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Chernoff's density is log-concave.

Fadoua Balabdaoui1, Jon A Wellner2

  • 1Centre de Recherche en Mathématiques de la Décision, Université Paris-Dauphine, Paris, France.

Bernoulli : Official Journal of the Bernoulli Society for Mathematical Statistics and Probability
|April 25, 2014
PubMed
Summary
This summary is machine-generated.

This study demonstrates that Chernoff's density is log-concave. Further evidence supports the conjecture that this density is strongly log-concave, also known as super-Gaussian.

Keywords:
Brownian motionPolya frequency functionPrekopa–Leindler theoremSchoenberg’s theoremairy functioncorrelation inequalitieshyperbolically monotonelog-concavemonotone function estimationslope processstrongly log-concave

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Area of Science:

  • Probability Theory
  • Information Theory
  • Statistical Inference

Background:

  • Chernoff's density, defined as Z = argmax{W(t) - t^2}, is a key concept in information theory.
  • Understanding the properties of Chernoff's density is crucial for various statistical applications.

Purpose of the Study:

  • To investigate the concavity properties of Chernoff's density.
  • To explore the conjecture that Chernoff's density is strongly log-concave (super-Gaussian).

Main Methods:

  • Mathematical analysis of the density function Z = argmax{W(t) - t^2}.
  • Derivation of log-concavity properties.

Main Results:

  • The study proves that Chernoff's density is log-concave.
  • Evidence is presented supporting the conjecture of strong log-concavity (super-Gaussian property).

Conclusions:

  • Chernoff's density exhibits log-concave properties.
  • The findings provide support for the super-Gaussian conjecture, opening avenues for further research.