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On the chernoff-savage theorem for dependent sequences

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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.

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References

  1. Billingsley, P. (1968).Convergence of Probability Measures, Wiley and Sons, New York.

    MATH  Google Scholar 

  2. Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric test statistics,Ann. Math. Statist.,29, 972–994.

    Article  MathSciNet  Google Scholar 

  3. Davydov, Y. A. (1968). Convergence of distribution generated by stationary stochastic processes,Theory Prob. Appl.,12, 691–696.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  6. Serfling, R. J. (1968). Contributions to central limit theorem for dependent variables,Ann. Math. Statist.,39, 1158–1175.

    Article  MathSciNet  Google Scholar 

  7. Withers, C. S. (1975). Convergence of empirical processes of mixing r.v.'s on [0, 1],Ann. Statist.,3, 1101–1108.

    Article  MathSciNet  Google Scholar 

  8. Yoshihara, K. (1974). Extension of Billingsley's theorems on weak convergence of empirical processes,Zeit. Wahrscheinlichkeitsth.,29, 87–92.

    Article  MathSciNet  Google Scholar 

  9. Yoshihara, K. (1976). Weak convergence of multidimensional empirical processes for strong mixing sequences of stochastic vectors,Zeit. Wahrscheinlichkeitsth.,33, 133–137.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Memphis State University, Memphis, USA

    Ibrahim A. Ahmad & Pi-Erh Lin

  2. Florida State University, Florida, USA

    Ibrahim A. Ahmad & Pi-Erh Lin

Authors
  1. Ibrahim A. Ahmad
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  2. Pi-Erh Lin
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Additional information

Research partially supported by the National Research Council of Canada under Grant No. A-3954.

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Ahmad, I.A., Lin, PE. On the chernoff-savage theorem for dependent sequences. Ann Inst Stat Math 32, 211–222 (1980). https://doi.org/10.1007/BF02480326

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  • DOI: https://doi.org/10.1007/BF02480326

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