Splitting circle method - Revision history

en>Myasuda: /* Basic numerical observation */ fixed wikilink

2014-05-10T03:55:42Z

Basic numerical observation: fixed wikilink

← Older revision		Revision as of 05:55, 10 May 2014
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	~~{{for\|~~the ~~process control topic\|Empirical process (process control model)}}~~		The name of the author is Numbers but it's not the most masucline title out there. California is our birth place. One of the things she enjoys most is to read comics and she'll be starting something else alongside with it. I am a meter reader but I strategy on altering it.<br><br>Stop by my weblog; [http://www.associazioneitalianafotografi.it/community/gruppi/points-to-know-when-confronted-with-candidiasis/ std testing at home]
	~~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, ''P''(''A'')(1 − ''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]~~]

en>Myasuda: /* Graeffe iteration */ added missing diacritic

2013-10-14T01:15:10Z

Graeffe iteration: added missing diacritic

← Older revision		Revision as of 03:15, 14 October 2013
Line 1:		Line 1:
	~~Alyson~~ is the ~~title people use to contact me~~ and I ~~think it sounds quite good when you say it~~. ~~Since he was free psychic readings~~ ([~~http:~~//~~www~~.~~seekavideo~~.~~com~~/~~playlist~~/~~2199~~/~~video~~/ ~~http~~://~~www~~.~~seekavideo.com~~/]) ~~18 he~~'~~s been working as~~ an ~~information officer but he plans on changing it~~. ~~The favorite pastime for him and his children~~ is to ~~play lacross and he would never give it up~~. ~~Alaska~~ is ~~exactly where he~~'s [~~http~~://~~www~~.~~taehyuna~~.~~net~~/xe/~~?document_srl~~=~~78721 accurate psychic predictions~~] ~~usually~~ been ~~residing.~~<br><br>~~Also visit my web page; psychics online~~ ([http://~~galab-work~~.cs.~~pusan~~.ac.kr/~~Sol09B~~/~~?document_srl=1489804 galab-work~~.cs.~~pusan~~.ac.kr])		{{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, ''P''(''A'')(1 − ''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]]

193.206.68.44 at 08:39, 17 April 2012

2012-04-17T08:39:29Z

New page

Alyson is the title people use to contact me and I think it sounds quite good when you say it. Since he was free psychic readings ([http://www.seekavideo.com/playlist/2199/video/ http://www.seekavideo.com/]) 18 he's been working as an information officer but he plans on changing it. The favorite pastime for him and his children is to play lacross and he would never give it up. Alaska is exactly where he's [http://www.taehyuna.net/xe/?document_srl=78721 accurate psychic predictions] usually been residing.<br><br>Also visit my web page; psychics online ([http://galab-work.cs.pusan.ac.kr/Sol09B/?document_srl=1489804 galab-work.cs.pusan.ac.kr])