= Empirical cumulative distribution function =

An '''empirical cumulative distribution function''' is a monotonic step function calculated from a sample.

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== Description ==

Given a sample of ''n'' observations drawn from a population that is distributed according to an unknown cumulative distribution function ''F(x)'' (i.e., ''F(x) = P[X < x]''), let ''1,,X<x,,'' be an [[Analysis/IndicatorFunctions|indicator function]] defined on the set of ''X,,i,,'' and an argument ''x''. It equals 1 if ''X,,i,,'' is less than ''x''. At a given level ''x'', the indicator function is a [[Analysis/BernoulliDistribution|Bernoulli distributed]] random variable with ''p = F(t)''.

An estimator for ''F(t)'' is calculated as:

{{attachment:emp.svg}}

It follows that ''nF̂(x)'' is a [[Statistics/SamplingDistribution|sample statistic]] following a [[Analysis/BinomialDistribution|binomial distribution]].



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CategoryRicottone