The Annals of Statistics

Estimation in exponential families on permutations

Sumit Mukherjee

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

Permanent link to this document
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

Keywords
Permutation normalizing constant Mallows model pseudo-likelihood

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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References

  • [1] Aas, K., Czado, C., Frigessi, A. and Bakken, H. (2009). Pair-copula constructions of multiple dependence. Insurance Math. Econom. 44 182–198.
  • [2] Andersen, H. C. and Diaconis, P. (2007). Hit and run as a unifying device. J. Soc. Fr. Stat. & Rev. Stat. Appl. 148 5–28.
  • [3] Awasthi, P., Blum, A., Sheffet, O. and Vijayaraghavan, A. (2014). Learning mixtures of ranking models. In Advances in Neural Information Processing Systems 27 2609–2617. Curran Associates, Inc. Montreal.
  • [4] Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. J. Roy. Statist. Soc. Ser. B 36 192–236.
  • [5] Besag, J. (1975). Statistical analysis of non-lattice data. J. R. Stat. Soc., Ser. D Stat. 24 179–195.
  • [6] Bhattacharya, B. and Mukherjee, S. (2015). Degree Sequence of Random Permutation Graphs. Preprint. Available at arXiv:1503.03582.
  • [7] Brigo, D., Pallavicini, A. and Torresetti, R. (2010). Credit Models and the Crisis: A Journey Into CDOs, Copulas, Correlations and Dynamic Models. Wiley, New York.
  • [8] Chen, H., Branavan, S. R. K., Barzilay, R. and Karger, D. R. (2009). Content modeling using latent permutations. J. Artificial Intelligence Res. 36 129–163.
  • [9] Critchlow, D. E. (1985). Metric Methods for Analyzing Partially Ranked Data. Lecture Notes in Statistics 34. Springer, Berlin.
  • [10] Critchlow, D. E., Fligner, M. A. and Verducci, J. S. (1991). Probability models on rankings. J. Math. Psych. 35 294–318.
  • [11] Csiszár, I. (1975). $I$-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 146–158.
  • [12] Deming, W. E. and Stephan, F. F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Stat. 11 427–444.
  • [13] Diaconis, P. (1988). Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series 11. IMS, Hayward, CA.
  • [14] Diaconis, P., Graham, R. and Holmes, S. P. (2001). Statistical problems involving permutations with restricted positions. In State of the Art in Probability and Statistics (Leiden, 1999). Institute of Mathematical Statistics Lecture Notes—Monograph Series 36 195–222. IMS, Beachwood, OH.
  • [15] Diaconis, P. and Ram, A. (2000). Analysis of systematic scan Metropolis algorithms using Iwahori–Hecke algebra techniques. Michigan Math. J. 48 157–190.
  • [16] Feigin, P. and Cohen, A. (1978). On a model of concordance between judges. J. R. Stat. Soc. Ser. B. Stat. Methodol. 40 203–213.
  • [17] Fienberg, S. (1971). Randomization and social affairs, the 1970 draft lottery. Science 171 255–261.
  • [18] Fligner, M. A. and Verducci, J. S. (1986). Distance based ranking models. J. Roy. Statist. Soc. Ser. B 48 359–369.
  • [19] Fligner, M. A. and Verducci, J. S. (1988). Multistage ranking models. J. Amer. Statist. Assoc. 83 892–901.
  • [20] Genest, C. and MacKay, J. (1986). The joy of copulas: Bivariate distributions with uniform marginals. Amer. Statist. 40 280–283.
  • [21] Hoeffding, W. (1951). A combinatorial central limit theorem. Ann. Math. Stat. 22 558–566.
  • [22] Hoppen, C., Kohayakawa, Y., Moreira, C. G., Ráth, B. and Menezes Sampaio, R. (2013). Limits of permutation sequences. J. Combin. Theory Ser. B 103 93–113.
  • [23] Huang, J., Guestrin, C. and Guibas, L. (2009). Fourier theoretic probabilistic inference over permutations. J. Mach. Learn. Res. 10 997–1070.
  • [24] Irurozki, E., Calvo, B. and Lozano, A. (2014). Sampling and learning the Mallows and Generalized Mallows models under the Cayley distance. Technical report. Available at https://addi.ehu.es/handle/10810/11239.
  • [25] Irurozki, E., Calvo, B. and Lozano, A. (2014). Sampling and learning the Mallows model under the Ulam distance. Technical report. Available at https://addi.ehu.es/handle/10810/11241.
  • [26] Irurozki, E., Calvo, B. and Lozano, A. (2014). Sampling and learning the Mallows and Weighted Mallows models under the Hamming distance. Technical report. Available at https://addi.ehu.es/handle/10810/11240.
  • [27] Jaworski, P., Durante, F., Härdle, W. and Rychlik, T. (2010). Copula theory and its applications. In Proceedings of the Workshop Held at the University of Warsaw, Warsaw, September 2526, 2009. Lecture Notes in Statistics—Proceedings 198. Springer, Heidelberg.
  • [28] Kondor, R., Howard, A. and Jebara, T. (2007). Multi-object tracking with representations of the symmetric group. In AISTATS 2 211–218.
  • [29] Kullback, S. (1968). Probability densities with given marginals. Ann. Math. Stat. 39 1236–1243.
  • [30] Lebanon, G. and Lafferty, J. (2002). Cranking: Combining rankings using conditional probability models on permutations. In Proceedings of the 19th International Conference on Machine Learning 363–370. Morgan Kaufmann, San Francisco, CA.
  • [31] Lebanon, G. and Mao, Y. (2008). Non-parametric modeling of partially ranked data. J. Mach. Learn. Res. 9 2401–2429.
  • [32] Lovász, L. (2012). Large Networks and Graph Limits. American Mathematical Society Colloquium Publications 60. Amer. Math. Soc., Providence, RI.
  • [33] Mai, J.-F. and Scherer, M. (2012). Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications. Series in Quantitative Finance 4. Imperial College Press, London.
  • [34] Mallows, C. L. (1957). Non-null ranking models. I. Biometrika 44 114–130.
  • [35] Marden, J. I. (1995). Analyzing and Modeling Rank Data. Monographs on Statistics and Applied Probability 64. Chapman & Hall, London.
  • [36] McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton Univ. Press, Princeton, NJ.
  • [37] McNeil, A. J. and Nešlehová, J. (2009). Multivariate Archimedean copulas, $d$-monotone functions and $l_{1}$-norm symmetric distributions. Ann. Statist. 37 3059–3097.
  • [38] Meila, M. and Bao, L. (2008). Estimation and clustering with infinite rankings. In Proceedings of the 24th Conference in Uncertainty in Artificial Intelligence 393–402. Helsinki.
  • [39] Meilă, M. and Bao, L. (2010). An exponential model for infinite rankings. J. Mach. Learn. Res. 11 3481–3518.
  • [40] Meila, M., Phadnis, K., Patterson, A. and Blimes, J. (2007). Consensus ranking under the exponential model, Technical Report 515, Dept. Statistics, Univ. Washington, Seattle, WA.
  • [41] Meucci, A. (2011). A new breed of copulas for risk and portfolio management. Risk 24 122–126.
  • [42] Mukherjee, S. (2015). Supplement to “Estimation in exponential families on permutations.” DOI:10.1214/15-AOS1389SUPP.
  • [43] Nelsen, R. B. (1999). An Introduction to Copulas. Lecture Notes in Statistics 139. Springer, New York.
  • [44] Rüschendorf, L. (1995). Convergence of the iterative proportional fitting procedure. Ann. Statist. 23 1160–1174.
  • [45] Ruschendorf, L., Schweizer, B. and Taylor, M. (1997). Distributions with Fixed Marginals & Related Topics. Lecture Notes—Monograph Series 28. IMS, Hayward, CA.
  • [46] Schweizer, B. and Wolff, E. F. (1981). On nonparametric measures of dependence for random variables. Ann. Statist. 9 879–885.
  • [47] Sinkhorn, R. (1964). A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Stat. 35 876–879.
  • [48] Sklar, M. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 229–231.
  • [49] Starr, S. (2009). Thermodynamic limit for the Mallows model on $S_{n}$. J. Math. Phys. 50 095208, 15.
  • [50] Trashorras, J. (2008). Large deviations for symmetrised empirical measures. J. Theoret. Probab. 21 397–412.
  • [51] Whitt, W. (1976). Bivariate distributions with given marginals. Ann. Statist. 4 1280–1289.

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].