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Strong consistency of density estimation by orthogonal series methods for dependent variables with applications

  • Ibrahim A. Ahmad
Article

Abstract

Among several widely use methods of nonparametric density estimation is the technique of orthogonal series advocated by several authors. For such estimate when the observations are assumed to have been taken from strong mixing sequence in the sense of Rosenblatt [7] we study strong consistency by developing probability inequality for bounded strongly mixing random variables. The results obtained are then applied to two estimates of the functional Δ(f)=∫f2(x)dx were strong consistency is established. One of the suggested two estimates of Δ(f) was recently studied by Schuler and Wolff [8] in the case of independent and identically distributed observations where they established consistency in the second mean of the estimate.

AMS subject classification

Primary: 62G05 Secondary: 60G10 

Key words and pharses

Density estimation orthogonal series strong mixing sequences strong consistency non-parametric inference 

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Copyright information

© The Institute of Statistical Mathematics, Tokyo 1979

Authors and Affiliations

  • Ibrahim A. Ahmad
    • 1
  1. 1.McMaster UniversityCanada

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