Geometry of logconcave density estimation springerlink. We find limiting distributions of the nonparametric maximum likelihood estimator mle of a logconcave density, i. In this paper, a much more flexible mixture model is considered, which assumes each component density to be logconcave. Symmetric logconcave density estimation and mixture modeling, august 2015, joint statistical meetings, seattle, washington. A faster algorithm for computing the log concave mle. The logconcave mle is the distribution with the uniform polyhedral su cient statistic. In recent years, log concave density estimation via maximum likelihood estimation has emerged as a fascinating alternative to traditional nonparametric smoothing techniques, such as kernel density estimation, which require the choice of one or more bandwidths. Now that we can compute the maximum likelihood estimator of a multi. Software is readily available as r cran package fmlogcondens.
Computing confidence intervals for logconcave densities. Nonparametric inference about a densitys mode and shape constraints. Cule, samworth, and stewart showed that the logarithm of the optimal logconcave density is. Pdf a fast algorithm for univariate logconcave density.
Maximum likelihood estimation of a logconcave density based. If f0 2p0 satis es f0x0 0, f0 00x0 estimation of a multivariate log concave density madeleine cule statistical laboratory, dpmms and emmanuel college university of cambridge this dissertation is submitted for the degree of doctor of philosophy 14 september 2009. Global rates of convergence in log concave density estimation arlene k. Onedimensional logconcave density estimation via maximum likelihood is discussed in d. We present a bayesian nonparametric approach to this problem based on an exponentiated dirichlet process mixture prior and show that the posterior distribution converges to the logconcave truth at the near minimax rate in hellinger. Density estimation of logconcave distributions has been extensively investigated in the statistics literature dr09, gw09, wal09, dw16, bjrp, cs, ks14, bd14 with a focus on analyzing the maximum likelihood estimator mle. Maximum likelihood estimation of a logconcave density has attracted considerable. This facilitates significant speed ups up to 30000x over stateoftheart approach. Samworth statistical laboratory, university of cambridge september 29, 2015 abstract the estimation of a log concave density on rd represents a central problem in the area of nonparametric inference under shape constraints. Maximum likelihood estimation of a logconcave density has attracted. It is in the multivariate case, however, where kernel density estimation is more di. The sum of two independent log concave random variables is log concave.
Pdf limit distribution theory for maximum likelihood. Recent progress in logconcave density estimation arxiv. Logconcavity and density estimation lutz dumbgen, kaspar ru. Although first uploaded to cran in 2006, a detailed description beyond the package manual. Under fairly general conditions, the log concave maximum likelihood estimator lcmle exists and is consistent. In recent years, logconcave density estimation via maximum likelihood estimation has emerged as a fascinating alternative to traditional nonparametric smoothing techniques, such as kernel density estimation, which require the choice of one or more bandwidths. Maximum likelihood estimation of a logconcave density based on censored data dominic schuhmacher institute of mathematical statistics and actuarial science university of bern joint work with lutz dumbgen and kaspar ru. Logconcdead logconcave density estimation in arbitrary dimensions. Nonparametric density estimation, shape constraint, log concave density, polya frequency function, strongly unimodal, iterative convex minorant algorithm, active set algorithm.
We study probability density functions that are logconcave. An e cient algorithm for highdimensional logconcave. Limit distribution theory for maximum likelihood estimation of a logconcave density. In this work, we study the problem of density estimation for the family of logconcave distributions on rd. A likelihood ratio test for the location of the maximum of a concave function, july 2015, european meeting of statisticians, amsterdam, netherlands. A polynomial time algorithm for maximum likelihood estimation. If a density is log concave, so is its cumulative distribution function cdf.
A similar rate is seen for the one dimensional nonparametric maximum likelihood estimator npmle of a logconcave density, which was recently proved to be n25, as long as the density is strictly log. Logconcave density estimation with symmetry or modal. Maximum likelihood estimation of a multidimensional log concave. Its main function is to compute the nonparametric maximum likelihood estimator of a log. Shapeconstrained density estimation is an important topic in mathematical statistics.
Exact solutions in logconcave maximum likelihood estimation. If a multivariate density is logconcave, so is the marginal density over any subset of variables. Recent progress in logconcave density estimation richard j. An e cient algorithm for highdimensional logconcave maximum. Estimation of a multidimensional logconcave density 547 fig. A polynomial time algorithm for maximum likelihood. Limit distribution theory for maximum likelihood estimation. We present a bayesian nonparametric approach to this problem based on an exponentiated dirichlet process mixture prior and show that the posterior distribution converges to the log concave truth at the near minimax rate in hellinger distance. Sep 03, 2016 adaptation in logconcave density estimation article pdf available in the annals of statistics 465 september 2016 with 40 reads how we measure reads. Likelihood estimation of a multivariate logconcave density madeleine cule, robert gramacy and richard samworth university of cambridge abstract in this document we introduce the r package logconcdead logconcave density estimation in arbitrary dimensions.
Now that we can compute the maximum likelihood estimator of a multidimensional logconcave density, we can carry this method through to its natural conclusion, i. This test is consistent, easy to implement, and has much improved. Computations related to univariate logconcave density estimation. Maximum likelihood estimation of a multi dimensional log. Expandingthescopeofourinvestigation, we now replace e. Adaptation in logconcave density estimation internet archive. The sum of two independent logconcave random variables is logconcave. A new fast algorithm for computing the nonparametric maximum likelihood estimate of a univariate log.
Inference and modeling with logconcave distributions. It is an extension of the constrained newton method for nonparametric mixture estimation. If a density is logconcave, so is its cumulative distribution function cdf. Pdf exact solutions in logconcave maximum likelihood. This follows from the fact that the convolution of two logconcave functions is logconcave. Samworth statistical laboratory, university of cambridge september 29, 2015 abstract the estimation of a logconcave density on rd represents a central problem in the area of nonparametric inference under shape constraints. Limit distribution theory for maximum likelihood estimation of a log concave density. For the continuous case, the sample complexity of the. Several algorithms have been proposed to estimate such a density. Nonparametric density estimation, shape constraint, logconcave density, polya frequency function, strongly unimodal, iterative convex minorant algorithm, active set algorithm. Cule, samworth, and stewart showed that the logarithm of the optimal log concave density is piecewise linear and supported on a regular. The log concave mle is the distribution with the uniform polyhedral su cient statistic. A new fast algorithm for computing the nonparametric maximum likelihood estimate of a univariate logconcave density is proposed and studied. It is an extension of the constrained newton method.
Smoothed logconcave maximum likelihood estimation 75 density. In this paper, a much more flexible mixture model is considered, which assumes each component density to be log concave. Two further straightforward stability properties are as follows. Despite the space of all such densities being infinitedimensional, the maximum likelihood estimate is the exponential of a piecewise linear function determined by finitely many quantities, namely the function values, or heights, at the data points.
Computations related to univariate logconcave density estimation lutz d umbgen university of bern kaspar ru bach university of zurich abstract maximum likelihood estimation of a logconcave density has attracted considerable attention over the last few years. We focus on densities on \\mathbb rd\ that are logconcave, and we study geometric properties of the maximum likelihood estimator mle for weighted samples. Global rates of convergence in logconcave density estimation arlene k. Despite the space of all such densities being infinitedimensional, t. Logconcave density estimation with symmetry or modal constraints by charles r. Section 4 is devoted to applications of the smoothed logconcave maximum likelihood estimator to classi. Logconcave densities probability density fon rdis logconcave if f exp. Tentlike structure of the graph of the logarithm of the maximum likelihood estimator for bivariate data that were described in groeneboom et al. Global rates of convergence in logconcave density estimation.
Under fairly general conditions, the logconcave maximum likelihood estimator lcmle exists and is consistent. Aug 24, 2018 shapeconstrained density estimation is an important topic in mathematical statistics. Maximum likelihood estimation of a logconcave density and its distribution function. It is in the multivariate case, however, where kernel density estimation is more di cult and parametric models less obvious, where a logconcave model may be most useful. Maximum likelihood estimation of a logconcave density and. Jun 01, 2009 we find limiting distributions of the nonparametric maximum likelihood estimator mle of a logconcave density, i. In recent years, logconcave density estimation via maximum likelihood estimation has emerged as a fascinating. Maximum likelihood estimation of the mixture of logconcave.
Mode estimation, logconcave density on r let x0 mf0 be the mode of the logconcave density f0, recalling that p0. Adaptation in logconcave density estimation article pdf available in the annals of statistics 465 september 2016 with 40 reads how we measure reads. Maximum likelihood estimation of the mixture of log. Background and motivation the estimation of a probability density function based on observed data is a classical and paradigmatic problem in statisticspearson1895 with a rich history see, e. We present theoretical properties of the logconcave maximum likelihood estimator of a density based on an independent and identically distributed sample in. Lower bound calculations using jongbloeds perturbation of 0 yields. The univariate case is by now well understood dl01, cdss14a, adls17, ks16, hw16. Maximum likelihood estimation of a multidimensional log. We explore in what sense exact solutions to this problem are possible.
This follows from the fact that the convolution of two log concave functions is log concave. If a multivariate density is log concave, so is the marginal density over any subset of variables. Fast multivariate logconcave density estimation sciencedirect. Logconcave density estimation is considered as an alternative for kernel density estimations which does not depend on tuning parameters. Maximum likelihood estimation of a logconcave density. Our study covers both the case where the true underlying density is logconcave, and where this model is misspecified. Fast nearoptimal solver for multivariate logconcave density estimation.
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