Electronic Journal of Statistics

Partition structure and the $A$-hypergeometric distribution associated with the rational normal curve

Shuhei Mano

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Abstract

A distribution whose normalization constant is an $A$-hypergeometric polynomial is called an $A$-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. In this paper, we will see that an $A$-hypergeometric distribution with a homogeneous matrix of two rows, especially, that associated with the rational normal curve, appears in inferences involving exchangeable partition structures. An exact sampling algorithm is presented for the general (any number of rows) $A$-hypergeometric distributions. Then, the maximum likelihood estimation of the $A$-hypergeometric distribution associated with the rational normal curve, which is an algebraic exponential family, is discussed. The information geometry of the Newton polytope is useful for analyzing the full and the curved exponential family. Algebraic methods are provided for evaluating the $A$-hypergeometric polynomials.

Article information

Source
Electron. J. Statist., Volume 11, Number 2 (2017), 4452-4487.

Dates
Received: August 2016
First available in Project Euclid: 17 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1510887943

Digital Object Identifier
doi:10.1214/17-EJS1361

Mathematical Reviews number (MathSciNet)
MR3724486

Zentralblatt MATH identifier
1382.62005

Subjects
Primary: 62E15: Exact distribution theory
Secondary: 13P25: Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) 60C05: Combinatorial probability

Keywords
A-hypergeometric system algebraic statistics Bayesian statistics exchangeability information geometry rational normal curve Newton polytope

Rights
Creative Commons Attribution 4.0 International License.

Citation

Mano, Shuhei. Partition structure and the $A$-hypergeometric distribution associated with the rational normal curve. Electron. J. Statist. 11 (2017), no. 2, 4452--4487. doi:10.1214/17-EJS1361. https://projecteuclid.org/euclid.ejs/1510887943


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References

  • [1] Gel’fand, I.M., Zelevinsky, A.V., Kapranov, M.M. (1990) Generalized Euler Integrals and $A$-hypergeometric functions., Adv. Math., 84, 255–271.
  • [2] Takayama N., Kuriki, S., Takemura, A. (2015) $A$-hypergeometric distributions and Newton polytope. arXiv:, 1510.02269.
  • [3] Charalambides, C.A. (2005), Combinatorial Methods in Discrete Distributions. New Jersey: Wiley.
  • [4] Hjort, N.L. Holmes, C, Mueller, P., Walker, S.G. (2010) Bayesian Nonparametrics., Camb. Ser. Stat. Probab. Math. Cambridge: Cambridge University Press.
  • [5] Crane, H. (2016) The ubiquitous Ewens sampling formula., Statist. Sci., 31, 1–19.
  • [6] Mano, S., Statistical Inferences with Random Combinatorial Models. JSS Research Series in Statistics, SpringerBriefs in Statistics, Springer, to appear.
  • [7] Aldous D.J. (1985) Exchangeability and related topics. In: Ecole d’Été de Probabilités de Saint Flour, Lecture Notes in Math., Vol. 1117. Berlin: Springer.
  • [8] Arratia, R., Barbour, A.D., Tavaré, S. (2003), Logarithmic Combinatorial Structures: a Probabilistic Approach. EMS Monogr. Math. Zürich: European Mathematical Society.
  • [9] Pitman, J. (2006) Combinatorial Stochastic Processes. In: Ecole d’Été de Probabilités de Saint Flour, Lecture Notes in Math., Vol. 1875. Berlin: Springer.
  • [10] Mano, S. (2017) Extreme sizes in the Gibbs-type exchangeable random partitions., Ann. Inst. Statist. Math., 69, 1–37.
  • [11] Nakayama, H., Nishiyama, K., Noro, M., Ohara, K., Sei, T., Takayama, N., Takemura, A. (2011) Holonomic gradient descent and its application to the Fisher-Bingham integral., Adv. Appl. Math., 47, 639–658.
  • [12] Ohara, K., Takayama, N.: Pfaffian systems of $A$-hypergeometric systems II – holonomic gradient method. arXiv:, 1505.02947.
  • [13] Takayama, N. References for the Holonomic Gradient Method (HGM) and the Holonomic Gradient Descent Method (HGD):, http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/ref-hgm.html
  • [14] Sibuya, M. (1993) A random-clustering process., Ann. Inst. Statist. Math., 45, 459–465.
  • [15] Comtet, L. (1974), Advanced Combinatorics. Dordrecht: Ridel.
  • [16] Hartshorne, R. (1977), Algebraic Geometry. Graduate Texts in Math., Vol. 52. New York: Springer.
  • [17] Cattani, E., D’Andrea, C., Dickenstein, A. (1999) Rational solutions of the A-hypergeometric system associated with a monomial curve., Duke Math. J., 99, 179–207.
  • [18] Saito, M., Sturmfels, B., Takayama, N. (2010), Gröbner deformations of Hypergeometric Differential Equations. Algorithms Comput. Math., Vol. 6. Berlin: Springer.
  • [19] Sturmfels, B. (1996) Gröbner Bases and Convex Polytopes., Univ. Lecture Ser., Vol. 8, Providence: American Mathematical Society.
  • [20] Hara, H., Takemura, A., Yoshida, R. (2010) On connectivity of fibers with positive marginals in multiple logistic regression., J. Multivariate Anal., 101, 99–925.
  • [21] Stanley, R.P. (1999), Ennumerative Combinatorics, Vol. 2. New York: Cambridge University Press.
  • [22] Lehmann, E.L., Romano, J.P. (2005), Testing Statistical Hypothesis, 3rd. edn. New York: Springer.
  • [23] Diaconis, P., Sturmfels, B. (1998) Algebraic algorithms for sampling from conditional distributions., Ann. Statist., 26, 363–397.
  • [24] Aoki, S., Hara, H., Takemura, A. (2012), Markov Bases in Algebraic Statistics. New York: Springer.
  • [25] Stewart, F.M. (1977) Computer algorithm for obtaining a random set of allele frequencies for a locus in an equilibrium population., Genetics, 86, 482–483.
  • [26] Diaconis, P., Eisenbud, B., Sturmfels, B. (1998) Lattice walks and primary decomposition. In: Mathematical Essays in Honor of Gian-Carlo Rota (Cambridge, MA, 1996)., Progr. Math., 161, pp. 173–193. Boston: Birkhäuser.
  • [27] Lee, J., Quintana, F.A., Müller, P. and Trippa, L. (2013) Defining predictive probability functions for species sampling models., Statist. Sci., 28, 209–222.
  • [28] Vershik, A.M. (1996) Statistical mechanics of combinatorial partitions, and their limit configurations., Funct. Anal. Appl., 30, 90–105.
  • [29] Keener, R., Rothman, E., Starr, N. (1978) Distribution of partitions., Ann. Statist., 15, 1466–1481.
  • [30] Lehmann, E.L., Casella, G. (1998), Theory of Point Estimation, 2nd. edn. New York: Springer.
  • [31] Gnedin, A., Pitman, J. (2005) Exchangeable Gibbs partitions and Stirling triangles., Zap. Nauchn. Sem. S. POMI, 325, 83–102.
  • [32] Pitman, J., Yor, M. (1997) The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator., Ann. Probab., 25, 855–900.
  • [33] Ishwaran, H., James, L.F. (2003) Generalized weighted Chinese restaurant processes for species sampling mixture models., Statist. Sinica, 13, 1211–1235.
  • [34] Pitman, J. (1995) Exchangeable and partially exchangeable random partitions., Probab. Theory Related Fields, 102, 145–158.
  • [35] Lijoi, A., Mena, R.H., Prünster, I. (2007) Bayesian nonparametric estimation of the probability of discovering new species., Biometrika, 94, 769–786.
  • [36] Sibuya, M. (2014) Prediction in Ewens-Pitman sampling formula and random samples from number partitions., Ann. Inst. Statist. Math., 66, 833–864.
  • [37] Brown, L.D. (1986), Fundermentals of Statistical Exponential Families with Applications in Statistical Decision Theory. Hayward: Institute of Mathematical Statistics.
  • [38] Barndorff-Nielsen, O.E. (2014), Information and Exponential Families in Statistical Theory. Chichester: Wiley.
  • [39] Drton, M. and Sullivant, S. (2007) Algebraic statistical models., Statist. Sinica., 17, 1273–1297.
  • [40] Amari, S., Nagaoka, H. (2000), Methods of Information Geometry. Transl. Math. Monogr. Providence: American Mathematical Society.
  • [41] Eriksson, E., Fienberg, S.E., Rinaldo, A., Sullivant, S. (2006) Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models., J. Symbolic. Comput., 41, 222–233 (2006).
  • [42] Shlyk, V.A. (2005) Polytopes of partitions of numbers., European J. Combin., 26, 1139–1153 (2005).
  • [43] Levin, B., Reeds, J. (1977) Compound multinomial likelihood functions are unimodal: proof of a conjecture of I.J. Good., Ann. Statist., 5, 79–87.
  • [44] Baayen, R.H. (2001), Word frequency distribution. Dordrecht: Kluwer Academic Publishers.
  • [45] Hibi, T. (eds) (2013), Gröbner Bases: Statistics and Computing. Tokyo: Springer.
  • [46] Goto, Y., Matsumoto, K. Pfaffian equations and contiguity relations of the hypergeometric function of type $(k+1,k+n+2)$ and their applications. arXiv: 1602.01637, to appear in Funkcialaj, Ekvacioj.
  • [47] Hashiguchi, H., Numata, Y., Takayama, N., Takemura, A. (2013) The holonomic gradient method for the distribution function of the largest root of a Wishart matrix., J. Multivariate Anal., 117, 296–312 (2013).