, Volume 81, Issue 6, pp 609–618 | Cite as

Local exact Bahadur efficiencies of two scale-free tests of normality based on a recent characterization

  • Ya. Yu. Nikitin


We consider two scale-free tests of normality based on the characterization of the symmetric normal law by Ahsanullah et al. (Normal and student’s t-distributions and their applications, Springer, Berlin, 2014). Both tests have an U-empirical structure, but the first one is of integral type, while the second one is of Kolmogorov type. We discuss the limiting behavior of the test statistics and calculate their local exact Bahadur efficiency for location, skew and contamination alternatives.


Testing of normality U-statistics Bahadur efficiency Kolmogorov test 

Mathematics Subject Classification

62G10 62G20 



The author is thankful to the Editor, to the associate editor and two referees for careful reading of the manuscript, and some important remarks.

Compliance with ethical standards

Conflict of interest

The author declares that there is no conflict of interests.


  1. Ahmad I, Mugdadi AR (2003) Testing normality using kernel methods. J Nonparametr Stat 15:273–288MathSciNetCrossRefMATHGoogle Scholar
  2. Ahsanullah M, Kibria BG, Shakil M (2014) Normal and student’s \(t\)-distributions and their applications. Springer, BerlinCrossRefMATHGoogle Scholar
  3. Allison JS, Pretorius C (2017) A Monte Carlo evaluation of the performance of two new tests for symmetry. Comput Stat 32:1323–1338MathSciNetCrossRefMATHGoogle Scholar
  4. Bahadur RR (1967) Rates of convergence of estimates and test statistics. Ann Math Stat 38:303–324MathSciNetCrossRefMATHGoogle Scholar
  5. Bahadur RR (1971) Some limit theorems in statistics. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  6. Bera AK, Galvao AF, Wang L, Xiao Z (2016) A new characterization of the normal distribution and test for normality. Econom Theor 32:1216–1252MathSciNetCrossRefMATHGoogle Scholar
  7. Csörgő M, Seshadri V, Yalovsky M (1975) Applications of characterizations in the area of goodness of fit. Stat Distrib Sci Work 2:79–90Google Scholar
  8. Durio A, Nikitin YY (2003) Local Bahadur efficiency of some goodness-of-fit tests under skew alternatives. J Stat Plan Inference 115:171–179MathSciNetCrossRefMATHGoogle Scholar
  9. Durio A, Nikitin YY (2016) Local efficiency of integrated goodness-of-fit tests under skew alternatives. Stat Probab Lett 117:136–143MathSciNetCrossRefMATHGoogle Scholar
  10. Helmers R, Janssen P, Serfling R (1988) Glivenko–Cantelli properties of some generalized empirical df’s and strong convergence of generalized \(L\)-statistics. Probab Theor Relat Fields 79:75–93MathSciNetCrossRefMATHGoogle Scholar
  11. Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19:293–395MathSciNetCrossRefMATHGoogle Scholar
  12. Hollander M, Proschan F (1972) Testing whether new is better than used. Ann Math Stat 43:1136–1146MathSciNetCrossRefMATHGoogle Scholar
  13. Korolyuk VS, Borovskikh YuV (1994) Theory of \(U\)-statistics. Kluwer, DordrechtCrossRefMATHGoogle Scholar
  14. Lin CC, Mudholkar GS (1980) A simple test for normality against asymmetric alternatives. Biometrika 67:455–461MathSciNetCrossRefMATHGoogle Scholar
  15. Litvinova VV, Nikitin YY (2016) Kolmogorov tests of normality based on some variants of Polya’s characterization. J Math Sci 5(219):782–788CrossRefMATHGoogle Scholar
  16. Muliere P, Nikitin YY (2002) Scale-invariant test of normality based on Polya’s characterization. Metron 60:21–33MathSciNetMATHGoogle Scholar
  17. Nikitin YY (1995) Asymptotic efficiency of nonparametric tests. Cambridge University Press, New YorkCrossRefMATHGoogle Scholar
  18. Nikitin YY (2010) Large deviations of \(U\)-empirical Kolmogorov–Smirnov tests, and their efficiency. J Nonparametr Stat 22:649–668MathSciNetCrossRefMATHGoogle Scholar
  19. Nikitin YY, Peaucelle I (2004) Efficiency and local optimality of distribution-free tests based on \(U\)- and \(V\)- statistics. Metron 62:185–200MathSciNetGoogle Scholar
  20. Nikitin YY, Ponikarov EV (1999) Rough large deviation asymptotics of Chernoff type for von Mises functionals and U-statistics. Proc Saint-Petersburg Math Soc 7:124–167; Engl. transl. in AMS Transl, ser 2, 2001, 203:107–146Google Scholar
  21. Sakata T (1977) A test of normality based on some characterization theorem. Mem Fac Sci Kyushu Univ Ser A Math 31:221–225MathSciNetMATHGoogle Scholar
  22. Villaseñor-Alva JA, Gonzalez-Estrada Y (2015) A correlation test for normality based on the Lévy characterization. Commun Stat Simul Comput 44:1225–1238CrossRefMATHGoogle Scholar
  23. Volkova KY, Nikitin YY (2009) On the asymptotic efficiency of normality tests based on the Shepp property. Vestnik St Petersb Univ Math 42:256–261MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Saint-Petersburg State UniversitySt. PetersburgRussia
  2. 2.Higher School of EconomicsNational Research UniversitySt. PetersburgRussia

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