Estimation in exponential families on permutations

TL;DR

This paper analyzes the asymptotic behavior of normalizing constants and estimators in exponential family models on permutations, including Mallows models with various distance metrics, providing theoretical guarantees and computational methods.

Contribution

It introduces a unified approach to analyze exponential families on permutations, including new results on consistency and convergence of estimators and algorithms.

Findings

MLE and approximations are consistent

Pseudo-likelihood estimator is $\

An iterative algorithm converges to the normalizing constant

Abstract

Asymptotics of the normalizing constant is computed for a class of one parameter exponential families on permutations which includes Mallows model with Spearmans's Footrule and Spearman's Rank Correlation Statistic. The MLE, and a computable approximation of the MLE are shown to be consistent. The pseudo-likelihood estimator of Besag is shown to be $\sqrt{n}$-consistent. An iterative algorithm (IPFP) is proved to converge to the limiting normalizing constant. The Mallows model with Kendall's Tau is also analyzed to demonstrate flexibility of the tools of this paper.