Randomized Empirical Processes and Confidence Bands via Virtual Resampling

TL;DR

This paper develops a new approach for estimating distribution functions and constructing confidence bands using virtual resampling and extends classical empirical process theory to this randomized setting.

Contribution

It introduces a novel framework for virtual resampling that generalizes empirical process theory to large datasets with random weights, under specific growth conditions.

Findings

Extends empirical process theory to weighted resampling schemes.

Provides conditions under which weak convergence holds for the new processes.

Enables construction of confidence bands using virtual resampling methods.

Abstract

Let $X,X_1,X_2,\cdots$ be independent real valued random variables with a common distribution function $F$, and consider $\{X_1,\cdots,X_N \}$, possibly a big concrete data set, or an imaginary random sample of size $N\geq 1$ on $X$. In the latter case, or when a concrete data set in hand is too big to be entirely processed, then the sample distribution function $F_N$ and the the population distribution function $F$ are both to be estimated. This, in this paper, is achieved via viewing $\{X_1,\cdots,X_N \}$ as above, as a finite population of real valued random variables with $N$ labeled units, and sampling its indices $\{1,\cdots,N \}$ with replacement $m_N:= \sum_{i=1}^N w_{i}^{(N)}$ times so that for each $1\leq i \leq N$, $w_{i}^{(N)}$ is the count of number of times the index $i$ of $X_i$ is chosen in this virtual resampling process. This exposition extends the Doob-Donsker classical theory of weak convergence of empirical processes to that of the thus created randomly weighted empirical processes when $N, m_N \rightarrow \infty$ so that $m_N=o(N^2)$.