Smoothed log-concave maximum likelihood estimation with applications

We study the smoothed log-concave maximum likelihood estimator of a probability distribution on R d . This is a fully automatic nonparametric density estimator, obtained as a canonical smoothing of the log-concave maximum likelihood estimator. We demonstrate its attractive features both through an analysis of its theoretical properties and a simulation study. Moreover, we show how the estimator can be used as an intermediate stage of more involved procedures, such as constructing a classifier or estimating a functional of the density. Here again, the use of the estimator can be justified both on theoretical grounds and through its finite sample performance, and we illustrate its use in a breast cancer diagnosis (classification) problem.

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

[2]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[3]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[4]  H. M. Möller,et al.  Invariant Integration Formulas for the n-Simplex by Combinatorial Methods , 1978 .

[5]  D. Freedman,et al.  Some Asymptotic Theory for the Bootstrap , 1981 .

[6]  W. Chan,et al.  Unimodality, convexity, and applications , 1989 .

[7]  Olvi L. Mangasarian,et al.  Nuclear feature extraction for breast tumor diagnosis , 1993, Electronic Imaging.

[8]  M. An.  Log-Concave Probability Distributions: Theory and Statistical Testing , 1996 .

[9]  J. Wellner,et al.  Estimation of a convex function: characterizations and asymptotic theory. , 2001 .

[10]  W. John Braun,et al.  Data Sharpening for Nonparametric Inference Subject to Constraints , 2001 .

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

[12]  J. Wellner,et al.  Estimation of a k-monotone density: limit distribution theory and the Spline connection , 2005, math/0509081.

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

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

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

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

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

[18]  S. Geer,et al.  Multivariate log-concave distributions as a nearly parametric model , 2008, Am. Math. Mon..

[19]  Lutz Duembgen,et al.  On an Auxiliary Function for Log-Density Estimation , 2008, 0807.4719.

[20]  Kellen Petersen August.  Real Analysis , 2009 .

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

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

[23]  G. Walther.  Inference and Modeling with Log-concave Distributions , 2009, 1010.0305.

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

[25]  R. Koenker,et al.  QUASI-CONCAVE DENSITY ESTIMATION , 2010, 1007.4013.

[26]  L. Dümbgen,et al.  Consistency of multivariate log-concave density estimators , 2010 .

[27]  Fadoua Balabdaoui,et al.  Estimation of a k‐monotone density: characterizations, consistency and minimax lower bounds , 2010, Statistica Neerlandica.

[28]  L. Dümbgen,et al.  logcondens: Computations Related to Univariate Log-Concave Density Estimation , 2011 .

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

[30]  Martin L. Hazelton,et al.  Assessing log-concavity of multivariate densities , 2011 .

[31]  Raymond J Carroll,et al.  Testing and Estimating Shape-Constrained Nonparametric Density and Regression in the Presence of Measurement Error , 2011, Journal of the American Statistical Association.

[32]  Jon A. Wellner,et al.  Nonparametric estimation of multivariate scale mixtures of uniform densities , 2010, J. Multivar. Anal..

[33]  M. Yuan,et al.  Independent component analysis via nonparametric maximum likelihood estimation , 2012, 1206.0457.

[34]  R. Samworth.  Optimal weighted nearest neighbour classifiers , 2011, 1101.5783.