Maximum likelihood estimation of a multi‐dimensional log‐concave density

@article{Cule2008MaximumLE,
  title={Maximum likelihood estimation of a multi‐dimensional log‐concave density},
  author={Madeleine L. Cule and Richard J. Samworth and Michael I. Stewart},
  journal={Journal of the Royal Statistical Society: Series B (Statistical Methodology)},
  year={2008},
  volume={72}
}
  • M. Cule, R. Samworth, M. Stewart
  • Published 24 April 2008
  • Mathematics
  • Journal of the Royal Statistical Society: Series B (Statistical Methodology)
Summary.  Let X1,…,Xn be independent and identically distributed random vectors with a (Lebesgue) density f. We first prove that, with probability 1, there is a unique log‐concave maximum likelihood estimator of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non‐constructive, we can reformulate the issue of computing in terms of a non‐differentiable… 
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References

SHOWING 1-10 OF 200 REFERENCES
Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density
We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in R d . Our study covers both the case where
Maximum likelihood estimation of a multivariate log-concave density
  • M. Cule
  • Computer Science, Mathematics
  • 2010
TLDR
This paper proposes an alternative approach using maximum likelihood under a qualitative assumption on the shape of the density, specifically log-concavity, and introduces a method for assessing the suitability of this shape constraint and applies it to several simulated datasets and one real dataset.
Limit Distribution Theory for Maximum Likelihood Estimation of a Log-Concave Density.
TLDR
It is shown that the limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density and its derivative are, under comparable smoothness assumptions, the same (up to sign) as in the convex density estimation problem.
Computing maximum likelihood estimators of a log-concave density function
TLDR
It is shown that the maximum likelihood estimator for the log-density is always a piecewise linear function with at most as many knots as observations, but typically much less, and can be exploited to design a linearly constrained optimization problem whose iteratively calculated solution yields the estimator.
Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency
We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two
Nonparametric likelihood based estimation for a multivariate Lipschitz density
NONPARAMETRIC ESTIMATION OF MULTIVARIATE CONVEX-TRANSFORMED DENSITIES.
We study estimation of multivariate densities p of the form p(x) = h(g(x)) for x ∈ ℝ(d) and for a fixed monotone function h and an unknown convex function g. The canonical example is h(y) = e(-y) for
Multivariate density estimation: A comparative study
TLDR
Simulations confirm the earlier conjecture that this last estimator proivdes a way of effectively estimating arbitrary and highly structured continuous densities on Rd, at least for small d, by using this estimator itself or by using it as a pilot estimator for a newly proposed plug-in estimator.
Multivariate log-concave distributions as a nearly parametric model
Abstract In this paper we show that the family Pd(lc) of probability distributions on ℝd with log-concave densities satisfies a strong continuity condition. In particular, it turns out that weak
LogConcDEAD: An R Package for Maximum Likelihood Estimation of a Multivariate Log-Concave Density
TLDR
The R package LogConcDEAD (Log-concave density estimation in arbitrary dimensions) is introduced, its main function is to compute the nonparametric maximum likelihood estimator of a log-conCave density.
...
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5
...