# Raking-ratio empirical process with auxiliary information learning

## TL;DR

This paper investigates the asymptotic properties of the raking-ratio empirical process when auxiliary information is estimated from a larger sample, with applications to statistical tests.

## Contribution

It extends the raking-ratio empirical process theory to cases with estimated auxiliary info from learning, under specific entropy and sample size conditions.

## Key findings

- Established strong approximation of the process under certain conditions.
- Proved weak convergence matches classical raking-ratio empirical process.
- Applied results to improve statistical tests like Z-test and chi-square goodness of fit.

## Abstract

The raking-ratio method is a statistical and computational method which adjusts the empirical measure to match the true probability of sets of a finite partition. We study the asymptotic behavior of the raking-ratio empirical process indexed by a class of functions when the auxiliary information is given by estimates. We suppose that these estimates result from the learning of the probability of sets of partitions from another sample larger than the sample of the statistician, as in the case of two-stage sampling surveys. Under some metric entropy hypothesis and conditions on the size of the information source sample, we establish the strong approximation of this process and show in this case that the weak convergence is the same as the classical raking-ratio empirical process. We also give possible statistical applications of these results like the strengthening of the $Z$-test and the chi-square goodness of fit test.