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GLOBAL RATES OF CONVERGENCE OF THE MLES OF LOG-CONCAVE AND s-CONCAVE DENSITIES.

ABSTRACT: We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and s-concave densities on ?. 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.

SUBMITTER: Doss CR 

PROVIDER: S-EPMC5619328 | biostudies-literature | 2016

REPOSITORIES: biostudies-literature

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GLOBAL RATES OF CONVERGENCE OF THE MLES OF LOG-CONCAVE AND <i>s</i>-CONCAVE DENSITIES.

Doss Charles R CR   Wellner Jon A JA  

Annals of statistics 20160411 3

We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and <i>s</i>-concave densities on ℝ. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than <i>n</i><sup>-2/5</sup> when -1 < <i>s</i> < ∞ where <i>s</i> = 0 corresponds to the log-concave case. We also show that the MLE does not exist for the classes of <i>s</i>-concave densities with <i>s</i> < -1. ...[more]

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