Empirical distribution function ($F_n$)

For real observations $X_1,\dots ,X_n$, the empirical distribution function is

The test rejects for large values of this tail-weighted empirical discrepancy. For a specified continuous null, its standard limiting statistic is ${\int }_0^1{B}^0(u{)}^2/[u(1-u)]du$, using a Brownian bridge. The reciprocal variance weight gives tail deviations more influence than the Cramér–von Mises criterion. If null parameters are estimated, fitting must be included when computing or simulating its reference law.

This goodness-of-fit criterion integrates the squared difference between the empirical distribution function and a specified continuous null distribution. Under a fully specified null, the probability integral transform makes its law distribution-free; its limiting law is ${\int }_0^1{B}^0(u{)}^{2}du$ for a Brownian bridge. Parameter fitting changes that calibration.

The empirical distribution process is the centred, scaled empirical distribution function of an independent sample with cumulative distribution function $G$. Its mean is zero and its covariance is $G(\min (x,y))-G(x)G(y)$. This follows by expanding the two indicator averages: different sample indices have zero covariance, while the same index contributes the covariance of the two nested events. The scaling by $\sqrt{n}$ keeps these covariances independent of $n$.

For an independent sample with cumulative distribution function $F$, the empirical distribution process converges in the supremum norm on ${\ell }^{\infty }(\mathbb{R})$ to the F-Brownian bridge. The convergence uses outer expectation and inner expectation when finite-sample Borel measurability fails. It is a full function-space theorem, stronger than the Finite-dimensional Brownian bridge limit of an empirical distribution process. No continuity of $F$ is required. The continuous mapping theorem gives convergence of the supremum statistic; for continuous $F$, its range contains $(0,1)$ and the bridge vanishes at zero and one, giving the Kolmogorov-Smirnov theorem. The function-space statement is stated here as a theorem; its tightness assertion is an additional ingredient beyond the finite-dimensional covariance calculation.

For fixed $x_1,\dots ,x_d$, the vector $({\alpha }_n(x_j){)}_{j=1}^d$ converges in distribution to $(B(G(x_j)){)}_{j=1}^d$, where $B$ is a standard Brownian bridge. To prove it, write the vector as $n^{-1/2}$ times the sum of independent centred bounded indicator vectors. Their covariance matrix is $G(\min (x_i,x_j))-G(x_i)G(x_j)$. The multivariate central limit theorem gives a centred multivariate normal distribution with this covariance, exactly the bridge vector. Singular covariance matrices, including endpoint times or repeated values of $G$, cause no problem. This is a finite-dimensional assertion; a full function-space limit additionally requires tightness.