The Annals of Statistics

Estimation in exponential families on permutations

Sumit Mukherjee

Abstract

Asymptotics of the normalizing constant are computed for a class of one parameter exponential families on permutations which include Mallows models 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 the flexibility of the tools of this paper.

Article information

Source
Ann. Statist., Volume 44, Number 2 (2016), 853-875.

Dates
Revised: September 2015
First available in Project Euclid: 17 March 2016

https://projecteuclid.org/euclid.aos/1458245737

Digital Object Identifier
doi:10.1214/15-AOS1389

Mathematical Reviews number (MathSciNet)
MR3476619

Zentralblatt MATH identifier
1341.62083

Subjects
Primary: 62F12: Asymptotic properties of estimators 60F10: Large deviations
Secondary: 05A05: Permutations, words, matrices

Citation

Mukherjee, Sumit. Estimation in exponential families on permutations. Ann. Statist. 44 (2016), no. 2, 853--875. doi:10.1214/15-AOS1389. https://projecteuclid.org/euclid.aos/1458245737

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Supplemental materials

• Supplement to “Estimation in exponential families on permutations”. The supplementary material contain the proofs of all theorems, corollaries, propositions and supporting lemmas. It also states Proposition 2.2, which deals with the joint limiting distribution of $\{\pi(1),\ldots,\pi(n)\}$ for $\pi$ from either the model $\mathbb{Q}_{n,f,\theta}$ of (1.1) or from the Mallows model with Kendall’s tau of Proposition 1.12. A short proof of this proposition is included using the more recent results of [6].