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{{Short description|Distribution function associated with the empirical measure of a sample}}

{{CSS image crop|Image = ECDF-100.png|bSize = 2500|cWidth = 250|cHeight = 250|oTop = {{#expr:{{#invoke:Random|number|9}}*250}}|oLeft = {{#expr:{{#invoke:Random|number|9}}*250}}|Location = right|Description = The green curve, which asymptotically approaches heights of 0 and 1 without reaching them, is the true [[Cumulative distribution function|cumulative distribution function]] of the standard normal distribution. The grey hash marks represent the observations in a particular [[Sample (statistics)|sample]] drawn from that distribution, and the horizontal steps of the blue step function (including the leftmost point in each step but not including the rightmost point) form the empirical distribution function of that sample. ({{purge|Click here to load a new graph.}})}}

In [[Statistics|statistics]], an '''empirical distribution function''' (commonly also called an '''empirical cumulative distribution function''', eCDF) is the [[Cumulative distribution function|distribution function]] associated with the [[Empirical measure|empirical measure]] of a [[Sampling (statistics)|sample]].<ref>{{Cite book|url=https://www.worldcat.org/oclc/262680588|title=A modern introduction to probability and statistics: Understanding why and how|date=2005|publisher=Springer|others=Michel Dekking|isbn=978-1-85233-896-1|location=London|pages=219|oclc=262680588}}</ref> This [[Cumulative distribution function|cumulative distribution function]] is a [[Step function|step function]] that jumps up by {{math|1/''n''}} at each of the {{math|''n''}} data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value.

The empirical distribution function is an [[Estimator|estimate]] of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution, according to the [[Glivenko–Cantelli theorem]]. A number of results exist to quantify the rate of [[Convergence of random variables#Convergence in distribution|convergence]] of the empirical distribution function to the underlying cumulative distribution function.

== Definition ==
Let {{math|(''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>)}} be independent, identically distributed real random variables with the common [[Cumulative distribution function|cumulative distribution function]] {{math|''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
  | url = https://archive.org/details/asymptoticstatis00vaar_017 | url-access = limited | year = 1998
  | publisher = Cambridge University Press
  | isbn = 0-521-78450-6
  | page = [https://archive.org/details/asymptoticstatis00vaar_017/page/n281 265]
  }}
</ref>
:<math>\widehat 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 <math style="vertical-align:-.3em">\mathbf{1}_{A}</math> is the [[Indicator function|indicator]] of [[Event (probability theory)|event]] {{math|''A''}}. For a fixed {{math|''t''}}, the indicator <math style="vertical-align:-.3em">\mathbf{1}_{X_i \le t}</math> is a [[Bernoulli distribution|Bernoulli random variable]] with parameter {{math|''p'' {{=}} ''F''(''t'')}}; hence <math style="vertical-align:-.3em">n \widehat F_n(t)</math> is a [[Binomial distribution|binomial random variable]] with [[Mean|mean]] {{math|''nF''(''t'')}} and [[Variance|variance]] {{math|''nF''(''t'')(1 − ''F''(''t''))}}. This implies that <math style="vertical-align:-.3em">\widehat F_n(t)</math> is an [[Bias of an estimator|unbiased]] estimator for {{math|''F''(''t'')}}.

However, in some textbooks, the definition is given as
:<math>\widehat F_n(t) = \frac{1}{n+1} \sum_{i=1}^n \mathbf{1}_{X_i \le t}</math><ref>Coles, S. (2001) ''An Introduction to Statistical Modeling of Extreme Values''. Springer, p. 36, Definition 2.4.  {{ISBN|978-1-4471-3675-0}}.</ref><ref>Madsen, H.O., Krenk, S., Lind, S.C. (2006) ''Methods of Structural Safety''. Dover Publications. p. 148-149. {{ISBN|0486445976}}</ref>

==Mean==

The [[Mean|mean]] of the empirical distribution is an [[Bias of an estimator|unbiased estimator]] of the mean of the population distribution.

:<math>E_n(X) = \frac{1}{n}\left (\sum_{i=1}^n{x_i}\right ) </math>

which is more commonly denoted <math>\bar{x}</math>

==Variance==

The [[Variance|variance]] of the empirical distribution times <math>\tfrac{n}{n-1}</math> is an unbiased estimator of the variance of the population distribution, for any distribution of X that has a finite variance.

:<math>\begin{align}
\operatorname{Var}(X) &= \operatorname{E}\left[(X - \operatorname{E}[X])^2\right] \\[4pt]
&= \operatorname{E}\left[(X - \bar{x})^2\right] \\[4pt]
&= \frac{1}{n}\left (\sum_{i=1}^n{(x_i-\bar{x})^2}\right )
\end{align}</math>

==Mean squared error==

The [[Mean squared error|mean squared error]] for the empirical distribution is as follows.

:<math>
\begin{align}
\operatorname{MSE}&=\frac{1}{n}\sum_{i=1}^n(Y_i-\hat{Y_i})^2\\[4pt]
&=\operatorname{Var}_{\hat\theta}(\hat{\theta})+ \operatorname{Bias}(\hat{\theta},\theta)^2
\end{align}</math>

Where <math>\hat{\theta}</math> is an estimator and <math>\theta</math> an unknown parameter

==[[Quantile|Quantiles]]==

For any real number <math>a</math> the notation <math>\lceil{a}\rceil</math> (read “ceiling of a”) denotes the least integer greater than or equal to <math>a</math>. For any real number a, the notation <math>\lfloor{a}\rfloor</math> (read “floor of a”) denotes the greatest integer less than or equal to <math>a</math>. 

If <math>nq</math> is not an integer, then the <math>q</math>-th quantile is unique and is equal to <math>x_{(\lceil{nq}\rceil)}</math>

If <math>nq</math>  is an integer, then the <math>q</math>-th quantile is not unique and is any real number <math>x</math> such that 

:<math>x_{({nq})}<x<x_{({nq+1})}</math>

==Empirical median==

If <math>n</math> is odd, then the empirical median is the number

:<math>\tilde{x} = x_{(\lceil{n/2}\rceil)}</math>

If <math>n</math> is even, then the empirical median is the number 

:<math>\tilde{x} =\frac{ x_{n/2} + x_{n/2+1}}{2}</math>

== Asymptotic properties ==
Since the ratio {{math|(''n''&nbsp;+&nbsp;1)/''n''}} approaches 1 as {{math|''n''}} goes to infinity, the asymptotic properties of the two definitions that are given above are the same.

By the strong law of large numbers, the estimator <math style="vertical-align:-.3em">\scriptstyle\widehat{F}_n(t)</math> converges to {{math|''F''(''t'')}} as {{math|''n'' → ∞}} almost surely, for every value of {{math|''t''}}:<ref name="vdv265" />
: <math>
    \widehat F_n(t)\ \xrightarrow{\text{a.s.}}\ F(t);
  </math>
thus the estimator <math style="vertical-align:-.3em">\scriptstyle\widehat{F}_n(t)</math> is [[Consistent estimator|consistent]]. This expression asserts the pointwise convergence of the empirical distribution function to the true cumulative distribution function. There is a stronger result, called the [[Glivenko–Cantelli theorem]], which states that the convergence in fact happens uniformly over {{math|''t''}}:<ref name="vdv266">{{cite book
  | last = van der Vaart | first = A.W.
  | title = Asymptotic statistics
  | url = https://archive.org/details/asymptoticstatis00vaar_017 | url-access = limited | year = 1998
  | publisher = Cambridge University Press
  | isbn = 0-521-78450-6
  | page = [https://archive.org/details/asymptoticstatis00vaar_017/page/n282 266]
  }}</ref>
: <math>
    \|\widehat F_n-F\|_\infty \equiv
    \sup_{t\in\mathbb{R}} \big|\widehat F_n(t)-F(t)\big|\ \xrightarrow{\text{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\widehat{F}_n(t)</math> and the assumed true cumulative distribution function {{math|''F''}}. Other [[Norm (mathematics)|norm function]]s may be reasonably used here instead of the sup-norm. For example, the L<sup>2</sup>-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|central limit theorem]] states that ''pointwise'', <math style="vertical-align:-.3em">\scriptstyle\widehat{F}_n(t)</math> has asymptotically normal distribution with the standard <math style="vertical-align:-.3em">\sqrt{n}</math> rate of convergence:<ref name="vdv265"/>
: <math>
    \sqrt{n}\big(\widehat 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|empirical process]]'' <math style="vertical-align:-.3em">\scriptstyle\sqrt{n}(\widehat{F}_n - F)</math>, viewed as a function indexed by <math style="vertical-align:-.3em">\scriptstyle t\in\mathbb{R}</math>, converges in distribution in the Skorokhod space <math style="vertical-align:-.3em">\scriptstyle D[-\infty, +\infty]</math> to the mean-zero [[Gaussian process]] <math style="vertical-align:-.3em">\scriptstyle G_F = B \circ F</math>, where {{math|''B''}} is the standard [[Brownian bridge]].<ref name="vdv266"/> The covariance structure of this Gaussian process is
: <math>
    \operatorname{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
  | url = https://archive.org/details/asymptoticstatis00vaar_017 | url-access = limited | year = 1998
  | publisher = Cambridge University Press
  | isbn = 0-521-78450-6
  | page = [https://archive.org/details/asymptoticstatis00vaar_017/page/n284 268]
  }}</ref>
: <math>
    \limsup_{n\to\infty} \frac{\sqrt{n}}{\ln^2 n} \big\| \sqrt{n}(\widehat 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}(\widehat{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}\|\widehat{F}_n-F\|_\infty</math>:<ref name="vdv268"/>
: <math>
    \Pr\!\Big( \sqrt{n}\|\widehat{F}_n-F\|_\infty > z \Big) \leq 2e^{-2z^2}.
  </math>
In fact, Kolmogorov has shown that if the cumulative distribution function {{math|''F''}} is continuous, then the expression <math style="vertical-align:-.3em">\scriptstyle\sqrt{n}\|\widehat{F}_n-F\|_\infty</math> converges in distribution to <math style="vertical-align:-.3em">\scriptstyle\|B\|_\infty</math>, which has the Kolmogorov distribution that does not depend on the form of {{math|''F''}}.

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

== Confidence intervals ==
[[File:Empirical CDF, CDF and Confidence Interval plots for various sample sizes of Normal Distribution.png|thumb|Empirical CDF, CDF and confidence interval plots for various sample sizes of [[Normal distribution|normal distribution]]]]
[[File:Cauchy emp .png|thumb|Empirical CDF, CDF and confidence interval plots for various sample sizes of [[Cauchy distribution|Cauchy distribution]]]]
[[File:Triangle emp.png|thumb|Empirical CDF, CDF and confidence interval plots for various sample sizes of [[Triangular distribution|triangle distribution]]]]
As per [[Dvoretzky–Kiefer–Wolfowitz inequality]] the interval that contains the true CDF, <math>F(x)</math>, with probability <math>1-\alpha</math> is specified as

:<math>F_n(x) - \varepsilon \le F(x) \le F_n(x) + \varepsilon \; \text{ where } \varepsilon = \sqrt{\frac{\ln{\frac{2}{\alpha}}}{2n}}.</math>

As per the above bounds, we can plot the Empirical CDF, CDF and confidence intervals for different distributions by using any one of the statistical implementations.

== Statistical implementation ==
A non-exhaustive list of software implementations of Empirical Distribution function includes:

* In [https://www.rdocumentation.org/packages/stats/versions/3.6.1/topics/ecdf R software], we compute an empirical cumulative distribution function, with several methods for plotting, printing and computing with such an “ecdf” object.
* In [https://www.mathworks.com/help/stats/cdfplot.html MATLAB] we can use Empirical cumulative distribution function (cdf) plot
* [https://www.jmp.com/support/help/14-2/cdf-plot.shtml jmp from SAS], the CDF plot creates a plot of the empirical cumulative distribution function. 
* [https://blog.minitab.com/blog/data-analysis-and-quality-improvement-and-stuff/the-empirical-cdf-part-2-software-vs-etch-a-sketch Minitab], create an Empirical CDF
* [http://www.mathwave.com/articles/distribution-fitting-graphs.html Mathwave], we can fit probability distribution to our data
* [https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/ecdfplot.htm Dataplot], we can plot Empirical CDF plot
* [https://scipy.github.io/devdocs/reference/generated/scipy.stats.ecdf.html Scipy], we can use scipy.stats.ecdf
* [https://www.statsmodels.org/dev/generated/statsmodels.distributions.empirical_distribution.ECDF.html#statsmodels.distributions.empirical_distribution.ECDF Statsmodels], we can use statsmodels.distributions.empirical_distribution.ECDF
* [https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ecdf.html Matplotlib], using the matplotlib.pyplot.ecdf function (new in version 3.8.0)<ref>https://matplotlib.org/stable/users/prev_whats_new/whats_new_3.8.0.html#axes-ecdf</ref>
* [https://seaborn.pydata.org/generated/seaborn.ecdfplot.html Seaborn], using the seaborn.ecdfplot function
*[https://plotly.com/python/ecdf-plots/ Plotly], using the plotly.express.ecdf function
* [https://www.statisticshowto.com/empirical-distribution-function/ Excel], we can plot Empirical CDF plot
* [[Software:ArviZ|ArviZ]], using the [https://python.arviz.org/en/latest/api/generated/arviz.plot_ecdf.html#arviz.plot_ecdf az.plot_ecdf] function

== See also ==
* [[Càdlàg]] functions
* [[Count data]]
* Distribution fitting
* [[Dvoretzky–Kiefer–Wolfowitz inequality]]
* [[Empirical probability]]
* [[Empirical process]]
* [[Quantile#Estimating quantiles from a sample|Estimating quantiles from a sample]]
* [[Frequency (statistics)]]
* [[Empirical likelihood]]
* [[Kaplan–Meier estimator]] for censored processes
* [[Survival function]]
* [[Q–Q plot]]

== References ==
{{reflist}}

== Further reading ==
* {{cite book
  | title = Empirical Processes with Applications to Statistics
  | year = 1986
  | publisher = Wiley
  | location = New York
  | isbn = 0-471-86725-X
  }}

== External links ==

{{-}}
{{Statistics|inference}}

{{DEFAULTSORT:Empirical Distribution Function}}
[[Category:Nonparametric statistics]]

{{Sourceattribution|Empirical distribution function}}