Journal of Statistical Theory and Practice

, Volume 11, Issue 3, pp 478–492 | Cite as

Self-consistency-based tests for bivariate distributions

  • Jochen EinbeckEmail author
  • Simos Meintanis


A novel family of tests based on the self–consistency property is developed. Our developments can be motivated by the well-known fact that a two–dimensional spherically symmetric distribution X is self–consistent with respect to the circle E||X||; that is, each point on that circle is the expectation of all observations that project onto that point. This fact allows the use of the self–consistency property in order to test for spherical symmetry. We construct an appropriate test statistic based on empirical characteristic functions, which turns out to have an appealing closed–form representation. Critical values of the test statistics are obtained empirically. The nominal level attainment of the test is verified in simulation, and the test power under several alternatives is studied. A similar test based on the self–consistency property is then also developed for the question of whether a given straight line corresponds to a principal component. The extendibility of this concept to further test problems for multivariate distributions is briefly discussed.


Self-consistency empirical characteristic functions spherical symmetry principal curves principal components 

AMS Subject Classification



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  1. Bierens, H. J. 1982. Consistent model specification tests. Journal of Econometrics 20:105–34.MathSciNetCrossRefGoogle Scholar
  2. Einmahl, J. H. J., and M. Gantner. 2012. Testing for bivariate spherical symmetry. TEST 21 (1):54–73.MathSciNetCrossRefGoogle Scholar
  3. Flury, B. D. 1990. Principal points. Biometrika 77:33–41.MathSciNetCrossRefGoogle Scholar
  4. Härdle, W. 1991. Smoothing techniques. With implementation in S. New York, NY: Springer Verlag.CrossRefGoogle Scholar
  5. Hastie, T., and W. Stuetzle. 1989. Principal curves. Journal of the American Statistical Association 84:502–16.MathSciNetCrossRefGoogle Scholar
  6. Henze, N., Z. Hlávka, and S. Meintanis. 2014. Testing for spherical symmetry via the empirical characteristic function. Statistics 48 (6):1282–96.MathSciNetCrossRefGoogle Scholar
  7. Koltchinskii, V., and L. Li. 1998. Testing for spherical symmetry of a multivariate distribution. Journal of Multivariate Analysis 65 (2):228–44.MathSciNetCrossRefGoogle Scholar
  8. R Core Team. 2016. R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.Google Scholar
  9. Tarpey, T., and B. Flury. 1996. Self-consistency: A fundamental concept in statistics. Statistical Science 11:229–43.MathSciNetCrossRefGoogle Scholar
  10. Tenreiro, C. 2009. On the choice of the smoothing parameter for the BHEP goodness-of-fit test. Computational Statistics and Data Analysis 53 (4):1038–53.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing, 20 Middlefield Ct, Greensboro, NC 27455 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesDurham UniversityDurham CityUnited Kingdom
  2. 2.Department of EconomicsNational and Kapodistrian University of AthensAthensGreece
  3. 3.Unit for Business Mathematics and InformaticsNorth-West UniversityPotchefstroomSouth Africa

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