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New dependence coefficients. Examples and applications to statistics
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  • Published: 27 December 2004

New dependence coefficients. Examples and applications to statistics

  • Jérôme Dedecker1 &
  • Clémentine Prieur2 

Probability Theory and Related Fields volume 132, pages 203–236 (2005)Cite this article

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  • 109 Citations

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Abstract.

To measure the dependence between a real-valued random variable X and a σ-algebra , we consider four distances between the conditional distribution function of X given and the distribution function of X. The coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations. In particular, we show that they are well adapted to functions of mixing sequences, iterated random functions and dynamical systems. Starting from a new covariance inequality, we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence. We obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis, under a minimal condition on the coefficients. Using recent results, we show that our coefficients may be also used to obtain various exponential inequalities, a concentration inequality for Lipschitz functions, and a Berry-Esseen type inequality.

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Author information

Authors and Affiliations

  1. Laboratoire de Statistique Théorique et Appliquée, Université Paris 6, 175 rue du Chevaleret, 75013, Paris, France

    Jérôme Dedecker

  2. Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse cedex 4, France

    Clémentine Prieur

Authors
  1. Jérôme Dedecker
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  2. Clémentine Prieur
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Corresponding author

Correspondence to Jérôme Dedecker.

Additional information

Mathematics Subject Classification (2000): 62G07, 60J10, 60E15, 37C30

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Dedecker, J., Prieur, C. New dependence coefficients. Examples and applications to statistics. Probab. Theory Relat. Fields 132, 203–236 (2005). https://doi.org/10.1007/s00440-004-0394-3

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  • Received: 23 June 2004

  • Revised: 23 September 2004

  • Published: 27 December 2004

  • Issue Date: June 2005

  • DOI: https://doi.org/10.1007/s00440-004-0394-3

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Key words

  • Dependence coefficients
  • Mixing
  • Covariance inequalities
  • Markov chains
  • Dynamical systems
  • Density estimation
  • Exponential inequalities
  • Concentration inequalities
  • Berry-Esseen inequality
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