Global rates of convergence of the MLEs of log-concave and $s$-concave densities
نویسندگان
چکیده
منابع مشابه
GLOBAL RATES OF CONVERGENCE OF THE MLES OF LOG-CONCAVE AND s-CONCAVE DENSITIES BY CHARLES
R d is log-concave if p = e where φ :Rd → [−∞,∞) is concave. We denote the class of all such densities p on R by Pd,0. Log-concave densities are always unimodal and have convex level sets. Furthermore, log-concavity is preserved under marginalization and convolution. Thus, the classes of log-concave densities can be viewed as natural nonparametric extensions of the class of Gaussian densities. ...
متن کاملGLOBAL RATES OF CONVERGENCE OF THE MLES OF LOG-CONCAVE AND s-CONCAVE DENSITIES.
We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and s-concave densities on ℝ. 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. We also show that the MLE does not exist for the classes of s-concave densities with s < -1.
متن کاملGLOBAL RATES OF CONVERGENCE OF THE MLES OF LOG-CONCAVE AND s-CONCAVE DENSITIES BY CHARLES R. DOSS
R d is log-concave if p = e where φ :Rd → [−∞,∞) is concave. We denote the class of all such densities p on R by Pd,0. Log-concave densities are always unimodal and have convex level sets. Furthermore, log-concavity is preserved under marginalization and convolution. Thus, the classes of log-concave densities can be viewed as natural nonparametric extensions of the class of Gaussian densities. ...
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We prove new entropy inequalities for log concave and s-concave functions that strengthen and generalize recently established reverse log Sobolev and Poincaré inequalities for such functions. This leads naturally to the concept of f -divergence and, in particular, relative entropy for s-concave and log concave functions. We establish their basic properties, among them the affine invariant valua...
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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, we study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach. We first show that no statistical procedure based on a sample of size n can estimate a log-concave density with res...
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ژورنال
عنوان ژورنال: The Annals of Statistics
سال: 2016
ISSN: 0090-5364
DOI: 10.1214/15-aos1394