Some theoretical results regarding the polygonal distribution

ABSTRACT Polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. We demonstrate that the densities of polygonal distributions are dense in the class of continuous and concave densities with bounded second derivatives. Furthermore, we prove that polygonal density functions provide O(g− 2) approximations (where g is the number of triangular distribution components), in the supremum distance, to any density function from the hypothesized class. Parametric consistency and Hellinger consistency results for the maximum likelihood (ML) estimator are obtained. A result regarding model selection via penalized ML estimation is proved.

[1]  S. Walker,et al.  Bayesian mode regression using mixtures of triangular densities , 2017 .

[2]  John Loustau,et al.  Finite Element Method, the Theory , 2016 .

[3]  Geoffrey J. McLachlan,et al.  Maximum likelihood estimation of triangular and polygonal distributions , 2016, Comput. Stat. Data Anal..

[4]  Charles R. Doss,et al.  GLOBAL RATES OF CONVERGENCE OF THE MLES OF LOG-CONCAVE AND s-CONCAVE DENSITIES. , 2013, Annals of statistics.

[5]  R. Nickl,et al.  Mathematical Foundations of Infinite-Dimensional Statistical Models , 2015 .

[6]  Ryan J. Vander Wielen,et al.  The General Segmented Distribution , 2015 .

[7]  Mats G. Larson,et al.  The Finite Element , 2013 .

[8]  Bertrand Michel,et al.  Slope heuristics: overview and implementation , 2011, Statistics and Computing.

[9]  Aurélie Fischer,et al.  On the number of groups in clustering , 2011 .

[10]  Cathy Maugis,et al.  A non asymptotic penalized criterion for Gaussian mixture model selection , 2011 .

[11]  A. Dasgupta.  Asymptotic Theory of Statistics and Probability , 2008 .

[12]  P. Massart,et al.  Statistical performance of support vector machines , 2008, 0804.0551.

[13]  Luzia Gonçalves,et al.  Triangular and Trapezoidal Distributions: Applications in the Genome Analysis , 2008 .

[14]  Dimitris Karlis,et al.  The Polygonal Distribution , 2008 .

[15]  Xiaolong Zhang,et al.  The Power of Doing: A Learning Exercise that Brings the Central Limit Theorem to Life , 2007 .

[16]  Raghu N. Kacker,et al.  Trapezoidal and triangular distributions for Type B evaluation of standard uncertainty , 2007 .

[17]  P. Massart,et al.  Minimal Penalties for Gaussian Model Selection , 2007 .

[18]  P. Massart,et al.  Concentration inequalities and model selection , 2007 .

[19]  M Cortina-Borja,et al.  Beyond beta: Other continuous families of distributions with bounded support and applications , 2006 .

[20]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[21]  Samuel Kotz,et al.  Beyond Beta: Other Continuous Families Of Distributions With Bounded Support And Applications , 2004 .

[22]  Kenneth S. Berenhaut,et al.  Advanced Calculus with Applications in Statistics , 2004 .

[23]  D. Hunter,et al.  A Tutorial on MM Algorithms , 2004 .

[24]  David P. Doane.  Using Simulation to Teach Distributions , 2004 .

[25]  Samuel Kotz,et al.  Generalized trapezoidal distributions , 2003 .

[26]  Alison L Gibbs,et al.  On Choosing and Bounding Probability Metrics , 2002, math/0209021.

[27]  S. Ghosal.  Convergence rates for density estimation with Bernstein polynomials , 2001 .

[28]  P. Massart,et al.  Gaussian model selection , 2001 .

[29]  D. Karlis,et al.  Robust Inference for Finite Poisson Mixtures , 2001 .

[30]  L. Wasserman,et al.  RATES OF CONVERGENCE FOR THE GAUSSIAN MIXTURE SIEVE , 2000 .

[31]  László Györfi,et al.  On piecewise linear density estimators , 1999 .

[32]  Ward Cheney,et al.  A course in approximation theory , 1999 .

[33]  A. V. D. Vaart,et al.  Asymptotic Statistics: Frontmatter , 1998 .

[34]  Huaiyu Zhu.  On Information and Sufficiency , 1997 .

[35]  P. J. Green,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[36]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[37]  R. Redner,et al.  Mixture densities, maximum likelihood, and the EM algorithm , 1984 .

[38]  R. Redner.  Note on the Consistency of the Maximum Likelihood Estimate for Nonidentifiable Distributions , 1981 .

[39]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[40]  U. Grenander.  On the theory of mortality measurement , 1956 .