Open Access
June, 1955 On Tests of Normality and Other Tests of Goodness of Fit Based on Distance Methods
M. Kac, J. Kiefer, J. Wolfowitz
Ann. Math. Statist. 26(2): 189-211 (June, 1955). DOI: 10.1214/aoms/1177728538


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.


Download Citation

M. Kac. J. Kiefer. J. Wolfowitz. "On Tests of Normality and Other Tests of Goodness of Fit Based on Distance Methods." Ann. Math. Statist. 26 (2) 189 - 211, June, 1955.


Published: June, 1955
First available in Project Euclid: 28 April 2007

zbMATH: 0066.12301
MathSciNet: MR70919
Digital Object Identifier: 10.1214/aoms/1177728538

Rights: Copyright © 1955 Institute of Mathematical Statistics

Vol.26 • No. 2 • June, 1955
Back to Top