The green curve, which asymptotically approaches heights of 0 and 1 without reaching them, is the truecumulative distribution function of thestandard normal distribution. The grey hash marks represent the observations in a particularsample 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. (Click here to load a new graph.)

Instatistics, anempirical distribution function (a.k.a. anempirical cumulative distribution function,eCDF) is thedistribution function associated with theempirical measure of asample.^[1] Thiscumulative distribution function is astep function that jumps up by1/n at each of then 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 anestimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution, according to theGlivenko–Cantelli theorem. A number of results exist to quantify the rate ofconvergence of the empirical distribution function to the underlying cumulative distribution function.

Let(X₁, …,X_n) beindependent, identically distributed real random variables with the commoncumulative distribution functionF(t). Then theempirical distribution function is defined as^[2]

${\displaystyle {\hat{F}}_n(t)=\frac{\text{number of elements in the sample}\le t}{n}=\frac{1}{n}\sum _{i=1}^n{\mathbf{1}}_{X_i\le t},}$

where${\displaystyle {\mathbf{1}}_A}$ is theindicator ofeventA. For a fixedt, the indicator${\displaystyle {\mathbf{1}}_{X_i\le t}}$ is aBernoulli random variable with parameterp =F(t); hence${\displaystyle n{\hat{F}}_n(t)}$ is abinomial random variable withmeannF(t) andvariancenF(t)(1 −F(t)). This implies that${\displaystyle {\hat{F}}_n(t)}$ is anunbiased estimator forF(t).

However, in some textbooks, the definition is given as

${\displaystyle {\hat{F}}_n(t)=\frac{1}{n+1}\sum _{i=1}^n{\mathbf{1}}_{X_i\le t}}$^[3][4]

Since the ratio(n + 1)/n approaches 1 asn goes to infinity, the asymptotic properties of the two definitions that are given above are the same.

By thestrong law of large numbers, the estimator${\displaystyle {\hat{F}}_n(t)}$ converges toF(t) asn → ∞almost surely, for every value oft:^[2]

${\displaystyle {\hat{F}}_n(t)\stackrel{\text{a.s.}}{\to }F(t);}$

thus the estimator${\displaystyle {\hat{F}}_n(t)}$ isconsistent. This expression asserts the pointwise convergence of the empirical distribution function to the true cumulative distribution function. There is a stronger result, called theGlivenko–Cantelli theorem, which states that the convergence in fact happens uniformly overt:^[5]

${\displaystyle \Vert {\hat{F}}_n-F{\Vert }_{\mathrm{\infty }}\equiv \underset{t\in \mathbb{R}}{sup}|{\hat{F}}_n(t)-F(t)|\stackrel{}{\to }0.}$

The sup-norm in this expression is called theKolmogorov–Smirnov statistic for testing the goodness-of-fit between the empirical distribution${\displaystyle {\hat{F}}_n(t)}$ and the assumed true cumulative distribution functionF. Othernorm functions may be reasonably used here instead of the sup-norm. For example, theL²-norm gives rise to theCramér–von Mises statistic.

The asymptotic distribution can be further characterized in several different ways. First, thecentral limit theorem states thatpointwise,${\displaystyle {\hat{F}}_n(t)}$ has asymptotically normal distribution with the standard${\displaystyle \sqrt{n}}$ rate of convergence:^[2]

${\displaystyle \sqrt{n}({\hat{F}}_n(t)-F(t))\stackrel{d}{\to }\mathcal{N}(0,F(t)(1-F(t))).}$

This result is extended by theDonsker’s theorem, which asserts that theempirical process${\displaystyle \sqrt{n}({\hat{F}}_n-F)}$, viewed as a function indexed by${\displaystyle t\in \mathbb{R}}$,converges in distribution in theSkorokhod space${\displaystyle D[-\mathrm{\infty },+\mathrm{\infty }]}$ to the mean-zeroGaussian process${\displaystyle G_F=B\circ F}$, whereB is the standardBrownian bridge.^[5] The covariance structure of this Gaussian process is

${\displaystyle \mathrm{E}[G_F(t_1)G_F(t_2)]=F(t_1\wedge t_2)-F(t_1)F(t_2).}$

The uniform rate of convergence in Donsker’s theorem can be quantified by the result known as theHungarian embedding:^[6]

Alternatively, the rate of convergence of${\displaystyle \sqrt{n}({\hat{F}}_n-F)}$ 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 theDvoretzky–Kiefer–Wolfowitz inequality provides bound on the tail probabilities of${\displaystyle \sqrt{n}\Vert {\hat{F}}_n-F{\Vert }_{\mathrm{\infty }}}$:^[6]

${\displaystyle Pr(\sqrt{n}\Vert {\hat{F}}_n-F{\Vert }_{\mathrm{\infty }}>z)\le 2e^{-2z^2}.}$

In fact, Kolmogorov has shown that if the cumulative distribution functionF is continuous, then the expression${\displaystyle \sqrt{n}\Vert {\hat{F}}_n-F{\Vert }_{\mathrm{\infty }}}$ converges in distribution to${\displaystyle \Vert B{\Vert }_{\mathrm{\infty }}}$, which has theKolmogorov distribution that does not depend on the form ofF.

Another result, which follows from thelaw of the iterated logarithm, is that^[6]

${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}\frac{\sqrt{n}\Vert {\hat{F}}_n-F{\Vert }_{\mathrm{\infty }}}{\sqrt{2\ln \ln n}}\le \frac{1}{2},\text{a.s.}}$

and

${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}\sqrt{2n\ln \ln n}\Vert {\hat{F}}_n-F{\Vert }_{\mathrm{\infty }}=\frac{\pi }{2},\text{a.s.}}$

As perDvoretzky–Kiefer–Wolfowitz inequality the interval that contains the true CDF,${\displaystyle F(x)}$, with probability${\displaystyle 1-\alpha }$ is specified as

${\displaystyle F_n(x)-\epsilon \le F(x)\le F_n(x)+\epsilon \text{ where }\epsilon =\sqrt{\frac{\ln \frac{2}{\alpha }}{2n}}.}$

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.

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

• InR software, we compute an empirical cumulative distribution function, with several methods for plotting, printing and computing with such an “ecdf” object.
• InMATLAB we can use Empirical cumulative distribution function (cdf) plot
• jmp from SAS, the CDF plot creates a plot of the empirical cumulative distribution function.
• Minitab, create an Empirical CDF
• Mathwave, we can fit probability distribution to our data
• Dataplot, we can plot Empirical CDF plot
• Scipy, we can use scipy.stats.ecdf
• Statsmodels, we can use statsmodels.distributions.empirical_distribution.ECDF
• Matplotlib, using the matplotlib.pyplot.ecdf function (new in version 3.8.0)^[7]
• Seaborn, using the seaborn.ecdfplot function
• Plotly, using the plotly.express.ecdf function
• Excel, we can plot Empirical CDF plot
• ArviZ, using theaz.plot_ecdf function

• Càdlàg functions
• Count data
• Distribution fitting
• Dvoretzky–Kiefer–Wolfowitz inequality
• Empirical probability
• Empirical process
• Estimating quantiles from a sample
• Frequency (statistics)
• Empirical likelihood
• Kaplan–Meier estimator for censored processes
• Survival function
• Q–Q plot

• Shorack, G.R.;Wellner, J.A. (1986).Empirical Processes with Applications to Statistics. New York: Wiley.ISBN 0-471-86725-X.

• Media related toEmpirical distribution functions at Wikimedia Commons

• Nonparametric statistics
• Empirical process

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• Commons category link is on Wikidata