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.
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| http://dx.doi.org/10.1214/15-AOS1394 | DOI Listing |
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