The local geometry of finite mixtures
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- by Elisabeth Gassiat and Ramon van Handel PDF
- Trans. Amer. Math. Soc. 366 (2014), 1047-1072 Request permission
Abstract:
We establish that for $q\ge 1$, the class of convex combinations of $q$ translates of a smooth probability density has local doubling dimension proportional to $q$. The key difficulty in the proof is to control the local geometric structure of mixture classes. Our local geometry theorem yields a bound on the (bracketing) metric entropy of a class of normalized densities, from which a local entropy bound is deduced by a general slicing procedure.References
- Charalambos D. Aliprantis and Kim C. Border, Infinite dimensional analysis, 3rd ed., Springer, Berlin, 2006. A hitchhiker’s guide. MR 2378491
- Patrice Assouad, Plongements lipschitziens dans $\textbf {R}^{n}$, Bull. Soc. Math. France 111 (1983), no. 4, 429–448 (French, with English summary). MR 763553, DOI 10.24033/bsmf.1997
- Ron Blei, Fuchang Gao, and Wenbo V. Li, Metric entropy of high dimensional distributions, Proc. Amer. Math. Soc. 135 (2007), no. 12, 4009–4018. MR 2341952, DOI 10.1090/S0002-9939-07-08935-6
- Bernd Carl, Ioanna Kyrezi, and Alain Pajor, Metric entropy of convex hulls in Banach spaces, J. London Math. Soc. (2) 60 (1999), no. 3, 871–896. MR 1753820, DOI 10.1112/S0024610799008005
- Gabriela Ciuperca, Likelihood ratio statistic for exponential mixtures, Ann. Inst. Statist. Math. 54 (2002), no. 3, 585–594. MR 1932403, DOI 10.1023/A:1022415228062
- D. Dacunha-Castelle and E. Gassiat, Testing the order of a model using locally conic parametrization: population mixtures and stationary ARMA processes, Ann. Statist. 27 (1999), no. 4, 1178–1209. MR 1740115, DOI 10.1214/aos/1017938921
- Fuchang Gao, Metric entropy of convex hulls, Israel J. Math. 123 (2001), 359–364. MR 1835305, DOI 10.1007/BF02784136
- Fuchang Gao, Entropy of absolute convex hulls in Hilbert spaces, Bull. London Math. Soc. 36 (2004), no. 4, 460–468. MR 2069008, DOI 10.1112/S0024609304003121
- Fuchang Gao, Wenbo V. Li, and Jon A. Wellner, How many Laplace transforms of probability measures are there?, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4331–4344. MR 2680059, DOI 10.1090/S0002-9939-2010-10448-3
- E. Gassiat and J. Rousseau, On the asymptotic behaviour of the posterior distribution in hidden Markov models, Bernoulli, to appear.
- Elisabeth Gassiat and Ramon van Handel, Consistent order estimation and minimal penalties, IEEE Trans. Inform. Theory 59 (2013), no. 2, 1115–1128. MR 3015722, DOI 10.1109/TIT.2012.2221122
- Christopher R. Genovese and Larry Wasserman, Rates of convergence for the Gaussian mixture sieve, Ann. Statist. 28 (2000), no. 4, 1105–1127. MR 1810921, DOI 10.1214/aos/1015956709
- Subhashis Ghosal and Aad W. van der Vaart, Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities, Ann. Statist. 29 (2001), no. 5, 1233–1263. MR 1873329, DOI 10.1214/aos/1013203453
- Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
- A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in function spaces, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 3–86 (Russian). MR 0112032
- Eugene Lukacs, Characteristic functions, Hafner Publishing Co., New York, 1970. Second edition, revised and enlarged. MR 0346874
- Pascal Massart, Concentration inequalities and model selection, Lecture Notes in Mathematics, vol. 1896, Springer, Berlin, 2007. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003; With a foreword by Jean Picard. MR 2319879
- Cathy Maugis and Bertrand Michel, A non asymptotic penalized criterion for Gaussian mixture model selection, ESAIM Probab. Stat. 15 (2011), 41–68. MR 2870505, DOI 10.1051/ps/2009004
- Sara A. van de Geer, Applications of empirical process theory, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 6, Cambridge University Press, Cambridge, 2000. MR 1739079
- Aad W. van der Vaart and Jon A. Wellner, Weak convergence and empirical processes, Springer Series in Statistics, Springer-Verlag, New York, 1996. With applications to statistics. MR 1385671, DOI 10.1007/978-1-4757-2545-2
- David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. MR 1155402, DOI 10.1017/CBO9780511813658
Additional Information
- Elisabeth Gassiat
- Affiliation: Laboratoire de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France
- Email: elisabeth.gassiat@math.u-psud.fr
- Ramon van Handel
- Affiliation: Operations Research and Financial Engineering Department, Sherrerd Hall, Room 227, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 761136
- Email: rvan@princeton.edu
- Received by editor(s): February 15, 2012
- Received by editor(s) in revised form: August 1, 2012
- Published electronically: August 8, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 1047-1072
- MSC (2010): Primary 41A46; Secondary 52A21, 52C17
- DOI: https://doi.org/10.1090/S0002-9947-2013-06041-2
- MathSciNet review: 3130325