Splitting circle method: Difference between revisions

Revision as of 02:15, 14 October 2013

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state. For a process in a discrete state space a population continuous time Markov chain^[1]^[2] or Markov population model^[3] is a process which counts the number of objects in a given state (without rescaling). In mean field theory, limit theorems (as the number of objects becomes large) are considered and generalise the central limit theorem for empirical measures. Applications of the theory of empirical processes arise in non-parametric statistics.^[4]

Definition

For X₁, X₂, ... X_n independent and identically-distributed random variables in R with common cumulative distribution function F(x), the empirical distribution function is defined by

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

where I_C is the indicator function of the set C.

For every (fixed) x, F_n(x) is a sequence of random variables which converge to F(x) almost surely by the strong law of large numbers. That is, F_n converges to F pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergence of F_n to F by the Glivenko–Cantelli theorem.^[5]

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 a map on measurable functions f given by

f \mapsto G_{n} f = \sqrt{n} (P_{n} - P) f = \sqrt{n} (\frac{1}{n} \sum_{i = 1}^{n} f (X_{i}) - 𝔼 f)

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, $G_{n} f$ converges in distribution to a normal random variable $N (0, 𝔼 (f - 𝔼 f)^{2})$ , provided that $𝔼 f$ and $𝔼 f^{2}$ exist.

Definition

(G_{n} (c))_{c \in 𝒞}

is called an empirical process indexed by

𝒞

, a collection of measurable subsets of S.

(G_{n} f)_{f \in ℱ}

is called an empirical process indexed by

ℱ

, a collection of measurable functions from S to

ℝ

.

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

Example

As an example, consider empirical distribution functions. For real-valued iid random variables X₁, 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 $𝒞 = {(- \infty, x] : x \in ℝ} .$ It has been shown that $𝒞$ is a Donsker class, in particular,

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

converges weakly in

ℓ^{\infty} (ℝ)

to a Brownian bridge B(F(x)) .

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

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.

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+{{for|the process control topic|Empirical process (process control model)}}
+In [[probability theory]], an '''empirical process''' is a stochastic process that describes the proportion of objects in a system in a given state.
+For a process in a discrete state space a '''population continuous time Markov chain'''<ref>{{cite doi|10.1016/j.peva.2013.01.001}}</ref><ref>{{cite doi|10.1007/978-3-642-30782-9_14}}</ref> or '''Markov population model'''<ref>{{cite doi|10.1002/nla.795}}</ref> is a process which counts the number of objects in a given state (without rescaling).
+In [[mean field theory]], limit theorems (as the number of objects becomes large) are considered and generalise the [[central limit theorem]] for [[empirical measure]]s. Applications of the theory of empirical processes arise in [[non-parametric statistics]].<ref>{{cite doi|10.1016/j.jspi.2006.02.016}}</ref>
+==Definition==
+For ''X''<sub>1</sub>, ''X''<sub>2</sub>, ... ''X''<sub>''n''</sub> [[independent and identically-distributed random variables]] in '''R''' with common [[cumulative distribution function]] ''F''(''x''), the empirical distribution function is defined by
+:<math>F_n(x)=\frac{1}{n}\sum_{i=1}^n I_{(-\infty,x]}(X_i),</math>
+where I<sub>''C''</sub> is the [[indicator function]] of the set ''C''.
+For every (fixed) ''x'', ''F''<sub>''n''</sub>(''x'') is a sequence of random variables which converge to ''F''(''x'') [[almost surely]] by the strong [[law of large numbers]]. That is, ''F''<sub>''n''</sub> converges to ''F'' [[pointwise convergence|pointwise]]. Glivenko and Cantelli strengthened this result by proving [[uniform convergence]] of ''F''<sub>''n''</sub> to ''F'' by the [[Glivenko–Cantelli theorem]].<ref>{{cite doi|10.1214/aoms/1177728852}}</ref>
+A centered and scaled version of the empirical measure is the [[signed measure]]
+:<math>G_n(A)=\sqrt{n}(P_n(A)-P(A))</math>
+It induces a map on measurable functions ''f'' given by
+:<math>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)</math>
+By the [[central limit theorem]], <math>G_n(A)</math> [[converges in distribution]] to a [[normal distribution|normal]] random variable ''N''(0,&nbsp;''P''(''A'')(1&nbsp;−&nbsp;''P''(''A''))) for fixed measurable set ''A''. Similarly, for a fixed function ''f'', <math>G_nf</math> converges in distribution to a normal random variable <math>N(0,\mathbb{E}(f-\mathbb{E}f)^2)</math>, provided that <math>\mathbb{E}f</math> and <math>\mathbb{E}f^2</math> exist.
+'''Definition'''
+:<math>\bigl(G_n(c)\bigr)_{c\in\mathcal{C}}</math> is called an ''empirical process'' indexed by <math>\mathcal{C}</math>, a collection of measurable subsets of ''S''.
+:<math>\bigl(G_nf\bigr)_{f\in\mathcal{F}}</math>  is called an ''empirical process'' indexed by <math>\mathcal{F}</math>, a collection of measurable functions from ''S'' to <math>\mathbb{R}</math>.
+A significant result in the area of empirical processes is [[Donsker's theorem]]. It has led to a study of [[Donsker classes]]: sets of functions with the useful property that empirical processes indexed by these classes [[weak convergence|converge weakly]] to a certain [[Gaussian process]]. While it can be shown that Donsker classes are [[Glivenko–Cantelli class]]es, the converse is not true in general.
+==Example==
+As an example, consider [[empirical distribution function]]s. For real-valued [[iid]] random variables ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub> they are given by
+:<math>F_n(x)=P_n((-\infty,x])=P_nI_{(-\infty,x]}.</math>
+In this case, empirical processes are indexed by a class <math>\mathcal{C}=\{(-\infty,x]:x\in\mathbb{R}\}.</math> It has been shown that <math>\mathcal{C}</math> is a Donsker class, in particular,
+:<math>\sqrt{n}(F_n(x)-F(x))</math> converges [[Weak convergence of measures|weakly]] in <math>\ell^\infty(\mathbb{R})</math> to a [[Brownian bridge]] ''B''(''F''(''x'')) .
+==See also==
+*[[Khmaladze transformation]]
+*[[Weak convergence of measures]]
+*[[Glivenko–Cantelli theorem]]
+==References==
+{{Reflist}}
+==Further reading==
+* {{cite book |first=P. |last=Billingsley |title=Probability and Measure |publisher=John Wiley and Sons |location=New York |edition=Third |year=1995 |isbn=0471007102 }}
+* {{cite doi|10.1214/aoms/1177729445}}
+* {{cite doi|10.1214/aop/1176995384}}
+* {{cite book |first=R. M. |last=Dudley |title=Uniform Central Limit Theorems |series=Cambridge Studies in Advanced Mathematics |volume=63 |publisher=Cambridge University Press |location=Cambridge, UK |year=1999 }}
+* {{cite doi|10.1007/978-0-387-74978-5}}
+* {{cite doi|10.1137/1.9780898719017}}
+* {{cite book |first=Aad W. |last=van der Vaart |first2=Jon A. |last2=Wellner |title=Weak Convergence and Empirical Processes: With Applications to Statistics |edition=2nd |publisher=Springer |year=2000 |isbn=978-0-387-94640-5 }}
+* {{cite doi|10.1007/BF01239992}}
+==External links==
+* [http://www.stat.yale.edu/~pollard/Books/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.
+{{Stochastic processes}}
+[[Category:Probability theory]]
+[[Category:Empirical process| ]]
+[[Category:Non-parametric statistics]]

Splitting circle method: Difference between revisions

Revision as of 02:15, 14 October 2013

Contents

Definition

Example

See also

References

Further reading

External links

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Splitting circle method: Difference between revisions

Revision as of 02:15, 14 October 2013

Definition

Example

See also

References

Further reading

External links

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