Empirical distribution function
In statistics, the empirical distribution function is the distribution function associated with the empirical measure of the sample. This cumulative distribution function is a step function that jumps up by 1/n at each of the n data points. The empirical distribution function estimates the cumulative distribution function underlying of the points in the sample and converges with probability 1 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.
where is the indicator of event A. For a fixed t, the indicator is a Bernoulli random variable with parameter p = F(t), hence is a binomial random variable with mean nF(t) and variance . This implies that is an unbiased estimator for F(t).
Since the ratio (n+1) / n approaches 1 as n goes to infinity, the asymptotic properties of the two definitions that are given above are the same.
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 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 cumulative distribution function 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 rate of convergence:
This result is extended by the Donsker’s theorem, which asserts that the empirical process , viewed as a function indexed by , converges in distribution in the Skorokhod space to the mean-zero Gaussian process , where B is the standard Brownian bridge. The covariance structure of this Gaussian process is
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.
- Càdlàg functions
- Dvoretzky–Kiefer–Wolfowitz inequality
- Empirical probability
- Empirical process
- Frequency (statistics)
- Kaplan–Meier estimator for censored processes
- Survival function
- Distribution fitting
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