Heart development - Revision history

en>Novangelis: rv promotional edit

2014-02-13T19:51:19Z

rv promotional edit

← Older revision		Revision as of 21:51, 13 February 2014
Line 1:		Line 1:
	~~In the theory~~ of ~~[[probability]] and [[statistics]],~~ the '''Dvoretzky–Kiefer–Wolfowitz inequality''' predicts how close an [[empirical distribution function\|empirically determined distribution function]] will be to the [[cumulative distribution function\|distribution function]] from which the empirical samples are drawn. It is ~~named after [[Aryeh Dvoretzky]], [[Jack Kiefer (mathematician)\|Jack Kiefer]], and [[Jacob Wolfowitz]], who in 1956 proved<ref name="Dvoretzky">{{citation~~		The name of the writer is Figures. He is truly fond of doing ceramics but he is having difficulties to discover time for it. Managing people has been his day job for a while. Years in the past we moved to North Dakota and I adore every working day residing here.<br><br>Also visit my web-site; [http://www.gaysphere.net/blog/167593 std home test]
	~~\| last1 = Dvoretzky~~
	~~\| first1 = A.~~
	~~\| authorlink1 = Aryeh Dvoretzky~~
	~~\| last2 = Kiefer~~
	~~\| first2 = J.~~
	~~\| authorlink2 = Jack Kiefer (mathematician)~~
	~~\| last3 = Wolfowitz~~
	~~\| first3 = J.~~
	~~\| authorlink3 = Jacob Wolfowitz~~
	~~\| title = Asymptotic minimax character~~ of ~~the sample distribution function and of the classical multinomial estimator~~
	~~\| journal = [[Annals of Mathematical Statistics]]~~
	~~\| volume = 27~~
	~~\| issue = 3~~
	~~\| year = 1956~~
	~~\| pages = 642–669~~
	~~\| url = http://projecteuclid~~.~~org/euclid.aoms/1177728174~~
	~~\| mr = 0083864~~
	~~\| doi = 10.1214/aoms/1177728174}}</ref>~~
	~~the inequality with~~ a ~~unspecified multiplicative constant ''C'' in front of the exponent on the right-hand side~~. ~~In 1990, Pascal Massart proved the inequality with the sharp constant ''C'' = 1, <ref name="Massart">{{citation~~
	~~\| last=Massart~~
	~~\| first = P.~~
	~~\| title = The tight constant~~ in the ~~Dvoretzky–Kiefer–Wolfowitz inequality~~
	~~\| journal = The Annals of Probability~~
	~~\| volume = 18~~
	~~\| issue = 3~~
	~~\| year = 1990~~
	~~\| pages = 1269–1283~~
	~~\| url = http://projecteuclid.org/euclid.aop/1176990746~~
	~~\| mr = 1062069~~
	~~\| doi=10.1214/aop/1176990746}}</ref> confirming a conjecture due~~ to ~~Birnbaum~~ and ~~McCarty.<ref>{{cite journal~~
	~~\| mr=0093874~~
	~~\| zbl = 0087.34002~~
	~~\| last = Birnbaum~~
	~~\| first = Z~~. W.
	~~\| last2 = McCarty~~
	~~\| first2 = R. C.~~
	~~\| title = A distribution-free upper confidence bound for Pr{Y<X}, based on independent samples of X and Y~~
	~~\| journal = Annals of Mathematical Statistics~~
	~~\| volume = 29~~
	~~\| year = 1958~~
	~~\| pages = 558–562~~
	~~\| doi = 10.1214/aoms/1177706631~~
	~~\| url = http://projecteuclid.org/euclid.aoms/1177706631}}</ref>~~

	~~==The DKW inequality==~~
	~~Given a natural number ''n'', let ''X''<sub>1</sub>, ''X''<sub>2~~<~~/sub~~>~~, …, ''X~~<~~sub~~>~~n</sub>'' be real~~-~~valued~~ [[independent and identically distributed]] [[random variable]]s with [[cumulative distribution function\|distribution function]] ''F''(·). Let ''F<sub>n</sub>'' denote the associated [[empirical distribution function]] defined by
	: ~~<math>~~
	~~F_n(x) = \frac1n \sum_{i=1}^n \mathbf{1}_{\{X_i\leq x\}},\qquad x\in\mathbb{R}.~~
	~~</math>~~

	~~The Dvoretzky–Kiefer–Wolfowitz inequality bounds the probability that the [[random function]] ''F<sub>n<~~/~~sub>'' differs from ''F'' by more than a given constant ''ε'' > 0 anywhere on the real line. More precisely, there is the one-sided estimate~~
	~~: <math>~~
	~~\Pr\Bigl(\sup_{x\in\mathbb R} \bigl(F_n(x) - F(x)\bigr) > \varepsilon \Bigr) \le e^{-2n\varepsilon^2}\qquad \text{for every }\varepsilon\geq\sqrt{\tfrac{1}{2n}\ln2},~~
	</~~math>~~

	~~which also implies a two-sided estimate~~
	~~<ref name="Kosorok">{{citation~~
	~~\| last1 = Kosorok~~
	~~\| first1 = M~~.R.
	~~\| title = Introduction to Empirical Processes and Semiparametric Inference~~
	~~\| year = 2008~~
	~~\| chapter = Chapter 11: Additional Empirical Process Results~~
	~~\| page = 210~~
	~~\| isbn = 9780387749778~~
	~~\| publisher=Springer }}<~~/~~ref>~~
	~~: <math>~~
	~~\Pr\Bigl(\sup_{x\in\mathbb R} \|F_n(x) - F(x)\| > \varepsilon \Bigr) \le 2e^{-2n\varepsilon^2}\qquad \text{for every }\varepsilon>0.~~
	</~~math>~~

	~~This strengthens the [[Glivenko–Cantelli theorem]] by quantifying the [[rate of convergence]] as ''n'' tends to infinity. It also estimates the tail probability of the [[Kolmogorov–Smirnov~~ test\|Kolmogorov–Smirnov statistic]]. The inequalities above follow from the case where ''F'' corresponds to be the [[uniform distribution (continuous)\|uniform distribution]] on [0,1] in view of the fact<ref name="Shorack">
	~~{{citation~~
	~~\| last1 = Shorack~~
	~~\| first1 = G.R.~~
	~~\| last2 = Wellner~~
	~~\| first2 = J.A.~~
	~~\| title = Empirical Processes with Applications to Statistics~~
	~~\| year = 1986 \|isbn=0-471-86725-X~~
	~~\| publisher=Wiley }}~~
	~~</ref>~~
	~~that ''F<sub>n</sub>'' has the same distributions as ''G<sub>n</sub>''(''F'') where ''G<sub>n</sub>'' is the empirical distribution of~~
	~~''U''<sub>1</sub>, ''U''<sub>2</sub>, …, ''U<sub>n</sub>'' where these are independent and Uniform(0,1), and noting that~~
	~~: <math>~~
	~~\sup_{x\in\mathbb R} \|F_n(x) - F(x)\|\stackrel{d}{=} \sup_{x \in \mathbb R} \| G_n (F(x)) - F(x) \| \le \sup_{0 \le t \le 1} \| G_n (t) -t \| ,~~
	~~</math>~~
	~~with equality if and only if ''F'' is continuous.~~

	~~==References==~~
	~~<references/>~~

	~~{{DEFAULTSORT:Dvoretzky-Kiefer-Wolfowitz inequality}}~~
	~~[[Category:Asymptotic statistical theory]]~~
	~~[[Category:Statistical inequalities]]~~
	~~[[Category:Empirical process]~~]

143.207.14.82: /* Initiation */

2013-12-10T23:37:47Z

Initiation

← Older revision		Revision as of 01:37, 11 December 2013
Line 1:		Line 1:
	~~Greetings! I am Myrtle Shroyer~~. ~~I am a meter reader~~. ~~Minnesota has usually been his home but his spouse wants them to transfer~~. ~~The thing she adores most is body developing~~ and ~~now she is attempting to make money~~ with it.<br><br>~~Feel free~~ to ~~surf to my web~~-~~site [~~http://~~www~~.~~gaysphere~~.~~net~~/~~blog~~/~~297540 http~~://~~www~~.~~gaysphere~~.~~net~~/]		In the theory of [[probability]] and [[statistics]], the '''Dvoretzky–Kiefer–Wolfowitz inequality''' predicts how close an [[empirical distribution function\|empirically determined distribution function]] will be to the [[cumulative distribution function\|distribution function]] from which the empirical samples are drawn. It is named after [[Aryeh Dvoretzky]], [[Jack Kiefer (mathematician)\|Jack Kiefer]], and [[Jacob Wolfowitz]], who in 1956 proved<ref name="Dvoretzky">{{citation
			\| last1 = Dvoretzky
			\| first1 = A.
			\| authorlink1 = Aryeh Dvoretzky
			\| last2 = Kiefer
			\| first2 = J.
			\| authorlink2 = Jack Kiefer (mathematician)
			\| last3 = Wolfowitz
			\| first3 = J.
			\| authorlink3 = Jacob Wolfowitz
			\| title = Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator
			\| journal = [[Annals of Mathematical Statistics]]
			\| volume = 27
			\| issue = 3
			\| year = 1956
			\| pages = 642–669
			\| url = http://projecteuclid.org/euclid.aoms/1177728174
			\| mr = 0083864
			\| doi = 10.1214/aoms/1177728174}}</ref>
			the inequality with a unspecified multiplicative constant ''C'' in front of the exponent on the right-hand side. In 1990, Pascal Massart proved the inequality with the sharp constant ''C'' = 1, <ref name="Massart">{{citation
			\| last=Massart
			\| first = P.
			\| title = The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequality
			\| journal = The Annals of Probability
			\| volume = 18
			\| issue = 3
			\| year = 1990
			\| pages = 1269–1283
			\| url = http://projecteuclid.org/euclid.aop/1176990746
			\| mr = 1062069
			\| doi=10.1214/aop/1176990746}}</ref> confirming a conjecture due to Birnbaum and McCarty.<ref>{{cite journal
			\| mr=0093874
			\| zbl = 0087.34002
			\| last = Birnbaum
			\| first = Z. W.
			\| last2 = McCarty
			\| first2 = R. C.
			\| title = A distribution-free upper confidence bound for Pr{Y<X}, based on independent samples of X and Y
			\| journal = Annals of Mathematical Statistics
			\| volume = 29
			\| year = 1958
			\| pages = 558–562
			\| doi = 10.1214/aoms/1177706631
			\| url = http://projecteuclid.org/euclid.aoms/1177706631}}</ref>

			==The DKW inequality==
			Given a natural number ''n'', let ''X''<sub>1</sub>, ''X''<sub>2</sub>, …, ''X<sub>n</sub>'' be real-valued [[independent and identically distributed]] [[random variable]]s with [[cumulative distribution function\|distribution function]] ''F''(·). Let ''F<sub>n</sub>'' denote the associated [[empirical distribution function]] defined by
			: <math>
			F_n(x) = \frac1n \sum_{i=1}^n \mathbf{1}_{\{X_i\leq x\}},\qquad x\in\mathbb{R}.
			</math>

			The Dvoretzky–Kiefer–Wolfowitz inequality bounds the probability that the [[random function]] ''F<sub>n</sub>'' differs from ''F'' by more than a given constant ''ε'' > 0 anywhere on the real line. More precisely, there is the one-sided estimate
			: <math>
			\Pr\Bigl(\sup_{x\in\mathbb R} \bigl(F_n(x) - F(x)\bigr) > \varepsilon \Bigr) \le e^{-2n\varepsilon^2}\qquad \text{for every }\varepsilon\geq\sqrt{\tfrac{1}{2n}\ln2},
			</math>

			which also implies a two-sided estimate
			<ref name="Kosorok">{{citation
			\| last1 = Kosorok
			\| first1 = M.R.
			\| title = Introduction to Empirical Processes and Semiparametric Inference
			\| year = 2008
			\| chapter = Chapter 11: Additional Empirical Process Results
			\| page = 210
			\| isbn = 9780387749778
			\| publisher=Springer }}</ref>
			: <math>
			\Pr\Bigl(\sup_{x\in\mathbb R} \|F_n(x) - F(x)\| > \varepsilon \Bigr) \le 2e^{-2n\varepsilon^2}\qquad \text{for every }\varepsilon>0.
			</math>

			This strengthens the [[Glivenko–Cantelli theorem]] by quantifying the [[rate of convergence]] as ''n'' tends to infinity. It also estimates the tail probability of the [[Kolmogorov–Smirnov test\|Kolmogorov–Smirnov statistic]]. The inequalities above follow from the case where ''F'' corresponds to be the [[uniform distribution (continuous)\|uniform distribution]] on [0,1] in view of the fact<ref name="Shorack">
			{{citation
			\| last1 = Shorack
			\| first1 = G.R.
			\| last2 = Wellner
			\| first2 = J.A.
			\| title = Empirical Processes with Applications to Statistics
			\| year = 1986 \|isbn=0-471-86725-X
			\| publisher=Wiley }}
			</ref>
			that ''F<sub>n</sub>'' has the same distributions as ''G<sub>n</sub>''(''F'') where ''G<sub>n</sub>'' is the empirical distribution of
			''U''<sub>1</sub>, ''U''<sub>2</sub>, …, ''U<sub>n</sub>'' where these are independent and Uniform(0,1), and noting that
			: <math>
			\sup_{x\in\mathbb R} \|F_n(x) - F(x)\|\stackrel{d}{=} \sup_{x \in \mathbb R} \| G_n (F(x)) - F(x) \| \le \sup_{0 \le t \le 1} \| G_n (t) -t \| ,
			</math>
			with equality if and only if ''F'' is continuous.

			==References==
			<references/>

			{{DEFAULTSORT:Dvoretzky-Kiefer-Wolfowitz inequality}}
			[[Category:Asymptotic statistical theory]]
			[[Category:Statistical inequalities]]
			[[Category:Empirical process]]

en>Yobot: WP:CHECKWIKI error fixes + general fixes using AWB (7896)

2012-01-02T04:45:46Z

WP:CHECKWIKI error fixes + general fixes using AWB (7896)

New page

Greetings! I am Myrtle Shroyer. I am a meter reader. Minnesota has usually been his home but his spouse wants them to transfer. The thing she adores most is body developing and now she is attempting to make money with it.<br><br>Feel free to surf to my web-site [http://www.gaysphere.net/blog/297540 http://www.gaysphere.net/]

← Older revision		Revision as of 21:51, 13 February 2014
Line 1:		Line 1:
	~~In the theory~~ of ~~[[probability]] and [[statistics]],~~ the '''Dvoretzky–Kiefer–Wolfowitz inequality''' predicts how close an [[empirical distribution function\|empirically determined distribution function]] will be to the [[cumulative distribution function\|distribution function]] from which the empirical samples are drawn. It is ~~named after [[Aryeh Dvoretzky]], [[Jack Kiefer (mathematician)\|Jack Kiefer]], and [[Jacob Wolfowitz]], who in 1956 proved<ref name="Dvoretzky">{{citation~~		The name of the writer is Figures. He is truly fond of doing ceramics but he is having difficulties to discover time for it. Managing people has been his day job for a while. Years in the past we moved to North Dakota and I adore every working day residing here.<br><br>Also visit my web-site; [http://www.gaysphere.net/blog/167593 std home test]
	~~\| last1 = Dvoretzky~~
	~~\| first1 = A.~~
	~~\| authorlink1 = Aryeh Dvoretzky~~
	~~\| last2 = Kiefer~~
	~~\| first2 = J.~~
	~~\| authorlink2 = Jack Kiefer (mathematician)~~
	~~\| last3 = Wolfowitz~~
	~~\| first3 = J.~~
	~~\| authorlink3 = Jacob Wolfowitz~~
	~~\| title = Asymptotic minimax character~~ of ~~the sample distribution function and of the classical multinomial estimator~~
	~~\| journal = [[Annals of Mathematical Statistics]]~~
	~~\| volume = 27~~
	~~\| issue = 3~~
	~~\| year = 1956~~
	~~\| pages = 642–669~~
	~~\| url = http://projecteuclid~~.~~org/euclid.aoms/1177728174~~
	~~\| mr = 0083864~~
	~~\| doi = 10.1214/aoms/1177728174}}</ref>~~
	~~the inequality with~~ a ~~unspecified multiplicative constant ''C'' in front of the exponent on the right-hand side~~. ~~In 1990, Pascal Massart proved the inequality with the sharp constant ''C'' = 1, <ref name="Massart">{{citation~~
	~~\| last=Massart~~
	~~\| first = P.~~
	~~\| title = The tight constant~~ in the ~~Dvoretzky–Kiefer–Wolfowitz inequality~~
	~~\| journal = The Annals of Probability~~
	~~\| volume = 18~~
	~~\| issue = 3~~
	~~\| year = 1990~~
	~~\| pages = 1269–1283~~
	~~\| url = http://projecteuclid.org/euclid.aop/1176990746~~
	~~\| mr = 1062069~~
	~~\| doi=10.1214/aop/1176990746}}</ref> confirming a conjecture due~~ to ~~Birnbaum~~ and ~~McCarty.<ref>{{cite journal~~
	~~\| mr=0093874~~
	~~\| zbl = 0087.34002~~
	~~\| last = Birnbaum~~
	~~\| first = Z~~. W.
	~~\| last2 = McCarty~~
	~~\| first2 = R. C.~~
	~~\| title = A distribution-free upper confidence bound for Pr{Y<X}, based on independent samples of X and Y~~
	~~\| journal = Annals of Mathematical Statistics~~
	~~\| volume = 29~~
	~~\| year = 1958~~
	~~\| pages = 558–562~~
	~~\| doi = 10.1214/aoms/1177706631~~
	~~\| url = http://projecteuclid.org/euclid.aoms/1177706631}}</ref>~~

	~~==The DKW inequality==~~
	~~Given a natural number ''n'', let ''X''<sub>1</sub>, ''X''<sub>2~~<~~/sub~~>~~, …, ''X~~<~~sub~~>~~n</sub>'' be real~~-~~valued~~ [[independent and identically distributed]] [[random variable]]s with [[cumulative distribution function\|distribution function]] ''F''(·). Let ''F<sub>n</sub>'' denote the associated [[empirical distribution function]] defined by
	: ~~<math>~~
	~~F_n(x) = \frac1n \sum_{i=1}^n \mathbf{1}_{\{X_i\leq x\}},\qquad x\in\mathbb{R}.~~
	~~</math>~~

	~~The Dvoretzky–Kiefer–Wolfowitz inequality bounds the probability that the [[random function]] ''F<sub>n<~~/~~sub>'' differs from ''F'' by more than a given constant ''ε'' > 0 anywhere on the real line. More precisely, there is the one-sided estimate~~
	~~: <math>~~
	~~\Pr\Bigl(\sup_{x\in\mathbb R} \bigl(F_n(x) - F(x)\bigr) > \varepsilon \Bigr) \le e^{-2n\varepsilon^2}\qquad \text{for every }\varepsilon\geq\sqrt{\tfrac{1}{2n}\ln2},~~
	</~~math>~~

	~~which also implies a two-sided estimate~~
	~~<ref name="Kosorok">{{citation~~
	~~\| last1 = Kosorok~~
	~~\| first1 = M~~.R.
	~~\| title = Introduction to Empirical Processes and Semiparametric Inference~~
	~~\| year = 2008~~
	~~\| chapter = Chapter 11: Additional Empirical Process Results~~
	~~\| page = 210~~
	~~\| isbn = 9780387749778~~
	~~\| publisher=Springer }}<~~/~~ref>~~
	~~: <math>~~
	~~\Pr\Bigl(\sup_{x\in\mathbb R} \|F_n(x) - F(x)\| > \varepsilon \Bigr) \le 2e^{-2n\varepsilon^2}\qquad \text{for every }\varepsilon>0.~~
	</~~math>~~

	~~This strengthens the [[Glivenko–Cantelli theorem]] by quantifying the [[rate of convergence]] as ''n'' tends to infinity. It also estimates the tail probability of the [[Kolmogorov–Smirnov~~ test\|Kolmogorov–Smirnov statistic]]. The inequalities above follow from the case where ''F'' corresponds to be the [[uniform distribution (continuous)\|uniform distribution]] on [0,1] in view of the fact<ref name="Shorack">
	~~{{citation~~
	~~\| last1 = Shorack~~
	~~\| first1 = G.R.~~
	~~\| last2 = Wellner~~
	~~\| first2 = J.A.~~
	~~\| title = Empirical Processes with Applications to Statistics~~
	~~\| year = 1986 \|isbn=0-471-86725-X~~
	~~\| publisher=Wiley }}~~
	~~</ref>~~
	~~that ''F<sub>n</sub>'' has the same distributions as ''G<sub>n</sub>''(''F'') where ''G<sub>n</sub>'' is the empirical distribution of~~
	~~''U''<sub>1</sub>, ''U''<sub>2</sub>, …, ''U<sub>n</sub>'' where these are independent and Uniform(0,1), and noting that~~
	~~: <math>~~
	~~\sup_{x\in\mathbb R} \|F_n(x) - F(x)\|\stackrel{d}{=} \sup_{x \in \mathbb R} \| G_n (F(x)) - F(x) \| \le \sup_{0 \le t \le 1} \| G_n (t) -t \| ,~~
	~~</math>~~
	~~with equality if and only if ''F'' is continuous.~~

	~~==References==~~
	~~<references/>~~

	~~{{DEFAULTSORT:Dvoretzky-Kiefer-Wolfowitz inequality}}~~
	~~[[Category:Asymptotic statistical theory]]~~
	~~[[Category:Statistical inequalities]]~~
	~~[[Category:Empirical process]~~]