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Journal of Statistical Theory and Practice

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

Self-consistency-based tests for bivariate distributions

Article

Abstract

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.

Keywords

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

AMS Subject Classification

62H15 

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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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