Skip to main content
SpringerLink
Log in

New \(U\)-empirical Tests of Symmetry Based on Extremal Order Statistics, and Their Efficiencies

  • Chapter
  • First Online:
Mathematical Statistics and Limit Theorems

Abstract

We use a characterization of symmetry in terms of extremal order statistics which enables to build several new nonparametric tests of symmetry. We discuss their limiting distributions and calculate their local exact Bahadur efficiency under location alternative which is mostly high.

Y.Y. Nikitin—Research supported by RFBR grant No. 13-01-00172, and by SPbGU grant No.6.38.672.2013.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Ahsanullah, M. (1992). On some characteristic property of symmetric distributions. Pakistan Journal of Statistics, 8, 19–22.

    MATH  MathSciNet  Google Scholar 

  • Azzalini A. with the collaboration of A. Capitanio (2014). The Skew-normal and related families. New York: Cambridge University Press.

    Google Scholar 

  • Bahadur, R. R. (1967). Rates of convergence of estimates and test statistics. Annals of Mathematical Statistics, 38, 303–324.

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Baringhaus, L., & Henze, N. (1992). A characterization of and new consistent tests for symmetry. Communications in Statistics - Theory, 21, 1555–1566.

    Article  MATH  MathSciNet  Google Scholar 

  • Halmos, P. R. (1946). The theory of unbiased estimation. Annals of Mathematical Statistics, 17, 34–43.

    Article  MATH  MathSciNet  Google Scholar 

  • Helmers, R., Janssen, P., & Serfling, R. (1988). Glivenko-Cantelli properties of some generalized empirical dfs and strong convergence of generalized L-statistics. Probability Theory and Related Fields, 79, 75–93.

    Article  MATH  MathSciNet  Google Scholar 

  • Henze, N., & Meintanis, S. (2002). Goodness-of-fit tests based on a new characterization of the exponential distribution. Communication in Statistics- Theory, 31, 1479–1497.

    Article  MATH  MathSciNet  Google Scholar 

  • Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Annals of Mathematical Statistics, 19, 293–325.

    Article  MATH  MathSciNet  Google Scholar 

  • Korolyuk, V. S., & Borovskikh, Y. V. (1994). Theory of U-statistics. Dordrecht: Kluwer.

    Book  MATH  Google Scholar 

  • Lambert, D., & Hall, W. J. (1982). Asymptotic lognormality of P-values. Annals of Statistics, 10, 44–64.

    Article  MATH  MathSciNet  Google Scholar 

  • Lee, A. J. (1990). U-statistics: Theory and practice. New York: Dekker.

    MATH  Google Scholar 

  • Linnik, Yu. V. (1953). Linear forms and statistical criteria. I, II. Ukrainian Mathematical Journal, 5(207–243), 247–290.

    Google Scholar 

  • Litvinova, V. V. (2001). New nonparametric test for symmetry and its asymptotic efficiency. Vestnik of St.Petersburg University. Mathematics, 34, 12–14.

    MathSciNet  Google Scholar 

  • Litvinova, V. V. (2004) Asymptotic properties of goodness-of-fit and symmetry tests based on characterizations. Ph.D. thesis. Saint-Petersburg University

    Google Scholar 

  • Morris, K., & Szynal, D. (2001). Goodness-of-fit tests using characterizations of continuous distributions. Applied Mathematics (Warsaw), 28, 151–168.

    Article  MATH  MathSciNet  Google Scholar 

  • Muliere, P., & Nikitin, Y. Y. (2002). Scale-invariant test of normality based on Polya’s characterization. Metron, 60, 21–33.

    MATH  MathSciNet  Google Scholar 

  • Nikitin, Y. (1995). Asymptotic efficiency of nonparametric tests. New York: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Nikitin, Y. Y. (1996a). On Baringhaus-Henze test for symmetry: Bahadur efficiency and local optimality for shift alternatives. Mathematical Methods of Statistics, 5, 214–226.

    Google Scholar 

  • Nikitin, Y. Y. (1996b). Bahadur efficiency of a test of exponentiality based on a loss-of-memory type functional equation. Journal of Nonparametric Statistic, 6, 13–26.

    Google Scholar 

  • Nikitin, Y. Y. (2010). Large deviations of U-empirical Kolmogorov-Smirnov tests, and their efficiency. Journal of Nonparametric Statistics, 22, 649–668.

    Article  MATH  MathSciNet  Google Scholar 

  • Nikitin, Y. Y., & Peaucelle, I. (2004). Efficiency and local optimality of distribution-free tests based on U- and V-statistics. Metron, LXII, 185–200.

    Google Scholar 

  • Nikitin, Y.Y, & Ponikarov, E.V. (1999) Rough large deviation asymptotics of Chernoff type for von Mises functionals and \(U\)-statistics. Proceedings of the St. Petersburg Mathematical Society (pp. 124–167) vol 7. (Y. Y. Nikitin, E. V. Ponikarov, AMS Transl, series 2, 203, pp. 107–146, Trans.)

    Google Scholar 

  • Nikitin, Y. Y., & Volkova, K. Y. (2010). Asymptotic efficiency of exponentiality tests based on order statistics characterization. Georgian Mathematical Journal, 17, 749–763.

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We are thankful to the referee and the editors for their comments on the paper leading to a substantial improvement of the presentation.

Author information

Authors and Affiliations

  1. Department of Mathematics and Mechanics, Saint-Petersburg State University, Universitetsky Pr. 28, Stary Peterhof, 198504, St. Petersburg, Russia

    Ya. Yu. Nikitin

  2. National Research University - Higher School of Economics, Souza Pechatnikov, 16, 190008, St. Petersburg, Russia

    Ya. Yu. Nikitin

  3. Department of Management Sciences, Rider University, Lawrenceville, NJ, 08648, USA

    M. Ahsanullah

Authors
  1. Ya. Yu. Nikitin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Ahsanullah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ya. Yu. Nikitin .

Editor information

Editors and Affiliations

  1. European Centre for Advanced Research in Economics and Statistics, Université Libre de Bruxelles, Bruxelles, Belgium

    Marc Hallin

  2. Department of Applied Economics and Statistics, University of Delaware, Newark, Delaware, USA

    David M. Mason

  3. School of Mathematics and Science, Carl von Ossietzky University, Oldenburg, Germany

    Dietmar Pfeifer

  4. Mathematical Institute, University of Cologne, Köln, Germany

    Josef G. Steinebach

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Nikitin, Y.Y., Ahsanullah, M. (2015). New \(U\)-empirical Tests of Symmetry Based on Extremal Order Statistics, and Their Efficiencies. In: Hallin, M., Mason, D., Pfeifer, D., Steinebach, J. (eds) Mathematical Statistics and Limit Theorems. Springer, Cham. https://doi.org/10.1007/978-3-319-12442-1_13

Download citation

Publish with us

Policies and ethics

Search

Navigation