Inferences based on a bivariate distribution with von Mises marginals

  • Grace S. Shieh
  • Richard A. Johnson
Directional Data


There is very little literature concerning modeling the correlation between paired angular observations. We propose a bivariate model with von Mises marginal distributions. An algorithm for generating bivariate angles from this von Mises distribution is given. Maximum likelihood estimation is then addressed. We also develop a likelihood ratio test for independence in paired circular data. Application of the procedures to paired wind directions is illustrated. Employing simulation, using the proposed model, we compare the power of the likelihood ratio test with six existing tests of independence.

Key words and phrases

Angular observations maximum likelihood estimation models of dependence power testing 


  1. Downs, T. D. (1974). Rotational angular correlation,Biorhythms, and Human Reproduction (eds. M. Ferin, F. Halberg and L. van der Wiele), 97–104, Wiley, New York.Google Scholar
  2. Fisher, N. I. (1995).Statistical Analysis of Circular Data, Cambridge University Press, Cambridge.Google Scholar
  3. Fisher, N. I. and Lee, A. J. (1982). Nonparametric measures of angular-angular correlation,Biometrika,69, 315–321.MATHCrossRefMathSciNetGoogle Scholar
  4. Fisher, N. I. and Lee, A. J. (1983). A correlation coefficient for circular data,Biometrika,70, 327–332.MATHCrossRefMathSciNetGoogle Scholar
  5. Fisher, N. I. and Lee, A. J. (1986). Correlation coefficients for random variables on a unit sphere or hypersphere,Biometrika,73, 159–164.MATHCrossRefMathSciNetGoogle Scholar
  6. Horimoto, K., Suyama, M., Toh, H., Mori, K. and Otsuka, J. (1998). A method for comparing circular genomes from gene locations: Application to mitochondrial genomes,Bioinformatics,14, 789–802.CrossRefGoogle Scholar
  7. Horimoto, K., Fukuchi, S. and Mori, K. (2001). Comprehensive comparison between locations of orthologous genes on archaeal and bacterial genomes,Bioinformatics,17, 791–802.CrossRefGoogle Scholar
  8. Johnson, R. A. and Shieh, G. S. (2002). On tests of independence for spherical data-invariance and centering.Statistics and Probability Letters,57, 327–335.MATHCrossRefMathSciNetGoogle Scholar
  9. Johnson, R. A. and Wehrly, T. (1977). Measures and models for angular correlation and angular-linear correlations,Journal of the Royal Statistical Society, Series B, Statistical Methodology,39, 222–229.MATHMathSciNetGoogle Scholar
  10. Jupp, P. J. and Mardia, K. V. (1980). A general correlation coefficient for directional data and related regression problems,Biometrika,67, 163–173.MATHCrossRefMathSciNetGoogle Scholar
  11. Jupp, P. J. and Mardia, K. V. (1989). A unified view of the theory of directional statistics,International Statistical Review,57, 1975–1988.Google Scholar
  12. MacKenzie, J. K. (1957). The estimation of an orientation relationship,Acta Crystallographica,10, 61–62.CrossRefMathSciNetGoogle Scholar
  13. Mardia, K. V. (1975a). Statistics of directional data,Journal of the Royal Statistical Society, Series B, Statistical Methodology,37, 349–393MATHMathSciNetGoogle Scholar
  14. Mardia, K. V. (1975b). Characterizations of directional distributions,Statistical Distributions in Scientific Work, 3 (eds. G. P. Patil, S. Kotz and J. K. Ord), 365–385, Reidel, Dordrecht.Google Scholar
  15. Mardia, K. V. and Jupp, P. J. (2000).Directional Statistics, John Wiley and Sons, New York.MATHGoogle Scholar
  16. Press, W. H. (1999).Numerical Recipes in Fortran, 2nd ed., Cambridge University Press, New York.Google Scholar
  17. Rivest, L.-P. (1988). A distribution for dependent unit vectors,Communications in Statistics. Theory and Methods,17, 461–483.MATHCrossRefMathSciNetGoogle Scholar
  18. Saw, J. C. (1983). Dependent unit vectors.Biometrika,70, 665–671.MATHCrossRefMathSciNetGoogle Scholar
  19. Self, S. G. and Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood, estimators and likelihood ratio tests under nonstandard conditions,Journal American Statistical Association,82, 605–610.MATHCrossRefMathSciNetGoogle Scholar
  20. Shieh, G. S., Johnson, R. A. and Frees, E. W. (1994). Testing independence of bivariate circular data and weighted degenerate U-statistics,Statistica Sinica,4, 729–747.MATHMathSciNetGoogle Scholar
  21. Shimizu, K. and Iida, K. (2002). Pearson type VII distribution on spheres,Communications in Statistics,31, 513–526.MATHMathSciNetCrossRefGoogle Scholar
  22. Singh, H., Hnizdo, V. and Demchuk, E. (2002). Probablistic model for two dependent circular variables.Biometrika,89, 719–723.MATHCrossRefMathSciNetGoogle Scholar
  23. Stephens, M. A. (1979). Vector correlation.Biometrika,66, 41–48.MATHCrossRefMathSciNetGoogle Scholar
  24. Wehrly, T. and Johnson, R. A. (1980). Bivariate models for dependence of angular observations and a related Markov process,Biometrika,67, 255–256.MATHCrossRefMathSciNetGoogle Scholar
  25. Zhou, J. L., Tits, A. L. and Lawrence, C. T. (1997). A fortran code for solving constrained nonlinear (minimax) optimization problems, generating iterates, satisfying all inequality and linear constrains, Technical Report, No. TR-92-107r2, Electrical Engineering Department and Institute for Systems Research, University of Maryland.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 2005

Authors and Affiliations

  • Grace S. Shieh
    • 1
  • Richard A. Johnson
    • 2
  1. 1.Institute of Statistical ScienceAcademia SinicaTaipeiTaiwan, R.O.C.
  2. 2.Department of StatisticsUniversity of Wisconsin-MadisonUSA

Personalised recommendations