Advertisement

Prediction Under Association

  • George G. Roussas
Part of the Statistics for Industry and Technology book series (SIT)

Abstract

Consider a discrete parameter time series Z n, n = 1,2, …,Z i+d-1 which is assumed to be strictly stationary and (either positively or negatively) associated. Let ϕ be a real-valued function defined on the real line, and let d be an integer greater than or equal 1. The predictor of Z i+d is defined to be the conditional expectation of Z i+d, given the random variables Z i,…, The problem discussed in this paper is the construction of an estimate of the proposed predictor, in terms of the random variables Z j, j =1,…, n+d , and the proof of its asymptotic normality under suitable conditions. The estimate considered is a kernel-type estimate, and a hint is made toward studying an alternative estimate

Keywords and phrases

Stationarity association discrete parameter time series prediction kernel estimate consistency asymptotic normality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bagai, I. and Prakasa Rao, B. L. S. (1991). Estimation of the survival function for stationary associated processes, Statist. Probab. Lett., 12, 385–391.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bagai, I. and Prakasa Rao, B. L. S. (1995). Kernel-type density and failure rate estimation for associated sequences, Ann. Inst. Statis. Math., 47, 253–266.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Birkel, Th. (1988). On the convergence rate in Central Limit Theorem for associated processes, Ann. Probab., 16, 1685–1698.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bulinski, A. V. (1996). On the convergence rates in the CLT for positively and negatively dependent random fields, In Probability Theory and Mathematical Statistics (Eds., I. A. Ibragimov and A. Yu. Zaitsev), Gordon and Breach.Google Scholar
  5. 5.
    Cai, Z. and Roussas, G. G. (1997). Smooth estimate of quantiles under association, Statist. Probab. Lett, 36, 275–287.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cai, Z. and Roussas, G. G. (1998a). Efficient estimation of a distribution function under quadrant dependence, Scand. J. Statist, 25, 211–224.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cai, Z. and Roussas, G. G. (1998b). Kaplan-Meier estimator under association, J. Multivariate Anal., 67, 318–348.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cai, Z. and Roussas, G. G. (1999a). Weak convergence for a smooth estimator of a distribution function under association, Stochastic Anal. Appl., 17, 145–168.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cai, Z. and Roussas, G. G. (1999b). Berry-Esseen bounds for smooth estimator of a distribution function under association, J. Nonparametric Statist 11, 79–106.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dewan, I. and Prakasa Rao, B. L. S. (1999). A general method of density estimation for associated random variables, J. Nonparametric Statist., 10, 405–420.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Doukhan, P. and Louhichi, S. (1999). Functional estimation of a density of a new weak dependence condition, Unpublished Manuscript Google Scholar
  12. 12.
    Loève, M. (1963). Probability Theory, 3rd edition, Princeton, New Jersey: Van Nostrand.zbMATHGoogle Scholar
  13. 13.
    Louhichi, S. (1998). Weak convergence for empirical processes of associated sequences, Prepublication #98.36, Université de Paris-Sud.Google Scholar
  14. 14.
    Masry, E. and Fan, J. (1997). Local polynomial estimation of regression functions for mixing processes, Scand. J. Statist. 24, 165–179.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Roussas, G. G. (1990). Nonparametric regression estimation under mixing conditions, Stochastic Process. Appl., 36, 107–116.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Roussas, G. G. (1991). Kernel estimates under association, Statist. Probab. Lett., 12, 393–403.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Roussas, G. G. (1993). Curve estimation in random fields of associated processes, J. Nonparametric Statist., 2, 215–224.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Roussas, G. G. (1994). Asymptotic normality of random fields of positively or negatively associated processes, J. Multivariate Anal., 50, 152–173.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Roussas, G. G. (1995). Asymptotic normality of a smooth estimate of a random field distribution function under association, Statist. Probab. Lett., 24, 77–90.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Roussas, G.G. (1999a). Positive and negative dependence with some statistical applications, In Asymptotics, Nonparametrics, and Time Series (Ed., S. Ghosh), New York: Marcel Dekker.Google Scholar
  21. 21.
    Roussas, G.G. (1999b). Asymptotic normality of a kernel estimate of a probability density function under association, Statist. Probab. Lett. (to appear).Google Scholar
  22. 22.
    Yu, H. (1993). Glivenko-Cantelli lemma and weak convergence for empirical processes of associated sequences, Probab. Theory Relat. Fields, 95, 357–370.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • George G. Roussas
    • 1
  1. 1.University of CaliforniaDavisUSA

Personalised recommendations