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

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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)=∫f 2 (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.

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Authors and Affiliations

  1. McMaster University, Canada

    Ibrahim A. Ahmad

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  1. Ibrahim A. Ahmad
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Research supported in part by the National Research Council of Canada and in part by McMaster University Research Board. Now at Memphis State University, Memphis, Tennessee 38152, U.S.A.

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Ahmad, I.A. Strong consistency of density estimation by orthogonal series methods for dependent variables with applications. Ann Inst Stat Math 31, 279–288 (1979). https://doi.org/10.1007/BF02480283

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  • DOI: https://doi.org/10.1007/BF02480283

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