Randomized empirical processes and confidence bands via virtual resampling

Abstract

A data set of N labeled units, or labeled units of a finite population, may on occasions be viewed as if they were random samples {X1,…,XN}, N≥1, the first N of the labeled units from an infinite sequence X,X1,X2,… of independent real valued random variables with a common distribution function F. In case of such a view of a finite population, or when an accordingly viewed data set in hand is too big to be entirely processed, then the sample distribution function FN and the population distribution function F are both to be estimated. This, in this paper, is achieved via sampling the indices {1,…,N} of {X1,…,XN} with replacement mN≔∑i=1Nwi(N) times so that for each 1≤i≤N, wi(N) is the count of the number of times the index i of Xi is chosen in this virtual resampling process. The classical theory of weak convergence of empirical processes is extended along these lines to that of the thus created randomly weighted empirical processes, via conditioning on the weights, when N,mN→∞ so that mN=o(N2).