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Empirical Distribution Function
The green curve, which asymptotically approaches heights of 0 and 1 without reaching them, is the true cumulative distribution function of the standard normal distribution. The grey hash marks represent the observations in a particular 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. (Click here to load a new graph.)
In statistics, an empirical distribution function is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by 1/n at each of the 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 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.
thus the estimator 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 t:
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 rate of convergence:
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 cumulative distribution function F is continuous, then the expression converges in distribution to , which has the Kolmogorov distribution that does not depend on the form of F.