Empirical distribution function

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. 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.

Let $(x 1, \dots, x n)$ be independent, identically distributed real random variables with the common cumulative distribution function $F (t)$ . Then the empirical distribution function is defined as

where $\mathbf {1} _{A}$ ${\mathbf {1}}_{{A}}$ is the indicator of event $A$ . For a fixed $t$ , the indicator $\mathbf {1} _{x_{i}\leq t}$ ${\mathbf {1}}_{{x_{i}\leq t}}$ is a Bernoulli random variable with parameter $p = F (t)$ , hence $\scriptstyle n{\hat {F}}_{n}(t)$ $\scriptstyle n{\hat F}_{n}(t)$ is a binomial random variable with mean $nF (t)$ and variance $nF (t)(1 - F (t))$ . This implies that $\scriptstyle {\hat {F}}_{n}(t)$ $\scriptstyle {\hat F}_{n}(t)$ is an unbiased estimator for $F (t)$ .

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