Global rates of convergence of the MLEs of log-concave and s-concave densities

Charles R. Doss; Jon A. Wellner

doi:10.1214/15-AOS1394

Global rates of convergence of the MLEs of log-concave and s-concave densities

Charles R. Doss, Jon A. Wellner

Statistics (Twin Cities)

Research output: Contribution to journal › Article › peer-review

42 Scopus citations

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.

Original language	English (US)
Pages (from-to)	954-981
Number of pages	28
Journal	Annals of Statistics
Volume	44
Issue number	3
DOIs	https://doi.org/10.1214/15-AOS1394
State	Published - Jun 2016

Bibliographical note

Publisher Copyright:
© Institute of Mathematical Statistics, 2016.

Keywords

Bracketing entropy
Consistency
Empirical processes
Global rate
Hellinger metric
Log-concave
S-concave

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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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AU - Wellner, Jon A.

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PY - 2016/6

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N2 - 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.

AB - 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.

KW - Bracketing entropy

KW - Consistency

KW - Empirical processes

KW - Global rate

KW - Hellinger metric

KW - Log-concave

KW - S-concave

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Global rates of convergence of the MLEs of log-concave and s-concave densities

Abstract

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Keywords

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