= Solution
For a fully specified continuous null distribution $F_0$, compare the <empirical distribution function> $F_n$ with $F_0$. The <probability integral transform> makes $F_0(Y_i)$ iid uniform under the null, giving distribution-free calibration. Three common discrepancies are
$$
D_n=\sup_x|F_n(x)-F_0(x)|,\qquad
C_n=n\int(F_n-F_0)^2dF_0,\qquad
A_n=n\int\frac{(F_n-F_0)^2}{F_0(1-F_0)}dF_0.
$$
These define the <Kolmogorov-Smirnov test>, <Cramér–von Mises criterion> and <Anderson–Darling test>, respectively. Reject for large values. The supremum detects the largest discrepancy, the unweighted integral combines discrepancies throughout the distribution, and the reciprocal weight emphasizes the tails.
The <Donsker theorem for empirical distribution functions> gives $\sqrt n(F_n-F_0)\Rightarrow B^0\circ F_0$ for a <Brownian bridge> $B^0$, with <covariance> $\min(u,v)-uv$. Consequently the standard continuous-null limits are $\sup_{0\leq u\leq1}|B^0(u)|$, $\int_0^1B^0(u)^2du$, and $\int_0^1B^0(u)^2/[u(1-u)]du$ for $\sqrt nD_n,C_n,A_n$, respectively; the weighted last limit also requires control near the endpoints. Exact finite-sample or asymptotic null <quantiles> provide critical values.
If parameters in $F_\theta$ are estimated from the same observations, these fully specified-null critical values generally fail. A first-order expansion is
$$
\sqrt n(F_n-F_{\widehat\theta})(x)
=\sqrt n(F_n-F_{\theta_0})(x)-\dot F_{\theta_0}(x)^T\sqrt n(\widehat\theta-\theta_0)+o_p(1),
$$
so the estimated parameter changes the limiting process. One must use the fitted-model null law, adjusted asymptotic theory, or a <parametric bootstrap> that re-estimates the parameters in each replicate.
Back to article page
