Maximum likelihood estimation of the mixture of log-concave densities

Finite mixture models are useful tools and can be estimated via the EM algorithm. A main drawback is the strong parametric assumption about the component densities. In this paper, a much more flexible mixture model is considered, which assumes each component density to be log-concave. Under fairly general conditions, the log-concave maximum likelihood estimator (LCMLE) exists and is consistent. Numeric examples are also made to demonstrate that the LCMLE improves the clustering results while comparing with the traditional MLE for parametric mixture models.

[1]  N. Campbell,et al.  A multivariate study of variation in two species of rock crab of the genus Leptograpsus , 1974 .

[2]  D. Hunter,et al.  Inference for mixtures of symmetric distributions , 2007, 0708.0499.

[3]  B. Lindsay,et al.  Bayesian Mixture Labeling by Highest Posterior Density , 2009 .

[4]  K. Rufibach.  Computing maximum likelihood estimators of a log-concave density function , 2007 .

[5]  R. Hathaway.  A Constrained Formulation of Maximum-Likelihood Estimation for Normal Mixture Distributions , 1985 .

[6]  Guenther Walther,et al.  Clustering with mixtures of log-concave distributions , 2007, Comput. Stat. Data Anal..

[7]  L. Hubert,et al.  Comparing partitions , 1985 .

[8]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[9]  Jayanta Kumar Pal,et al.  Estimating a Polya Frequency Function , 2006 .

[10]  David R. Hunter,et al.  mixtools: An R Package for Analyzing Mixture Models , 2009 .

[11]  Fadoua Balabdaoui,et al.  Inference for a mixture of symmetric distributions under log-concavity , 2014, 1411.4708.

[12]  Laurent Bordes,et al.  Semiparametric Estimation of a Two-component Mixture Model where One Component is known , 2006 .

[13]  Yong Wang,et al.  Estimation of finite mixtures with symmetric components , 2011, Statistics and Computing.

[14]  D. Hunter,et al.  mixtools: An R Package for Analyzing Mixture Models , 2009 .

[15]  L. Duembgen,et al.  APPROXIMATION BY LOG-CONCAVE DISTRIBUTIONS, WITH APPLICATIONS TO REGRESSION , 2010, 1002.3448.

[16]  Weixin Yao,et al.  A profile likelihood method for normal mixture with unequal variance , 2010 .

[17]  R. Samworth,et al.  Smoothed log-concave maximum likelihood estimation with applications , 2011, 1102.1191.

[18]  W. Yao,et al.  Flexible estimation of a semiparametric two-component mixture model with one parametric component , 2015 .

[19]  J. Wellner,et al.  Limit Distribution Theory for Maximum Likelihood Estimation of a Log-Concave Density. , 2007, Annals of statistics.

[20]  L. Bordes,et al.  SEMIPARAMETRIC ESTIMATION OF A TWO-COMPONENT MIXTURE MODEL , 2006, math/0607812.

[21]  Hajo Holzmann,et al.  Semiparametric location mixtures with distinct components , 2013 .

[22]  M. Stephens.  Dealing with label switching in mixture models , 2000 .

[23]  G. McLachlan,et al.  The EM algorithm and extensions , 1996 .

[24]  Weixin Yao,et al.  Label switching and its solutions for frequentist mixture models , 2015 .

[25]  M. Cule,et al.  Maximum likelihood estimation of a multi‐dimensional log‐concave density , 2008, 0804.3989.

[26]  L. Duembgen,et al.  Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency , 2007, 0709.0334.

[27]  Jiahua Chen,et al.  INFERENCE FOR NORMAL MIXTURES IN MEAN AND VARIANCE , 2008 .

[28]  Fadoua Balabdaoui,et al.  Limit distribution theory for maximum likelihood estimation of a log-concave density , 2009 .

[29]  Cristina Butucea,et al.  Semiparametric Mixtures of Symmetric Distributions , 2011, 1111.2247.

[30]  Arlene K. H. Kim,et al.  Global rates of convergence in log-concave density estimation , 2014, 1404.2298.

[31]  Robert B. Gramacy,et al.  Maximum likelihood estimation of a multivariate log-concave density , 2010 .

[32]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[33]  W. Yao,et al.  Minimum profile Hellinger distance estimation for a semiparametric mixture model , 2014 .

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

[35]  M. Cule,et al.  Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density , 2009, 0908.4400.

[36]  Laurent Bordes,et al.  Semiparametric two-component mixture model with a known component: An asymptotically normal estimator , 2010 .

[37]  Paul D. McNicholas,et al.  Parsimonious Gaussian mixture models , 2008, Stat. Comput..

[38]  Jayanta Kumar Pal,et al.  Estimating a Polya frequency function$_2$ , 2007, 0708.1064.

[39]  P. Deb.  Finite Mixture Models , 2008 .

[40]  G. Walther.  Detecting the Presence of Mixing with Multiscale Maximum Likelihood , 2002 .