Raking-ratio empirical process with auxiliary information learning

Mickael Albertus

doi:10.1051/ps/2020017

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. The asymptotic behavior of the raking-ratio empirical process indexed by a class of functions is studied when the auxiliary information is given by estimates. These estimates are supposed to 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, the strong approximation of this process and in particular the weak convergence are established. Under these conditions, the asymptotic behavior of the new process is the same as the classical raking-ratio empirical process. Some possible statistical applications of these results are also given, like the strengthening of the Z-test and the chi-square goodness of fit test.

Highlights

The raking-ratio method is a statistical and computational method aiming to incorporate auxiliary information given by the knowledge of probability of a set of several partitions
Under the conditions that all initial frequencies are strictly positive, if the ratio step are cycling through a finite number of partitions, the method converges to a frequency table satisfying the expected values – see [12]. It is the “raking” step of the algorithm. The goal of these operations is to improve the quality of estimators or the power of statistical tests based on the exploitation of the sample frequency table by lowering the quadratic risk when the sample size is large enough
The main statistical question of this article is whether the statistician can still apply the raking-ratio method by using the estimate of inclusion probabilities rather than the true ones as auxiliary information

Summary

Introduction

The raking-ratio method is a statistical and computational method aiming to incorporate auxiliary information given by the knowledge of probability of a set of several partitions. For an example of a simple statistic using the new weights from the raking-ratio method see Appendix A. The main statistical question of this article is whether the statistician can still apply the raking-ratio method by using the estimate of inclusion probabilities rather than the true ones as auxiliary information.

Results

Conclusion