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Biometrika 1972 59(3):605-611; doi:10.1093/biomet/59.3.605
© 1972 by Biometrika Trust
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Quadratic forms in order statistics used as goodness-of-fit criteria

H. O. HARTLEY and R. C. PFAFFENBERGER

Institute of Statistics, Texas A & M University

Consider a sample vector, z, whosen elements z(i)are the probability transforms z(i) {theta} P{x(i)} of the order statistics x(i) of a random sample of size n drawn from a cumulative distribution function P(x). Most nonparametric goodness-of-fit criteria concerned with testing whether the specified P(x) is the true one are computed from z. The criterion considered here is the x2-type criterion S2 {theta} (z – µ)'V–1(z – µ), where µ {theta} E(z) and V {theta} E{(z –;µ)(z – µ)'}. The extract small-sample distribution for S2 is derived for small n and tabulated for n = 2(1) 16 with the help of an exact recurrence formula. For larger n, a Peason Type V fit to the exact moments of S2 is validated by extensive Monte-Carlo computations. For the more general S2 criteria based on subsets of k < n of the z(i) asymptotic optimality properties are proved.

Key Words: Goodness-of-fit teste • Uniform order statistics • Spacings • Fitting by Pearson curves • Randomness of points on a line • Quadratic forms in order statistics


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