Limit distribution theory for maximum likelihood estimation of a log concave density. The logconcave maximum likelihood estimator of a density on the real line based on a sample of size n is known to attain the minimax optimal rate of convergence of on. Computations related to univariate logconcave density estimation. Although first uploaded to cran in 2006, a detailed description beyond the package manual. Global rates of convergence in logconcave density estimation arlene k. Jun 01, 2009 we find limiting distributions of the nonparametric maximum likelihood estimator mle of a logconcave density, i. Pdf limit distribution theory for maximum likelihood.
Aug 24, 2018 shapeconstrained density estimation is an important topic in mathematical statistics. Recent progress in log concave density estimation richard j. Maximum likelihood estimation of a multi dimensional log. Logconcavity and density estimation lutz dumbgen, kaspar ru. Despite the space of all such densities being infinitedimensional, t. Section 4 is devoted to applications of the smoothed logconcave maximum likelihood estimator to classi. 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. This test is consistent, easy to implement, and has much improved. Adaptation in logconcave density estimation article pdf available in the annals of statistics 465 september 2016 with 40 reads how we measure reads. Inference and modeling with logconcave distributions. Shapeconstrained density estimation is an important topic in mathematical statistics. Two further straightforward stability properties are as follows. If a multivariate density is logconcave, so is the marginal density over any subset of variables.
The sum of two independent logconcave random variables is logconcave. It is an extension of the constrained newton method. Recent progress in logconcave density estimation richard j. Global rates of convergence in log concave density estimation arlene k. Maximum likelihood estimation of the mixture of log. Logconcave densities probability density fon rdis logconcave if f exp. We study probability density functions that are logconcave. A polynomial time algorithm for maximum likelihood estimation. Lower bound calculations using jongbloeds perturbation of 0 yields. Mode estimation, logconcave density on r let x0 mf0 be the mode of the logconcave density f0, recalling that p0. Now that we can compute the maximum likelihood estimator of a multi.
Limit distribution theory for maximum likelihood estimation. Maximum likelihood estimation of the mixture of logconcave. Its main function is to compute the nonparametric maximum likelihood estimator of a log. Cule, samworth, and stewart showed that the logarithm of the optimal logconcave density is. 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.
A new fast algorithm for computing the nonparametric maximum likelihood estimate of a univariate logconcave density is proposed and studied. Maximum likelihood estimation of a logconcave density. Pdf exact solutions in logconcave maximum likelihood. Symmetric logconcave density estimation and mixture modeling, august 2015, joint statistical meetings, seattle, washington. Logconcave density estimation is considered as an alternative for kernel density estimations which does not depend on tuning parameters. We focus on densities on \\mathbb rd\ that are logconcave, and we study geometric properties of the maximum likelihood estimator mle for weighted samples. 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. Logconcave density estimation with symmetry or modal. In recent years, logconcave density estimation via maximum likelihood estimation has emerged as a fascinating. Under fairly general conditions, the logconcave maximum likelihood estimator lcmle exists and is consistent. Under fairly general conditions, the log concave maximum likelihood estimator lcmle exists and is consistent.
If a density is log concave, so is its cumulative distribution function cdf. We present theoretical properties of the logconcave maximum likelihood estimator of a density based on an independent and identically distributed sample in. Maximum likelihood estimation of a multidimensional log concave. In this paper, a much more flexible mixture model is considered, which assumes each component density to be logconcave. 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. In this paper, a much more flexible mixture model is considered, which assumes each component density to be log concave.
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. For the continuous case, the sample complexity of the. 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. Pdf a fast algorithm for univariate logconcave density. Nonparametric density estimation, shape constraint, log concave density, polya frequency function, strongly unimodal, iterative convex minorant algorithm, active set algorithm. 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. 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. An e cient algorithm for highdimensional logconcave. If a multivariate density is log concave, so is the marginal density over any subset of variables. This follows from the fact that the convolution of two log concave functions is log concave.
Adaptation in logconcave density estimation internet archive. Maximum likelihood estimation of a logconcave density and. 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. In this work, we study the problem of density estimation for the family of logconcave distributions on rd. 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. The univariate case is by now well understood dl01, cdss14a, adls17, ks16, hw16. Exact solutions in logconcave maximum likelihood estimation. The logconcave mle is the distribution with the uniform polyhedral su cient statistic. Our study covers both the case where the true underlying density is logconcave, and where this model is misspecified. Density estimation of logconcave densities has been extensively investigated. A likelihood ratio test for the location of the maximum of a concave function, july 2015, european meeting of statisticians, amsterdam, netherlands. Software is readily available as r cran package fmlogcondens.
Estimation of a multidimensional logconcave density 547 fig. 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. 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. Geometry of logconcave density estimation springerlink. We find limiting distributions of the nonparametric maximum likelihood estimator mle of a logconcave density, i. It is an extension of the constrained newton method for nonparametric mixture estimation. It is in the multivariate case, however, where kernel density estimation is more di. Tentlike structure of the graph of the logarithm of the maximum likelihood estimator for bivariate data that were described in groeneboom et al. Nonparametric density estimation, shape constraint, logconcave density, polya frequency function, strongly unimodal, iterative convex minorant algorithm, active set algorithm. Recent progress in logconcave density estimation arxiv. A faster algorithm for computing the log concave mle. 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. Limit distribution theory for maximum likelihood estimation of a logconcave density. Fast multivariate logconcave density estimation sciencedirect.
Logconcdead logconcave density estimation in arbitrary dimensions. A polynomial time algorithm for maximum likelihood. This facilitates significant speed ups up to 30000x over stateoftheart approach. We explore in what sense exact solutions to this problem are possible. The sum of two independent log concave random variables is log concave.
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. Maximum likelihood estimation of a logconcave density has attracted considerable. This follows from the fact that the convolution of two logconcave functions is logconcave. Global rates of convergence in logconcave density estimation. Nonparametric inference about a densitys mode and shape constraints.
A new fast algorithm for computing the nonparametric maximum likelihood estimate of a univariate log. The log concave mle is the distribution with the uniform polyhedral su cient statistic. Logconcave density estimation with symmetry or modal constraints by charles r. Several algorithms have been proposed to estimate such a density. 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. Maximum likelihood estimation of a logconcave density has attracted. Maximum likelihood estimation of a logconcave density and its distribution function. Smoothed logconcave maximum likelihood estimation 75 density. Maximum likelihood estimation of a multidimensional log. 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.
Computing confidence intervals for logconcave densities. 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. Maximum likelihood estimation of a logconcave density based. An e cient algorithm for highdimensional logconcave maximum. Fast nearoptimal solver for multivariate logconcave density estimation. Expandingthescopeofourinvestigation, we now replace e. Cule, samworth, and stewart showed that the logarithm of the optimal log concave density is piecewise linear and supported on a regular. Onedimensional logconcave density estimation via maximum likelihood is discussed in d. A main drawback is the strong parametric assumption about the component densities. If a density is logconcave, so is its cumulative distribution function cdf.