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2013-12-30T16:32:35Z

208.65.144.234: added "Centrifugal" to the line describing the types of devices, in this case, pumps, that are subject to cavitation at the impeller. Other pumps ddo not have impellers.

[[File:Empirical CDF.png|thumb|300px|The blue line shows an empirical distribution function. The black bars represent the samples corresponding to the ecdf and the gray line is the true cumulative distribution function.]]

In [[statistics]], the '''empirical distribution function''', or '''empirical cdf''', is the [[cumulative distribution function]] associated with the [[empirical measure]] of the [[sample (statistics)|sample]]. This cdf is a [[step function]] that jumps up by 1/''n'' at each of the ''n'' data points. The empirical distribution function estimates the true underlying cdf of the points in the sample and converges with probability 1 according to the [[Glivenko–Cantelli theorem]]. A number of results exist to quantify the rate of convergence of the empirical cdf to the underlying cdf.

== Definition ==
Let (''x''1, …, ''x''''n'') be [[iid]] real random variables with the common [[Cumulative distribution function|cdf]] ''F''(''t''). Then the '''empirical distribution function''' is defined as <ref name="vdv265">{{cite book
| last = van der Vaart | first = A.W.
| title = Asymptotic statistics
| year = 1998
| publisher = Cambridge University Press
| isbn = 0-521-78450-6
| page = 265
}}</ref><ref>[http://planetmath.org/encyclopedia/EmpiricalDistributionFunction.html PlanetMath]</ref>
: <math>
\hat F_n(t) = \frac{ \mbox{number of elements in the sample} \leq t}n =
\frac{1}{n} \sum_{i=1}^n \mathbf{1}\{x_i \le t\},
</math>
where '''1'''{''A''} is the [[indicator function|indicator]] of [[event (probability theory)|event]] ''A''. For a fixed ''t'', the indicator '''1'''{''xi'' ≤ ''t''} is a [[Bernoulli distribution|Bernoulli]] random variable with parameter {{nowrap|''p'' {{=}} ''F''(''t'')}}, hence <math style="vertical-align:-.3em">\scriptstyle n \hat F_n(t)</math> is a [[binomial distribution|binomial]] random variable with [[mean]] ''nF''(''t'') and [[variance]] {{nowrap|''nF''(''t'')(1 − ''F''(''t''))}}. This implies that <math style="vertical-align:-.3em">\scriptstyle \hat F_n(t)</math> is an [[bias of an estimator|unbiased]] estimator for ''F''(''t'').

== Asymptotic properties ==
By the [[strong law of large numbers]], the estimator <math style="vertical-align:-.3em">\scriptstyle\hat{F}_n(t)</math> converges to ''F''(''t'') as {{nowrap|''n'' → ∞}} [[almost sure convergence|almost surely]], for every value of ''t'': <ref name="vdv265" />
: <math>
\hat F_n(t)\ \xrightarrow{a.s.}\ F(t),
</math>
thus the estimator <math style="vertical-align:-.3em">\scriptstyle\hat{F}_n(t)</math> is [[consistent estimator|consistent]]. This expression asserts the pointwise convergence of the empirical distribution function to the true cdf. There is a stronger result, called the [[Glivenko–Cantelli theorem]], which states that the convergence in fact happens uniformly over ''t'': <ref name="vdv266">{{cite book
| last = van der Vaart | first = A.W.
| title = Asymptotic statistics
| year = 1998
| publisher = Cambridge University Press
| isbn = 0-521-78450-6
| page = 266
}}</ref>
: <math>
\|\hat F_n-F\|_\infty \equiv
\sup_{t\in\mathbb{R}} \big|\hat F_n(t)-F(t)\big|\ \xrightarrow{a.s.}\ 0.
</math>
The sup-norm in this expression is called the [[Kolmogorov–Smirnov test|Kolmogorov–Smirnov statistic]] for testing the goodness-of-fit between the empirical distribution <math style="vertical-align:-.3em">\scriptstyle\hat{F}_n(t)</math> and the assumed true cdf ''F''. Other [[norm (mathematics)|norm function]]s may be reasonably used here instead of the sup-norm. For example, the [[Lp norm|L²-norm]] gives rise to the [[Cramér–von Mises criterion|Cramér–von Mises statistic]].

The asymptotic distribution can be further characterized in several different ways. First, the [[central limit theorem]] states that ''pointwise'', <math style="vertical-align:-.3em">\scriptstyle\hat{F}_n(t)</math> has asymptotically normal distribution with the standard ''√n'' rate of convergence:<ref name="vdv265"/>
: <math>
\sqrt{n}\big(\hat F_n(t) - F(t)\big)\ \ \xrightarrow{d}\ \ \mathcal{N}\Big( 0, F(t)\big(1-F(t)\big) \Big).
</math>
This result is extended by the [[Donsker’s theorem]], which asserts that the ''[[empirical process]]'' <math style="vertical-align:-.3em">\scriptstyle\sqrt{n}(\hat{F}_n - F)</math>, viewed as a function indexed by {{nowrap|''t'' ∈ '''R'''}}, [[convergence in distribution|converges in distribution]] in the [[Skorokhod space]] {{nowrap|''D''[−∞, +∞]}} to the mean-zero [[Gaussian process]] {{nowrap|''GF'' {{=}} ''B''∘''F''}}, where ''B'' is the standard [[Brownian bridge]].<ref name="vdv266"/> The covariance structure of this Gaussian process is
: <math>
\mathrm{E}[\,G_F(t_1)G_F(t_2)\,] = F(t_1\wedge t_2) - F(t_1)F(t_2).
</math>
The uniform rate of convergence in Donsker’s theorem can be quantified by the result, known as the [[Hungarian embedding]]: <ref name="vdv268">{{cite book
| last = van der Vaart | first = A.W.
| title = Asymptotic statistics
| year = 1998
| publisher = Cambridge University Press
| isbn = 0-521-78450-6
| page = 268
}}</ref>
: <math>
\limsup_{n\to\infty} \frac{\sqrt{n}}{\ln^2 n} \big\| \sqrt{n}(\hat F_n-F) - G_{F,n}\big\|_\infty < \infty, \quad \text{a.s.}
</math>

Alternatively, the rate of convergence of <math style="vertical-align:-.3em">\scriptstyle\sqrt{n}(\hat{F}_n-F)</math> can also be quantified in terms of the asymptotic behavior of the sup-norm of this expression. Number of results exist in this venue, for example the [[Dvoretzky–Kiefer–Wolfowitz inequality]] provides bound on the tail probabilities of <math style="vertical-align:-.3em">\scriptstyle\sqrt{n}\|\hat{F}_n-F\|_\infty</math>:<ref name="vdv268"/>
: <math>
\Pr\!\Big( \sqrt{n}\|\hat{F}_n-F\|_\infty > z \Big) \leq 2e^{-2z^2}.
</math>
In fact, Kolmogorov has shown that if the cdf ''F'' is continuous, then the expression <math style="vertical-align:-.3em">\scriptstyle\sqrt{n}\|\hat{F}_n-F\|_\infty</math> converges in distribution to ||''B''||∞, which has the [[Kolmogorov distribution]] that does not depend on the form of ''F''.

Another result, which follows from the [[law of the iterated logarithm]], is that <ref name="vdv268"/>
: <math>
\limsup_{n\to\infty} \frac{\sqrt{n}\|\hat{F}_n-F\|_\infty}{\sqrt{2\ln\ln n}} \leq \frac12, \quad \text{a.s.}
</math>
and
: <math>
\liminf_{n\to\infty} \sqrt{2n\ln\ln n} \|\hat{F}_n-F\|_\infty = \frac{\pi}{2}, \quad \text{a.s.}
</math>

== See also ==
* [[Càdlàg]] functions
* [[Dvoretzky–Kiefer–Wolfowitz inequality]]
* [[Empirical probability]]
* [[Empirical process]]
* [[Kaplan–Meier estimator]] for censored processes
* [[Survival function]]
* [[Distribution fitting]]

== References ==
{{reflist}}

== Further reading ==
* {{cite book
| last1 = Shorack | first1 = G.R.
| last2 = Wellner | first2 = J.A.
| title = Empirical Processes with Applications to Statistics
| year = 1986
| publisher = Wiley
| location = New York
| isbn = 0-471-86725-X
| ref = harv
}}

== External links ==
{{commonscat|Cumulative frequency distribution|Empirical distribution functions}}

{{DEFAULTSORT:Empirical Distribution Function}}
[[Category:Data analysis]]
[[Category:Non-parametric statistics]]
[[Category:Empirical process]]

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