Global rates of convergence in log-concave density estimation
نویسندگان
چکیده
منابع مشابه
Global Rates of Convergence in Log-concave Density Estimation by Arlene
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...
متن کاملGLOBAL 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.
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Assume that one has to estimate a density f on R e from X1, ..., Xn, a sequence of independent random vectors with common density f A density estimate is a sequence (f,) of Borel measurable mappings: R a ( n + ~ R ; for fixed n, f (x) is estimated by f,~(x)=f,(x, X 1 . . . . , X,). In this note, we take a look at the rate of convergence of E(~ ]f , (x)f (x) lPdx) ( p > l ) for all density estim...
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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ژورنال
عنوان ژورنال: The Annals of Statistics
سال: 2016
ISSN: 0090-5364
DOI: 10.1214/16-aos1480