CS(19) Empirical Distribution Function (EDF)

The Empirical Distribution Function (EDF) is a fundamental concept in statistics, particularly in the context of Monte Carlo methods and non-parametric inference. It serves as an estimator for the cumulative distribution function (CDF) of a population based on sample data.

Definition of the Empirical Distribution Function

Suppose we have a random sample X1, X2, …, Xn drawn from a population with an unknown distribution function F(x). The empirical distribution function Fn(x) is defined as:

Where I(Xi ≤ x) is an indicator function that equals 1 if Xi ≤ x, and 0 otherwise. Intuitively, Fn(x) represents the proportion of sample observations less than or equal to x. It is a step function that jumps at each observed data point.

Properties of the EDF

1. Consistency: The EDF is a consistent estimator of the true CDF F(x). As the sample size n → ∞, Fn(x) converges to F(x) for all x.
2. Unbiasedness: Fn(x) is an unbiased estimator of F(x).
3. Variance: For a given x, the variance of Fn(x) is: Var(Fn(x))= F(x).(1-F(x))/n