ecdf

The empirical cumulative distribution function (ECDF) is a nonparametric estimator of a population’s distribution function based on a sample. Given a sample x1, x2, ..., xn drawn from a distribution with cumulative distribution function F, the ECDF is defined as F_n(x) = (1/n) sum_{i=1}^n I{xi ≤ x}, where I{.} is the indicator function.

The ECDF is a step function that is nondecreasing and right-continuous. It satisfies 0 ≤ F_n(x) ≤ 1 for all x, and it jumps by 1/n at each observed data point. If the data are ordered as X_(1) ≤ X_(2) ≤ ... ≤ X_(n), then F_n(X_(i)) = i/n and F_n(x) = k/n for x between X_(k) and X_(k+1). The ECDF provides a direct, data-driven view of the distribution without assuming a particular parametric form.

Key properties include that the expectation E[F_n(x)] equals the true CDF F(x) for any x, making the ECDF an unbiased estimator of F at each point. The Glivenko-Cantelli theorem states that F_n converges to F uniformly almost surely as the sample size n grows, ensuring consistency of the ECDF as an estimator of the population distribution.

Common uses include exploratory data analysis to understand distributional shape, estimating quantiles and percentiles, and nonparametric goodness-of-fit tests such as the Kolmogorov-Smirnov test. Practical considerations include limited resolution for small samples and the presence of ties, which produce larger jumps at repeated values. The ECDF is often plotted as a staircase to summarize the data distribution.