Statistics and Computing

, Volume 26, Issue 6, pp 1307–1317 | Cite as

Parametric bootstrap goodness-of-fit testing for Wehrly–Johnson bivariate circular distributions

  • Arthur PewseyEmail author
  • Shogo Kato


The Wehrly–Johnson family of bivariate circular distributions is by far the most general one currently available for modelling data on the torus. It allows complete freedom in the specification of the marginal circular densities as well as the binding circular density which regulates any dependence that might exist between them. We propose a parametric bootstrap approach for testing the goodness-of-fit of Wehrly–Johnson distributions when the forms of their marginal and binding densities are assumed known. The approach admits the use of any test for toroidal uniformity, and we consider versions of it incorporating three such tests. Simulation is used to illustrate the operating characteristics of the approach when the underlying distribution is assumed to be bivariate wrapped Cauchy. An analysis of wind direction data recorded at a Texan weather station illustrates the use of the proposed goodness-of-fit testing procedure.


Bingham-type test Bivariate von Mises distribution  Bivariate wrapped Cauchy distribution Data-driven test  Rayleigh-type test Sobolev test Toroidal uniformity 



We are most grateful to Davy Paindaveine for raising an insightful question during the ADISTA14 Workshop that prompted us to reconsider how best to establish the significance of \((\hat{T}_{10},\,\hat{T}_{20}).\) Financial support for the research which led to the production of this paper was received by Pewsey from the Spanish Ministry of Science and Education and the Junta de Extremadura, and by Kato from the Japan Society for the Promotion of Science.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Mathematics Department, Escuela PolitécnicaUniversity of ExtremaduraCáceresSpain
  2. 2.Institute of Statistical MathematicsTachikawaJapan

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