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

原文摘要

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 x2-type criterion S2 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 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.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）