Empirical Distribution Function - Asymptotic Properties

Asymptotic Properties

By the strong law of large numbers, the estimator converges to F(t) as n → ∞ almost surely, for every value of t:

thus the estimator is 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:

The sup-norm in this expression is called the Kolmogorov–Smirnov statistic for testing the goodness-of-fit between the empirical distribution and the assumed true cdf F. 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, has asymptotically normal distribution with the standard √n rate of convergence:

This result is extended by the Donsker’s theorem, which asserts that the empirical process, viewed as a function indexed by t ∈ R, converges in distribution in the Skorokhod space D to the mean-zero Gaussian process G_F = B∘F, where B is the standard Brownian bridge. The covariance structure of this Gaussian process is

The uniform rate of convergence in Donsker’s theorem can be quantified by the result, known as the Hungarian embedding:

Alternatively, the rate of convergence of 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 :

In fact, Kolmogorov has shown that if the cdf F is continuous, then the expression 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

and