Empirical Distribution Function - Definition

Definition

Let (x1, …, xn) be iid real random variables with the common cdf F(t). Then the empirical distribution function is defined as

\hat F_n(t) = \frac{ \mbox{number of elements in the sample} \leq t}n = \frac{1}{n} \sum_{i=1}^n \mathbf{1}\{x_i \le t\},

where 1{A} is the indicator of event A. For a fixed t, the indicator 1{xi ≤ t} is a Bernoulli random variable with parameter p = F(t), hence is a binomial random variable with mean nF(t) and variance nF(t)(1 − F(t)). This implies that is an unbiased estimator for F(t).

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