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

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.