Statistics and Computing

, Volume 24, Issue 2, pp 165–179 | Cite as

Fitting Kent models to compositional data with small concentration



Compositional data can be transformed to directional data by the square root transformation and then modelled by using the Kent distribution. The current approach for estimating the parameters in the Kent model for compositional data relies on a large concentration assumption which assumes that the majority of the transformed data is not distributed too close to the boundaries of the positive orthant. When the data is distributed close to the boundaries with large variance significant folding may result. To treat this case we propose new estimators of the parameters derived based on the actual folded Kent distribution which are obtained via the EM algorithm. We show that these new estimators significantly reduce the bias in the current estimators when both the sample size and amount of folding is moderately large. We also propose using a saddlepoint density approximation for the Kent distribution normalising constant in order to more accurately estimate the shape parameters when the concentration is small or only moderately large.


Compositional data Directional data EM algorithm Folding Kent distribution Maximum likelihood estimation Saddlepoint approximation Square root transformation 


  1. Aitchison, J.: The statistical analysis of compositional data (with discussion). J. R. Stat. Soc., Ser. B, Stat. Methodol. 44, 139–177 (1982) MATHMathSciNetGoogle Scholar
  2. Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman and Hall, London (1986) CrossRefMATHGoogle Scholar
  3. Chen, M., Kianifard, F.: Estimation of treatment difference and standard deviation with blinded data in clinical trials. Biom. J. 45, 135–142 (2003) CrossRefMathSciNetGoogle Scholar
  4. Cuesta-Albertos, J.A., Cuevas, A., Fraiman, R.: On projection-based tests for directional and compositional data. Stat. Comput. 19, 367–380 (2009) CrossRefMathSciNetGoogle Scholar
  5. Jung, S., Foskey, M., Marron, J.S.: Principal arc analysis on direct product manifolds. Ann. Appl. Stat. 5, 578–603 (2011) CrossRefMATHMathSciNetGoogle Scholar
  6. Kent, J.T.: The Fisher-Bingham distribution on the sphere. J. R. Stat. Soc., Ser. B, Stat. Methodol. 44, 71–80 (1982) MATHMathSciNetGoogle Scholar
  7. Kent, J.T., Mardia, K.V., McDonnell, P.: The complex Bingham quartic distribution and shape analysis. J. R. Stat. Soc., Ser. B, Stat. Methodol. 68, 747–765 (2006) CrossRefMATHMathSciNetGoogle Scholar
  8. Kume, A., Walker, S.G.: Sampling from compositional and directional distributions. Stat. Comput. 16, 261–265 (2006) CrossRefMathSciNetGoogle Scholar
  9. Kume, A., Wood, A.T.A.: Saddlepoint approximations for the Bingham and Fisher-Bingham normalising constants. Biometrika 92, 465–476 (2005) CrossRefMATHMathSciNetGoogle Scholar
  10. Rivest, L.: On the information matrix for symmetric distributions on the hypersphere. Ann. Stat. 12, 1085–1089 (1984) CrossRefMATHMathSciNetGoogle Scholar
  11. Matz, A.W.: Maximum likelihood parameter estimation for the quartic exponential distribution. Technometrics 20, 475–484 (1978) CrossRefMATHGoogle Scholar
  12. McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions, 2nd edn. Wiley, New Jersey (2008) CrossRefMATHGoogle Scholar
  13. Scealy, J.L.: Modelling techniques for compositional data using distributions defined on the hypersphere. PhD thesis, Australian National University (2010) Google Scholar
  14. Scealy, J.L., Welsh, A.H.: Regression for compositional data by using distributions defined on the hypersphere. J. R. Stat. Soc., Ser. B, Stat. Methodol. 73, 351–375 (2011) CrossRefMathSciNetGoogle Scholar
  15. Stephens, M.A.: Use of the von mises distribution to analyse continuous proportions. Biometrika 69, 197–203 (1982) CrossRefMathSciNetGoogle Scholar
  16. Sundberg, R.: On estimation and testing for the folded normal distribution. Commun. Stat., Theory Methods 3, 55–72 (1974) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Australian National UniversityCanberraAustralia

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