Empirical distribution function

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In statistics, an empirical distribution function is a cumulative probability distribution function that concentrates probability 1/n at each of the n numbers in a sample.

Let X₁, …, X_n be iid real random variables with the cdf F(x). The empirical distribution function F̂_n(x) is a step function defined by

where I(A) is the indicator of event A.

For fixed x, I(X_i ≤ x) is a Bernoulli random variable with parameter p = F(x), hence nF̂_n(x) is a binomial random variable with mean nF(x) and variance nF(x)(1 − F(x)).

Asymptotical properties

• By the strong law of large numbers,

for fixed x (a.s. denotes almost sure convergence).

In other words, F̂_n(x) is a consistent unbiased estimator of the cumulative distribution function F(x).

• By the central limit theorem,

converges in distribution to a normal distribution N(0, F(x)(1 − F(x))) for fixed x.

The Berry–Esséen theorem provides the rate of this convergence.

• By the Glivenko–Cantelli theorem F̂_n(x) → F(x) uniformly over x, that is

The Dvoretzky–Kiefer–Wolfowitz inequality provides the rate of this convergence.

• Kolmogorov showed that

converges in distribution to the Kolmogorov distribution, provided that F(x) is continuous.

The Kolmogorov–Smirnov test for goodness-of-fit is based on this fact.

• By Donsker's theorem,

as a process indexed by x, converges in law in the Skorokhod space to a Gaussian process B(F(x)), where B(t) is the Brownian bridge.

See also

• Càdlàg functions
• Empirical probability
• Empirical process