Doss Charles R, Wellner Jon A
University of Minnesota and University of Washington.
Ann Stat. 2016;44(3):954-981. doi: 10.1214/15-AOS1394. Epub 2016 Apr 11.
We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and -concave densities on ℝ. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than when -1 < < ∞ where = 0 corresponds to the log-concave case. We also show that the MLE does not exist for the classes of -concave densities with < -1.
我们建立了关于实数域上对数凹和凹密度的最大似然估计量（MLEs）的全局收敛速率。主要发现是，在Hellinger度量下，MLE的收敛速率不超过 ，其中 -1 < < ∞，且 = 0对应对数凹情形。我们还表明，对于 < -1的凹密度类，MLE不存在。
