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

  • Ibrahim A. Ahmad


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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  1. [1]
    Billingsley, P. (1968).Convergence of Probability Measures. Wiley, New York.zbMATHGoogle Scholar
  2. [2]
    Cencov, N. N. (1962). Estimation of an unknown density function from observations,Dokl. Akad. Nauk, SSSR,147, 45–48 (in Russian).MathSciNetGoogle Scholar
  3. [3]
    Dvoretzky, A. (1972). Asymptotic normality for sums of dependent random variables,Proc. Sixth Berkeley Symp. Math. Statist. Prob.,2, 513–535.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables,J. Amer. Statist. Ass.,58, 13–30.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Ibragimov, I. A. and Linnik, Yu, V. (1971).Independent and Stationary Sequences of Random Variables, Walters-Noordhoff, the Netherlands.zbMATHGoogle Scholar
  6. [6]
    Krornmal, R. and Tarter, M. (1968). The estimation of probability densities and cumulative by fourier series methods,J. Amer. Statist. Ass.,63, 925–952.MathSciNetzbMATHGoogle Scholar
  7. [7]
    Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition.Proc. Nat. Acad. Sci. USA,42, 43–47.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Schuler, L. and Wolff, H. (1976). Zur Schatzung eines dichtefunktionals,Metrika,23, 149–153.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Schwartz, S. C. (1967). Estimation of a probability density by an orthogonal series,Ann. Math. Statist.,38, 1262–1265.MathSciNetGoogle Scholar
  10. [10]
    Watson, G. S. (1969). Density estimation by orthogonal series,Ann. Math. Statist.,40, 1496–1498.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1979

Authors and Affiliations

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

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