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Empirical process

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Stochastic process in probability theory

For the process control topic, seeProcess control § Control model.

Inprobability theory, anempirical process is astochastic process that characterizes the deviation of theempirical distribution function from its expectation.Inmean field theory, limit theorems (as the number of objects becomes large) are considered and generalise thecentral limit theorem forempirical measures. Applications of the theory of empirical processes arise innon-parametric statistics.^[1]

Definition

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ForX₁,X₂, ...X_nindependent and identically-distributed random variables with values in $\mathbb {R}$ andcumulative distribution functionF(x), the empirical distribution function is defined as

F_{n}(x)={\frac {1}{n}}\sum _{i=1}^{n}I_{(-\infty ,x]}(X_{i}),

where I_C is theindicator function of the setC.

For every (fixed)x,F_n(x) is a sequence of random variables which converge toF(x)almost surely by the stronglaw of large numbers. That is,F_n converges toFpointwise. Glivenko and Cantelli strengthened this result by provinguniform convergence ofF_n toF by theGlivenko–Cantelli theorem.^[2]

A centered and scaled version of the empirical measure is thesigned measure

G_{n}(A)={\sqrt {n}}(P_{n}(A)-P(A))

It induces a map on measurable functionsf given by

f\mapsto 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\right)

By thecentral limit theorem, $G_{n}(A)$ converges in distribution to anormal random variableN(0, P(A)(1 − P(A))) for fixed measurable setA. Similarly, for a fixed functionf, $G_{n}f$ 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

{\bigl (}G_{n}(c){\bigr )}_{c\in {\mathcal {C}}}

is called anempirical process indexed by

{\mathcal {C}}

, a collection of measurable subsets ofS.

{\bigl (}G_{n}f{\bigr )}_{f\in {\mathcal {F}}}

is called anempirical process indexed by

{\mathcal {F}}

, a collection of measurable functions fromS to

\mathbb {R}

A significant result in the area of empirical processes isDonsker's theorem. It has led to a study ofDonsker classes: sets of functions with the useful property that empirical processes indexed by these classesconverge weakly to a certainGaussian process. While it can be shown that Donsker classes areGlivenko–Cantelli classes, the converse is not true in general.

Example

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As an example, considerempirical distribution functions. For real-valuediid random variablesX₁,X₂, ...,X_n they are given by

F_{n}(x)=P_{n}((-\infty ,x])=P_{n}I_{(-\infty ,x]}.

In this case, empirical processes are indexed by a class ${\mathcal {C}}=\{(-\infty ,x]:x\in \mathbb {R} \}.$ It has been shown that ${\mathcal {C}}$ is a Donsker class, in particular,

{\sqrt {n}}(F_{n}(x)-F(x))

convergesweakly in

\ell ^{\infty }(\mathbb {R} )

to aBrownian bridgeB(F(x)) .

References

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^Mojirsheibani, M. (2007). "Nonparametric curve estimation with missing data: A general empirical process approach".Journal of Statistical Planning and Inference.137 (9):2733–2758.doi:10.1016/j.jspi.2006.02.016.
^Wolfowitz, J. (1954)."Generalization of the Theorem of Glivenko-Cantelli".The Annals of Mathematical Statistics.25:131–138.doi:10.1214/aoms/1177728852.

External links

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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.

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Empirical process

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