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Approximate asymptotic Bahadur efficiency of independence tests with random sample size

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Summary

We compare various kind of independence tests based on samples with random size, in order to provide practitioners with some guidance for their choice based on approximate Bahadur efficiency. Such results are obtained for a wide class of distributions of the random index; the efficiency slopes of the statistics we consider are then expressed in terms of the parameters of these same distributions.

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References

  • Allen, J., Beekman, J. (1967), On the distribution of the M. Kac statistic.Ann. Math. Stat., 38, 6, 1919–1923.

    Article  MATH  MathSciNet  Google Scholar 

  • Bahadur, R. R. (1960), Stochastic comparison of tests.Ann. Math. Stat., 31, 276–295.

    Article  MATH  MathSciNet  Google Scholar 

  • Bahadur, R. R. (1971),Some Limit Theorems in Statistics, Philadelphia: SIAM.

    MATH  Google Scholar 

  • Bajorski, P. (1987), Local Bahadur optimality of some rank tests of independence.Stat. and Prob. Letters, 5, 6, 255–262.

    Article  MATH  MathSciNet  Google Scholar 

  • Beghin, L. (2000), Weak convergence of some randomly indexed empirical processes, Technical Reports Dipart. di Stat. Prob. e Stat. Appi., Univ. di Roma, «La Sapienza», Serie A — Ricerche, 28, (submitted for publication).

  • Conti, P. L., Nikitin, Ya. Yu. (1999), Asymptotic efficiency of independence tests based on Gini’s rank association coefficient, Spearman’s footrule and their generalizations.Commun. Statist. — Theory Meth., 28, 2, 453–465.

    Article  MATH  Google Scholar 

  • Csörgo, M. (1968), On the strong law of large numbers and the central limit theorems for martingales.Trans. Am. Math. Soc., 131, 259–275.

    Article  Google Scholar 

  • Csörgo, M. (1972), Distribution results for distance functions based on the modified empirical distribution function of Kac.Ann. Inst. Stat. Math., 24, 101.

    Article  Google Scholar 

  • Csörgo, S. (1981), Strong approximations of empirical Kac processes.Ann. Inst. Stat. Math., 33, 3, 417–423.

    Article  Google Scholar 

  • Dudley, R. M. (1999),Uniform Central Limit Theorems, New York: Cambridge Studies in Adv. Mathem., Cambridge Univ. Press.

  • Farlie, D. J. (1960), The performance of some correlation coefficients for a general bivariate distribution.Biometrika, 47, 307–323.

    MATH  MathSciNet  Google Scholar 

  • Gnedenko, B. V., Korolev, V. Yu. (1996),Random Summation: Limit Theorems and Applications, Boca Raton-New York-London-Tokyo: CRC Press.

    MATH  Google Scholar 

  • Gumbel, E. J. (1958), Distributions à plusieurs variables dont les marges sont données.C.R. Acad. Sci., 246, 2717–1719.

    MATH  MathSciNet  Google Scholar 

  • Kac, M. (1949), On deviations between theoretical and empirical distributions.Proc. Nat. Acad. Sci. U.S.A., 35, 252–257.

    Article  MATH  MathSciNet  Google Scholar 

  • Klaassen, C. A., Wellner, J. A. (1992), Kac empirical processes and the bootstrap.Proc. of the Eight Intern. Conf. on Prob. in Banach Spaces, 411–429, (eds. M. Hahn and J. Kuelbs) New York: Birkäuser.

    Google Scholar 

  • Marcus, M. B., Shepp, L.A. (1971), Sample behaviour of Gaussian processes.Proc. Sixth Berkeley Symp. Math. Stat. Prob., 2, 423–442.

    Google Scholar 

  • Morgenstern, D. (1956), Einfache Beispiele zweidimensionaler Verteilungen.Mitt. Mathem. Statist., 8, 234–235.

    MathSciNet  Google Scholar 

  • Nikitin, Ya. Yu. (1981), Limit distributions and comparative asymptotic efficiency of the Kolmogorov-Smirnov statistics with random index.Journal of Soviet Math., 2, 1042–1049.

    Article  Google Scholar 

  • Nikitin, Ya. Yu. (1995),Asymptotic Efficiency of Nonparametric Tests, New York: Cambridge Univ. Press.

    Book  MATH  Google Scholar 

  • Nikitin, Ya. Yu., Pankrashova, A. (1990), Bahadur efficiency and local asymptotic optimality of certain nonparametric tests of independence.Journal of Soviet Math., 52, 2942–2955.

    Article  MathSciNet  Google Scholar 

  • Paulauskas, V. I. (1972), On the sum of a random number of multidimensional random vectors.Litov. Mat. Sb., 12, no. 2, 109–131, in Russian.

    MATH  MathSciNet  Google Scholar 

  • Suzuki, G. (1972), Distribution of Kac-statistics.Ann. Inst. Stat. Math., 24, 415–421.

    Article  MATH  Google Scholar 

  • Van der Vaart, A. W., Wellner, J. A. (1996),Weak Convergence and Empirical Processes, New York-Berlin-Heidelberg: Springer-Verlag.

    MATH  Google Scholar 

  • Wieand, H. S. (1976), A condition under which the Pitman and Bahadur approaches to efficiency coincide.Ann. Stat., 4, 1003–1011.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Statistica, Probabilità e Statistiche Applicate, Università di Roma «La Sapienza», P.le Aldo Moro, 5, I-00185, Roma, Italy

    L. Beghin

  2. Department of Mathematics and Mechanics, St. Petersburg State University, 198904, St. Petersburg, Russia

    Y. Y. Nikitin

Authors
  1. L. Beghin
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  2. Y. Y. Nikitin
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Additional information

Research partially supported by RFBR (grant no. 99-01-0724) and ICER (Turin).

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Beghin, L., Nikitin, Y.Y. Approximate asymptotic Bahadur efficiency of independence tests with random sample size. J. Ital. Statist. Soc. 8, 1 (1999). https://doi.org/10.1007/BF03178938

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  • DOI: https://doi.org/10.1007/BF03178938

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