Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes

The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function $F(x)$. If $F_n(x)$ is the empirical cumulative distribution function and $\psi(t)$ is some nonnegative weight function $(0 \leqq t \leqq 1)$, we consider $n^{\frac{1}{2}} \sup_{-\infty<x<\infty} \{| F(x) - F_n(x) | \psi^\frac{1}{2}\lbrack F(x) \rbrack\}$ and $n\int^\infty_{-\infty}\lbrack F(x) - F_n(x) \rbrack^2 \psi\lbrack F(x)\rbrack dF(x).$ A general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations. For certain weight functions including $\psi = 1$ and $\psi = 1/\lbrack t(1 - t) \rbrack$ we give explicit limiting distributions. A table of the asymptotic distribution of the von Mises $\omega^2$ criterion is given.