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Asymptotic Normality of Binned Kernel Density Estimators for Non-stationary Dependent Random Variables

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Abstract

We establish the asymptotic normality of binned kernel density estimators for a sequence of dependent and nonstationary random variables converging to a sequence of stationary random variables. We compute the asymptotic variance of a suitably normalized binned kernel density estimator and study its absolute third-order moment. Then, we show that its characteristic function tends to that of a zero-mean Gaussian random variable (rv). We illustrate our results with a simulation experiment.

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Acknowledgments

We would like to thank the Editor and two anonymous referees whose comments led to improve the paper.

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Authors and Affiliations

  1. ÉSPÉ de Limoges, Limoges, 87036, Cedex, France

    Michel Harel

  2. Université Paul Sabatier, Toulouse, 31062, Cedex, France

    Michel Harel

  3. Faculté des Sciences et Techniques, Limoges, 87060, Cedex, France

    Jean-François Lenain

  4. EHESP de Rennes, Institut Élie Cartan de Lorraine, Université de Lorraine, Vandoeuvre-lès-Nancy, 54506, Cedex, France

    Joseph Ngatchou-Wandji

Authors
  1. Michel Harel
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  2. Jean-François Lenain
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  3. Joseph Ngatchou-Wandji
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Corresponding author

Correspondence to Joseph Ngatchou-Wandji .

Editor information

Editors and Affiliations

  1. European Centre for Advanced Research in Economics and Statistics, Université Libre de Bruxelles, Bruxelles, Belgium

    Marc Hallin

  2. Department of Applied Economics and Statistics, University of Delaware, Newark, Delaware, USA

    David M. Mason

  3. School of Mathematics and Science, Carl von Ossietzky University, Oldenburg, Germany

    Dietmar Pfeifer

  4. Mathematical Institute, University of Cologne, Köln, Germany

    Josef G. Steinebach

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Harel, M., Lenain, JF., Ngatchou-Wandji, J. (2015). Asymptotic Normality of Binned Kernel Density Estimators for Non-stationary Dependent Random Variables. In: Hallin, M., Mason, D., Pfeifer, D., Steinebach, J. (eds) Mathematical Statistics and Limit Theorems. Springer, Cham. https://doi.org/10.1007/978-3-319-12442-1_10

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