## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 23, Number 2 (1952), 193-212.

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

T. W. Anderson and D. A. Darling

#### Abstract

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.

#### Article information

**Source**

Ann. Math. Statist., Volume 23, Number 2 (1952), 193-212.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177729437

**Digital Object Identifier**

doi:10.1214/aoms/1177729437

**Mathematical Reviews number (MathSciNet)**

MR50238

**Zentralblatt MATH identifier**

0048.11301

**JSTOR**

links.jstor.org

#### Citation

Anderson, T. W.; Darling, D. A. Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes. Ann. Math. Statist. 23 (1952), no. 2, 193--212. doi:10.1214/aoms/1177729437. https://projecteuclid.org/euclid.aoms/1177729437