## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 26, Number 2 (1955), 189-211.

### On Tests of Normality and Other Tests of Goodness of Fit Based on Distance Methods

M. Kac, J. Kiefer, and J. Wolfowitz

#### Abstract

The authors study the problem of testing whether the distribution function (d.f.) of the observed independent chance variables $x_1, \cdots, x_n$ is a member of a given class. A classical problem is concerned with the case where this class is the class of all normal d.f.'s. For any two d.f.'s $F(y)$ and $G(y)$, let $\delta(F, G) = \sup_y | F(y) - G(y) |$. Let $N(y \mid \mu, \sigma^2)$ be the normal d.f. with mean $\mu$ and variance $\sigma^2$. Let $G^\ast_n(y)$ be the empiric d.f. of $x_1, \cdots, x_n$. The authors consider, inter alia, tests of normality based on $\nu_n = \delta(G^\ast_n(y), N(y \mid \bar{x}, s^2))$ and on $w_n = \int (G^\ast_n(y) - N(y \mid \bar{x}, s^2))^2 d_yN (y \mid \bar{x}, s^2)$. It is shown that the asymptotic power of these tests is considerably greater than that of the optimum $\chi^2$ test. The covariance function of a certain Gaussian process $Z(t), 0 \leqq t \leqq 1$, is found. It is shown that the sample functions of $Z(t)$ are continuous with probability one, and that $\underset{n\rightarrow\infty}\lim P\{nw_n < a\} = P\{W < a\}, \text{where} W = \int^1_0 \lbrack Z(t)\rbrack^2 dt$. Tables of the distribution of $W$ and of the limiting distribution of $\sqrt{n}\nu_n$ are given. The role of various metrics is discussed.

#### Article information

**Source**

Ann. Math. Statist., Volume 26, Number 2 (1955), 189-211.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177728538

**Digital Object Identifier**

doi:10.1214/aoms/1177728538

**Mathematical Reviews number (MathSciNet)**

MR70919

**Zentralblatt MATH identifier**

0066.12301

**JSTOR**

links.jstor.org

#### Citation

Kac, M.; Kiefer, J.; Wolfowitz, J. On Tests of Normality and Other Tests of Goodness of Fit Based on Distance Methods. Ann. Math. Statist. 26 (1955), no. 2, 189--211. doi:10.1214/aoms/1177728538. https://projecteuclid.org/euclid.aoms/1177728538