GLOBAL RATES OF CONVERGENCE OF THE MLES OF LOG-CONCAVE AND s-CONCAVE DENSITIES BY CHARLES R. DOSS
نویسنده
چکیده
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. The classes of log-concave densities on R and R are special cases of the classes of s-concave densities studied and developed by [5–7] and [30]. Dharmadhikari and Joag-Dev [11], pages 84–99, gives a useful summary. These classes are defined by the generalized means of order s as follows. Let
منابع مشابه
GLOBAL 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 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 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 کاملTheoretical properties of the log-concave maximum likelihood estimator of a multidimensional density
Abstract: We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in R. Our study covers both the case where the true underlying density is log-concave, and where this model is misspecified. We begin by showing that for a sequence of log-concave densities, convergence in distribution implies much s...
متن کاملAPPROXIMATION AND ESTIMATION OF s-CONCAVE DENSITIES VIA RÉNYI DIVERGENCES.
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 c...
متن کامل