The probability of large deviations for the sum functions of spacings

Abstract

Let0=U0,n≤U1,n≤⋯≤Un−1,n≤Un,n=1be an ordered sample from uniform[0,1]distribution, andDin=Ui,n−Ui−1,n,i=1,2,…,n;n=1,2,…,be their spacings, and letf1n,…,fnnbe a set of measurable functions. In this paper, the probabilities of the moderate and Cramer-type large deviation theorems for statisticsRn(D)=f1n(nD1n)+⋯+fnn(nDnn)are proved. Application of these theorems for determination of the intermediate efficiencies of the tests based onRn(D)-type statistic is presented here too.