Multiplicative models for frequency data, estimation and testing

TL;DR

This paper explores multiplicative probability models with sum-to-one constraints, analyzing their geometric properties, proposing a new estimation algorithm, and evaluating hypothesis testing methods through simulations.

Contribution

It introduces a novel algorithm for maximum likelihood estimation and provides new insights into the geometric structure of these models.

Findings

New algorithm for MLE computation

Geometric analysis of the models as curved exponential families

Asymptotic distributions for hypothesis testing statistics

Abstract

This paper is about models for a vector of probabilities whose elements must have a multiplicative structure and sum to 1 at the same time; in certain applications, as basket analysis, these models may be seen as a constrained version of quasi-independence. After reviewing the basic properties of these models, their geometric features as a curved exponential family are investigated. A new algorithm for computing maximum likelihood estimates is presented and new insights are provided on the underlying geometry. The asymptotic distribution of three statistics for hypothesis testing are derived and a small simulation study is presented to investigate the accuracy of asymptotic approximations.