Approximation by log-concave distributions, with applications to regression

Dümbgen, Lutz; Samworth, Richard J.; Schuhmacher, Dominic

doi:10.1214/10-aos853

Cited by 93 publications

(155 citation statements)

References 23 publications

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“…For log-concave densities on ℝ it has been explored in more detail by Dümbgen and Rufibach [2009], Balabdaoui, Rufibach and Wellner [2009], and recent results for estimation of log-concave densities on ℝ d have been obtained by Cule and Samworth [2010], Cule, Samworth and Stewart [2010], Dümbgen, Samworth and Schuhmacher [2011]. Cule, Samworth and Stewart [2010] formulate the problem of computing the maximum likelihood estimator of a multidimensional log-concave density as a non-differentiable convex optimization problem and propose an algorithm that combines techniques of computational geometry with Shor's r-algorithm to produce a sequence that converges to the estimator.…”

Section: Some Open Problems and Further Connections With Log-concamentioning

confidence: 99%

Log-concavity and strong log-concavity: A review

Saumard¹,

Wellner²

2014

Statist. Surv.

205

154

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We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on ℝ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

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Section: Some Open Problems and Further Connections With Log-concamentioning

confidence: 99%

Log-concavity and strong log-concavity: A review

Saumard¹,

Wellner²

2014

Statist. Surv.

205

154

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show abstract

“…Specifically, the epi-splines computed from {(P n p,m )} ∞ n=1 tend to a point in the Kullback-Leibler projection, relative to the soft information constraint set, of the true density on the class of epi-splines under consideration. We refer to [24,14,44] for related results on model misspecfication. The second part shows that if the true density is not excluded by the soft information, then {(P n p,m )} ∞ n=1 eventually yields the true density, or possibly a closely related one that deviates at most on m.…”

Section: Theorem (Consistency)mentioning

confidence: 99%

Fusion of Hard and Soft Information in Nonparametric Density Estimation

Røyset¹,

Wets²

2015

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Abstract.This article discusses univariate density estimation in situations when the sample (hard information) is supplemented by "soft" information about the random phenomenon. These situations arise broadly in operations research and management science where practical and computational reasons severely limit the sample size, but problem structure and past experiences could be brought in. In particular, density estimation is needed for generation of input densities to simulation and stochastic optimization models, in analysis of simulation output, and when instantiating probability models. We adopt a constrained maximum likelihood estimator that incorporates any, possibly random, soft information through an arbitrary collection of constraints. We illustrate the breadth of possibilities by discussing soft information about shape, support, continuity, smoothness, slope, location of modes, symmetry, density values, neighborhood of known density, moments, and distribution functions. The maximization takes place over spaces of extended real-valued semicontinuous functions and therefore allows us to consider essentially any conceivable density as well as convenient exponential transformations. The infinite dimensionality of the optimization problem is overcome by approximating splines tailored to these spaces. To facilitate the treatment of small samples, the construction of these splines is decoupled from the sample. We discuss existence and uniqueness of the estimator, examine consistency under increasing hard and soft information, and give rates of convergence. Numerical examples illustrate the value of soft information, the ability to generate a family of diverse densities, and the effect of misspecification of soft information.

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“…Log-concave densities have many favorable properties as described by Balabdaoui et al (2009). To estimate the log-density ϕ ( x ), Dümbgen et al (2011) proposed an estimator by maximizing a log-likelihood-type functional:

L (ϕ, Q) = \int ϕ d Q - \int exp {ϕ (x)} d x + 1,

where Q ∈ 𝒬 , 𝒬 is the family of all d -dimensional distributions, ϕ ∈ Φ and Φ is the family of all concave function. For linear regression with log-concave error density, Dümbgen et al (2011) proposed an estimator by maximizing:

\hat{L} (ϕ, β, Q) = \frac{1}{n} \sum_{i = 1}^{n} ϕ false(y_{i} - x_{i}^{T} bold-italicβ false) - \int exp {ϕ (x)} d x + 1.

…”

Section: Introductionmentioning

confidence: 99%

The robust EM-type algorithms for log-concave mixtures of regression models

Yao

2017

Computational Statistics & Data Analysis

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Finite mixture of regression (FMR) models can be reformulated as incomplete data problems and they can be estimated via the expectation-maximization (EM) algorithm. The main drawback is the strong parametric assumption such as FMR models with normal distributed residuals. The estimation might be biased if the model is misspecified. To relax the parametric assumption about the component error densities, a new method is proposed to estimate the mixture regression parameters by only assuming that the components have log-concave error densities but the specific parametric family is unknown. Two EM-type algorithms for the mixtures of regression models with log-concave error densities are proposed. Numerical studies are made to compare the performance of our algorithms with the normal mixture EM algorithms. When the component error densities are not normal, the new methods have much smaller MSEs when compared with the standard normal mixture EM algorithms. When the underlying component error densities are normal, the new methods have comparable performance to the normal EM algorithm.

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Approximation by log-concave distributions, with applications to regression

Cited by 93 publications

References 23 publications

Log-concavity and strong log-concavity: A review

Log-concavity and strong log-concavity: A review

Fusion of Hard and Soft Information in Nonparametric Density Estimation

The robust EM-type algorithms for log-concave mixtures of regression models

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