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

M. Kac, J. Kiefer and J. Wolfowitz
The Annals of Mathematical Statistics
Vol. 26, No. 2 (Jun., 1955), pp. 189-211
Stable URL: http://www.jstor.org/stable/2236876
Page Count: 23

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Abstract

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 $\delta(F, G) = \sup_y| F(y) - G(y)|$. Let N(y ∣ μ, σ2) be the normal d.f. with mean μ and variance σ2. Let G* n(y) be the empiric d.f. of x1, ⋯, xn. The authors consider, inter alia, tests of normality based on νn = δ(G* n(y), N(y ∣ x̄, s2)) and on wn = ∫ (G* n(y) - N(y ∣ x̄, s2))2 dyN (y ∣ x̄, s2). It is shown that the asymptotic power of these tests is considerably greater than that of the optimum χ2 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 $\underset{n\rightarrow\infty}\lim P\{nw_n < a\} = P\{W < a\}, \text{where} W = \int^1_0 \lbrack Z(t)\rbrack^2 dt$. 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.

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