Asymptotic formula for sum of moment mean deviation for order statistics from uniform distribution

摘要

Assume that n mobile sensors are thrown uniformly and independently at random with the uniform distribution on the unit interval. We study the expected sum over all sensors i from 1 to n, where the contribution of the ith sensor is its displacement from the current location to the anchor equidistant point ti = i n ? 1 2n, raised to the ath power, when a is an odd natural number.As a consequence, we derive the following asymptotic identity. Fix a positive integer. Let Xi:n denote the ith order statistic from a random sample of size n from the Uniform(0, 1) population. Then ∑i=1nE[|X i:n ?E[Xi:n]|a] = Γ a 2 + 1 2a 2(1 + a) n na 2 + O n na 2 , where Γ(z) is the Gamma function.