Empirical distribution function explained

See also: Frequency distribution.

In statistics, an empirical distribution function (commonly also called an empirical cumulative distribution function, eCDF) is the distribution function associated with the empirical measure of a sample.^[1] This cumulative distribution function is a step function that jumps up by at each of the 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 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 the empirical distribution function to the underlying cumulative distribution function.

Definition

Let be independent, identically distributed real random variables with the common cumulative distribution function . Then the empirical distribution function is defined as^[2]

\widehat F _{n(t)} = (numberofelementsinthesample \leq t)/(n) = (1)/(n) \sum ^{n} _{i =1} 1 _{X _{i} \le t} , where $\backslash mathbf\_$ is the indicator of event . For a fixed, the indicator $\backslash mathbf\_$ is a Bernoulli random variable with parameter ; hence $n\; \backslash widehat\; F\_n(t)$ is a binomial random variable with mean and variance . This implies that $\backslash widehat\; F\_n(t)$ is an unbiased estimator for .

However, in some textbooks, the definition is given as

\widehat F _{n(t)} = (1)/(n +1) \sum ^{n} _{i =1} 1 _{X _{i} \le t} ^{[3] [4]}

Asymptotic properties

Since the ratio approaches 1 as 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 $\backslash scriptstyle\backslash widehat\_n(t)$ converges to as almost surely, for every value of :

\widehat F _{n(t) \xrightarrow{a.s.} }\ F(t); thus the estimator $\backslash scriptstyle\backslash widehat\_n(t)$ is 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 :^[5] \|\widehat F _{n-F\|} _{infty} \equiv \sup _{t\inR} | \widehat F _{n(t)-F(t)| \xrightarrow{} 0.} The sup-norm in this expression is called the Kolmogorov–Smirnov statistic for testing the goodness-of-fit between the empirical distribution $\backslash scriptstyle\backslash widehat\_n(t)$ and the assumed true cumulative distribution function . Other norm functions may be reasonably used here instead of the sup-norm. For example, the L²-norm gives rise to the Cramér–von Mises statistic.

The asymptotic distribution can be further characterized in several different ways. First, the central limit theorem states that pointwise, $\backslash scriptstyle\backslash widehat\_n(t)$ has asymptotically normal distribution with the standard $\backslash sqrt$ rate of convergence:^[2]

\sqrt{n} ( \widehat F _{n(t)} - F(t) ) \xrightarrow{d} l{N} ( 0 , F(t) ( 1- F(t) ) ) . This result is extended by the Donsker’s theorem, which asserts that the empirical process $\backslash scriptstyle\backslash sqrt(\backslash widehat\_n\; -\; F)$, viewed as a function indexed by $\backslash scriptstyle\; t\backslash in\backslash mathbb$, converges in distribution in the Skorokhod space $\backslash scriptstyle\; D[-\backslash infty,\; +\backslash infty]$ to the mean-zero Gaussian process $\backslash scriptstyle\; G\_F\; =\; B\; \backslash circ\; F$, where is the standard Brownian bridge.^[5] The covariance structure of this Gaussian process is\operatorname{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 the Hungarian embedding:^[6] \limsup _{n\toinfty} (\sqrt{n)/() } \big\| \sqrt(\widehat F_n-F) - G_\big\|_\infty < \infty, \quad \text

Alternatively, the rate of convergence of $\backslash scriptstyle\backslash sqrt(\backslash widehat\_n-F)$ 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 the Dvoretzky–Kiefer–Wolfowitz inequality provides bound on the tail probabilities of $\backslash scriptstyle\backslash sqrt\backslash |\backslash widehat\_n-F\backslash |\_\backslash infty$:^[6]

\Pr ( \sqrt{n}\|\widehat{F} _{n-F\|} _{infty} > z ) \leq 2 e ^{-2 z ^{2}} . In fact, Kolmogorov has shown that if the cumulative distribution function is continuous, then the expression $\backslash scriptstyle\backslash sqrt\backslash |\backslash widehat\_n-F\backslash |\_\backslash infty$ converges in distribution to $\backslash scriptstyle\backslash |B\backslash |\_\backslash infty$, which has the Kolmogorov distribution that does not depend on the form of .

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

\limsup _{n\toinfty} (\sqrt{n)/(\|\widehat{F}) _{n-F\|} _{infty}{\sqrt{2lnln} n}} \leq (12 ,)/() a.s. and\liminf _{n\toinfty} \sqrt{ 2 n lnln n} \|\widehat{F} _{n-F\|} _{infty} = (\pi)/(2) , a.s.

Confidence intervals

As per Dvoretzky–Kiefer–Wolfowitz inequality the interval that contains the true CDF,

F(x) , with probability 1- \alpha is specified asF _{n(x)} - \varepsilon \le F(x) \le F _{n(x)} + \varepsilon where \varepsilon = \sqrt{ (ln { (2)/(\alpha))/() }}.

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 R software, we compute an empirical cumulative distribution function, with several methods for plotting, printing and computing with such an “ecdf” object.
• In MATLAB 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 the az.plot_ecdf function

See also

• 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

Further reading

• Book: Shorack . G.R. . Galen R. Shorack . Wellner . J.A. . Jon A. Wellner . Empirical Processes with Applications to Statistics . 1986 . Wiley . New York . 0-471-86725-X .

Notes and References

1. Book: A modern introduction to probability and statistics: Understanding why and how. 2005. Springer. Michel Dekking. 978-1-85233-896-1. London. 219. 262680588.
2. Book: van der Vaart, A.W. . Asymptotic statistics . limited . 1998 . Cambridge University Press . 0-521-78450-6 . 265 .
3. Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values. Springer, p. 36, Definition 2.4. .
4. Madsen, H.O., Krenk, S., Lind, S.C. (2006) Methods of Structural Safety. Dover Publications. p. 148-149.
5. Book: van der Vaart, A.W. . Asymptotic statistics . limited . 1998 . Cambridge University Press . 0-521-78450-6 . 266 .
6. Book: van der Vaart, A.W. . Asymptotic statistics . limited . 1998 . Cambridge University Press . 0-521-78450-6 . 268 .
7. Web site: What's new in Matplotlib 3.8.0 (Sept 13, 2023) — Matplotlib 3.8.3 documentation .