The step function that assigns to each real value x the fraction of sample observations less than or equal to x; a nonparametric estimator of the underlying cumulative distribution.

Principle

For a sample of size n, ECDF(x) = (1/n) ∑_{i=1}^n 1_{X_i ≤ x}; it uses sample ranks directly and converges to the true cumulative distribution uniformly as sample size increases.

Demonstration

For sample {2,3,5}, the ECDF equals 0 for x<2, 1/3 for 2≤x<3, 2/3 for 3≤x<5, and 1 for x≥5, stepping up by 1/n at each observed value.

Misapplication

Treating the ECDF as an estimator of the density without smoothing (differentiating the step function) yields inconsistent or noisy density estimates; also, comparing ECDFs from dependent samples without adjustment can mislead.

Consequence

Provides a distribution-free estimator for cumulative probabilities, underpins nonparametric goodness-of-fit and two-sample procedures, and forms the basis for bootstrap and rank methods.

Reversal

A parametric cumulative distribution is specified by a finite set of parameters and a smooth functional form; it delivers interpolated probability statements but imposes structural assumptions absent in the ECDF.

Boundary

Defined naturally for scalar-valued observations with a total order; multivariate generalization is nonunique and requires choosing margins, projections, or partial orders.

Semantic Tension

Kernel density estimators produce smooth density estimates by smoothing observations, while the ECDF estimates cumulative probabilities without smoothing; the trade-off is between unbiased exact ranks and smoothed density bias/variance.

Synthesis

The ECDF is the sample-based stepwise estimator of the cumulative distribution that assigns empirical probabilities to thresholds, converging to the true distribution as sample size grows.