The Annals of Statistics

Estimation in exponential families on permutations

Sumit Mukherjee

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Permutation normalizing constant Mallows model pseudo-likelihood


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

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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].