Auxiliary information : the raking-ratio empirical process

TL;DR

This paper analyzes the asymptotic and nonasymptotic properties of a modified empirical measure adjusted by the raking-ratio method, providing theoretical insights and bounds relevant for finite samples and large iterations.

Contribution

It establishes asymptotic behavior, Gaussian approximation bounds, and explicit covariance formulas for the raking-ratio empirical process, including special cases like contingency tables.

Findings

Asymptotic properties of the raking-ratio empirical process are characterized.

Nonasymptotic Gaussian approximation bounds are derived, depending on sample size and iterations.

Explicit formulas for limiting covariance matrices are provided, especially for contingency tables.

Abstract

We study the empirical measure associated to a sample of size $n$ and modified by $N$ iterations of the raking-ratio method. This empirical measure is adjusted to match the true probability of sets in a finite partition which changes each step. We establish asymptotic properties of the raking-ratio empirical process indexed by functions as $n\rightarrow +\infty$, for $N$ fixed. We study nonasymptotic properties by using a Gaussian approximation which yields uniform Berry-Esseen type bounds depending on $n, N$ and provides estimates of the uniform quadratic risk reduction. A closed-form expression of the limiting covariance matrices is derived as $N\rightarrow +\infty$. In the two-way contingency table case the limiting process has a simple explicit formula.