Advertisement

On the chernoff-savage theorem for dependent sequences

  • Ibrahim A. Ahmad
  • Pi-Erh Lin
Article

Summary

Given a sequence of ϕ-mixing random variables not necessarily stationary, a Chernoff-Savage theorem for two-sample linear rank statistics is proved using the Pyke-Shorack [5] approach based on weak convergence properties of empirical processes in an extended metric. This result is a generalization of Fears and Mehra [4] in that the stationarity is not required and that the condition imposed on the mixing numbers is substantially relaxed. A similar result is shown to hold for strong mixing sequences under slightly stronger conditions on the mixing numbers.

AMS 1970 Subject Classifications

Primary 60F05, 62E20 Secondary 62G10 

Key words and phrases

ϕ-mixing process two-sample linear rank statistics weak convergence of empirical processes 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Billingsley, P. (1968).Convergence of Probability Measures, Wiley and Sons, New York.zbMATHGoogle Scholar
  2. [2]
    Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric test statistics,Ann. Math. Statist.,29, 972–994.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Davydov, Y. A. (1968). Convergence of distribution generated by stationary stochastic processes,Theory Prob. Appl.,12, 691–696.CrossRefGoogle Scholar
  4. [4]
    Fears, T. R. and Mehra, K. L. (1974). Weak convergence of a two-sample empirical process and a Chernoff-Savage theorem for ϕ-mixing sequences,Ann. Statist.,2, 586–596.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Pyke, R. and Shorack, G. (1968). Weak convergence of two-sample empirical process and a new proof to Chernoff-Savage theorems,Ann. Math. Statist.,39, 755–771.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Serfling, R. J. (1968). Contributions to central limit theorem for dependent variables,Ann. Math. Statist.,39, 1158–1175.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Withers, C. S. (1975). Convergence of empirical processes of mixing r.v.'s on [0, 1],Ann. Statist.,3, 1101–1108.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Yoshihara, K. (1974). Extension of Billingsley's theorems on weak convergence of empirical processes,Zeit. Wahrscheinlichkeitsth.,29, 87–92.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Yoshihara, K. (1976). Weak convergence of multidimensional empirical processes for strong mixing sequences of stochastic vectors,Zeit. Wahrscheinlichkeitsth.,33, 133–137.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1980

Authors and Affiliations

  • Ibrahim A. Ahmad
    • 1
    • 2
  • Pi-Erh Lin
    • 1
    • 2
  1. 1.Memphis State UniversityUSA
  2. 2.Florida State UniversityUSA

Personalised recommendations