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Active Set and EM Algorithms for Log-Concave Densities Based on Complete and Censored Data
TLDR
An active set algorithm for the maximum likelihood estimation of a log-concave density based on complete data and an EM algorithm to treat arbitrarily censored or binned data are developed.Abstract:
We develop an active set algorithm for the maximum likelihood estimation of a log-concave density based on complete data. Building on this fast algorithm, we indidate an EM algorithm to treat arbitrarily censored or binned data.read more
Citations
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Maximum likelihood estimation of a multi‐dimensional log‐concave density
TL;DR: In this article, the kernel estimator is used in conjunction with the expectation-maximization algorithm to fit finite mixtures of log-concave densities, which is shown to have smaller mean integrated squared error compared with kernel-based methods.
Journal ArticleDOI
Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency
Lutz Duembgen,Kaspar Rufibach +1 more
TL;DR: In this article, the authors study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function, and show that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least
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Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density
TL;DR: In this paper, the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in R d was studied and it was shown that the estimator converges to the log concave density that is closest in the Kullback-Leibler sense to the true density.
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Global rates of convergence in log-concave density estimation
TL;DR: In this paper, the authors study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach, showing that when the loss function is squared Hellinger loss with supremum risk smaller than order n −4/5 and order n−2/(d+1) when $d = 2,3, then the log-Concave maximum likelihood estimator achieves the minimax optimal rate.
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Inference and Modeling with Log-concave Distributions
TL;DR: In this article, a review of the literature concerning the theory and applications of log-concave distributions is presented, and the MLE can be computed with readily available algorithms.
References
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Journal ArticleDOI
The Empirical Distribution Function with Arbitrarily Grouped, Censored, and Truncated Data
TL;DR: In this paper, a simple algorithm is constructed and shown to converge monotonically to yield a maximum likelihood estimate of a distribution function when the data are incomplete due to grouping, censoring and/or truncation.
Book ChapterDOI
Log-concave probability and its applications
TL;DR: In this article, a series of theorems relating log-concavity and/or logconvexity of probability density functions, distribution functions, reliability functions, and their integrals are presented.