The asymptotic distribution of order statistics

Abstract

AbstractFor each n., X1(n), X2(n), …, Xn(n) are IID, with common pdf fn(x). y1(n) < … < Yn (n) are the ordered values of X1 (n), …, Xn(n). Kn is a positive integer, with lim Kn = ∞. Under certain conditions on Kn and fn (x), it was shown in an earlier paper that the joint distribution of a special set of Kn + 1 of the variables Y1 (n), …, Yn (n) can be assumed to be normal for all asymptotic probability calculations. In another paper, it was shown that if fn (x) approaches the pdf which is uniform over (0, 1) at a certain rate as n increases, then the conditional distribution of the order statistics not in the special set can be assumed to be uniform for all asymptotic probability calculations. The present paper shows that even if fn (x) does not approach the uniform distribution as n increases, the distribution of the order statistics contained between order statistics in the special set can be assumed to be the distribution of a quadratic function of uniform random variables, for all asymptotic probability calculations. Applications to statistical inference are given.