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Summary
This summary is machine-generated.

We determined the convergence rates for Maximum Likelihood Estimators (MLEs) of log-concave and s-concave densities. The MLE

Keywords:
Bracketing entropyHellinger metricPrimary 62G07consistencyempirical processesglobal ratelog-concaves-concavesecondary 62G05, 62G20

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Area of Science:

  • Statistical theory
  • Probability theory
  • Nonparametric statistics

Background:

  • Maximum Likelihood Estimators (MLEs) are fundamental in statistical inference.
  • Log-concave and s-concave densities are important classes of probability distributions with applications in various fields.
  • Understanding the convergence rates of estimators is crucial for assessing their efficiency.

Purpose of the Study:

  • To establish global rates of convergence for MLEs of log-concave and s-concave densities on the real line (ℝ).
  • To analyze the impact of the parameter 's' on the convergence rate.
  • To investigate the existence of MLEs for different ranges of 's'.

Main Methods:

  • Theoretical analysis of Maximum Likelihood Estimators.
  • Derivation of convergence rates in the Hellinger metric.
  • Examination of density classes defined by concavity properties.

Main Results:

  • Established global convergence rates for MLEs of log-concave (s=0) and s-concave densities for -1 < s < ∞.
  • The convergence rate in the Hellinger metric is shown to be no worse than n^(-2/5).
  • Demonstrated that MLEs do not exist for s-concave densities when s < -1.

Conclusions:

  • The MLE provides a reliable estimation method for log-concave and s-concave densities within a specific range of 's'.
  • The convergence rate is robust across a wide range of s-concavity.
  • The non-existence of MLEs for s < -1 highlights limitations for certain density classes.