ''Empirical processes'' provide 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
:
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'
: is called ''empirical process'' indexed by $mathcal\{C\}$, a collection of measurable subsets of ''S''.
: is called ''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.

★ J. Wolfowitz, Generalization of the theorem of Glivenko-Cantelli. Annals of Mathematical Statistics, '25', 131-138, 1954.

External links

★ Empirical Processes: Theory and Applications, by David Pollard, a textbook available online.

★ Introduction to Empirical Processes and Semiparametric Inference, by Michael Kosorok, another textbook available online.