Global Rates of Convergence of the MLEs of Log-concave and s-concave Densities

2013/06/06 by Doss, Charles R., Wellner, Jon A.
#62G05 #62G07 #62G20 #FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1306.1438

Abstract

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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