APPROXIMATION AND ESTIMATION OF s-CONCAVE DENSITIES VIA RÉNYI DIVERGENCES.

@article{Han2016APPROXIMATIONAE,
  title={APPROXIMATION AND ESTIMATION OF s-CONCAVE DENSITIES VIA R{\'E}NYI DIVERGENCES.},
  author={Qiyang Han and Jon A. Wellner},
  journal={Annals of statistics},
  year={2016},
  volume={44 3},
  pages={
          1332-1359
        }
}
In this paper, we study the approximation and estimation of s-concave densities via Rényi divergence. We first show that the approximation of a probability measure Q by an s-concave density exists and is unique via the procedure of minimizing a divergence functional proposed by [Ann. Statist.38 (2010) 2998-3027] if and only if Q admits full-dimensional support and a first moment. We also show continuity of the divergence functional in Q: if Qn → Q in the Wasserstein metric, then the projected… Expand
APPROXIMATION AND ESTIMATION OF S-CONCAVE
In this paper, we study the approximation and estimation of s- concave densities via Renyi divergence. We first show that the ap- proximation of a probability measure Q by an s-concave density existsExpand
A new computational framework for log-concave density estimation
In Statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statisticalExpand
High-dimensional nonparametric density estimation via symmetry and shape constraints
We tackle the problem of high-dimensional nonparametric density estimation by taking the class of log-concave densities on $\mathbb{R}^p$ and incorporating within it symmetry assumptions, whichExpand
Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression
TLDR
It is proved that estimating a log-concave density - even a uniform distribution on a convex set - up to a fixed accuracy requires the number of samples at least exponential in the dimension. Expand
Inference for the mode of a log-concave density
We study a likelihood ratio test for the location of the mode of a log-concave density. Our test is based on comparison of the log-likelihoods corresponding to the unconstrained maximum likelihoodExpand
A Polynomial Time Algorithm for Maximum Likelihood Estimation of Multivariate Log-concave Densities
TLDR
This work studies the problem of computing the maximum likelihood estimator (MLE) of multivariate log-concave densities and presents the first computationally efficient algorithm for this problem, based on a natural convex optimization formulation of the underlying problem and a projected stochastic subgradient method. Expand
Multivariate convex regression: global risk bounds and adaptation
We study the problem of estimating a multivariate convex function defined on a convex body in a regression setting with random design. We are interested in optimal rates of convergence under aExpand
GLOBAL RATES OF CONVERGENCE OF THE MLES OF LOG-CONCAVE AND s-CONCAVE DENSITIES.
TLDR
The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than n-2/5 when -1 < s < ∞ where s = 0 corresponds to the log-concave case. Expand
Log-concave density estimation with symmetry or modal constraints
We study nonparametric maximum likelihood estimation of a log-concave density function $f_0$ which is known to satisfy further constraints, where either (a) the mode $m$ of $f_0$ is known, or (b)Expand
S-Concave Distributions: Towards Broader Distributions for Noise-Tolerant and Sample-Efficient Learning Algorithms
TLDR
New convex geometry tools to study the properties of s-concave distributions are introduced and these properties are used to provide bounds on quantities of interest to learning including the probability of disagreement between two halfspaces, disagreement outside a band, and disagreement coefficient. Expand
...
1
2
3
4
...

References

SHOWING 1-10 OF 53 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 whereExpand
APPROXIMATION BY LOG-CONCAVE DISTRIBUTIONS, WITH APPLICATIONS TO REGRESSION
We study the approximation of arbitrary distributions P on d-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback―Leibler-type functional. We showExpand
Maximum likelihood estimation of a multidimensional log-concave density
Let X_1, ..., X_n be independent and identically distributed random vectors with a log-concave (Lebesgue) density f. We first prove that, with probability one, there exists a unique maximumExpand
Global rates of convergence in log-concave density estimation
The estimation of a log-concave density on $\mathbb{R}^d$ represents a central problem in the area of nonparametric inference under shape constraints. In this paper, we study the performance ofExpand
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. Expand
Estimation of a k-monotone density: limit distribution theory and the Spline connection
We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a kmonotone density g0 at a fixed point x0 when k> 2. We find that the jth derivative of the estima- tors atExpand
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 weakExpand
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) forExpand
logcondens: Computations Related to Univariate Log-Concave Density Estimation
Maximum likelihood estimation of a log-concave density has attracted considerable attention over the last few years. Several algorithms have been proposed to estimate such a density. Two of thoseExpand
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 twoExpand
...
1
2
3
4
5
...