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

The authors study the problem of testing whether the distribution function (d. f. ) of the observed independent chance variables x₁, , xₙ 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 (F, G) = ᵧ | F (y) - G (y) |. Let N (y, ²) be the normal d. f. with mean and variance ². Let G^ₙ (y) be the empiric d. f. of x₁, , xₙ. The authors consider, inter alia, tests of normality based on ₙ = (G^ₙ (y), N (y x, s²) ) and on wₙ = (G^ₙ (y) - N (y x, s²) ) ² dᵧN (y x, s²). It is shown that the asymptotic power of these tests is considerably greater than that of the optimum ² test. The covariance function of a certain Gaussian process Z (t), 0 t 1, is found. It is shown that the sample functions of Z (t) are continuous with probability one, and that n P\nwₙ < a\ = P\W < a\, where W = ¹₀ Z (t) ² dt. Tables of the distribution of W and of the limiting distribution of nₙ are given. The role of various metrics is discussed.