Distribution of an analog of Sherman's statistics under rank-censored observations

Abstract

Let U_n(1),..., U_n(n) be a variational series constructed from a sequence of n aggregate-independent random variables distributed uniformly on (0, 1). Let 0 = k₀, k₁,..., k_m, k_m+1= n+1 be an increasing sequence of nonnegative integers, λ= k_r+1−k_r, r=0,..., m, and

$$\xi _n = \frac{1}{2}\sum\nolimits_{r = 0}^m {\left| {U_n (k_{r + 1} ) - U_n^\prime (k_r ) - \frac{{k_{r + 1} - k_r }}{{n + 1}}} \right|.}$$

Under certain restrictions on the numbers λ_r= k{r+1}−k_r, in this paper we have shown the asymptotic normality (with an appropriate norming) of the quantity ξ_n as n, m →∞ such that lim sup (m/√n) ar ∞.