Asymptotic Normality in Nonparametric Methods

Let U₁, U₂, \, UN be a random sample from a population with a continuous distribution function and Rᵢ, i = 1, \, N, be the rank of Uᵢ among the N observations. Asymptotic normality is studied for the statistics of the type *\0. 1 \N₈=₁ \N₉=₁ c₈₉aN (Rᵢ/N, Rⱼ/N), * where constants c₈₉ satisfy certain negligibility conditions and the score function aN (\, \) is derived from a function a (\, \) satisfying certain monotonicity and integrability conditions. It is shown that the statistic (0. 1) is asymptotically equivalent to *\0. 2 \N₈=₁ \N₉=₁ c₈₉a (Uᵢ, Uⱼ), * so that the problem is reduced to a simpler one, viz. studying the asymptotic distribution of (0. 2). Similar results are obtained for the two sample analog of (0. 1) viz. *\0. 3 \N₈=₁ \M₉=₁ c₈₉a₍₌ (Rᵢ/N, Sⱼ/M) * where Sⱼ, j = 1, \, M, are the ranks corresponding to another independent random sample of size M from some other population. Few more variants of the above and applications of these statistics are given. The present study is a generalization of a paper by Hajek (1961).