Quadratic forms in order statistics used as goodness-of-fit criteria

Abstract

Consider a sample vector, z, whosen elements z (i) are the probability transforms z (i) θ 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 x 2 -type criterion S 2 θ (z – μ)′ V–1 (z – μ), where μ θ E (z) and V θ E{(z –;μ)(z – μ)′}. The extract small-sample distribution for S 2 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 S 2 is validated by extensive Monte-Carlo computations. For the more general S 2 criteria based on subsets of k < n of the z( i ) asymptotic optimality properties are proved.