Sine-Cosine Weighted Circular Distributions
Abstract
This paper introduces a new family of multimodal and skew-symmetric circular distributions, namely, the sine-cosine weighted circular distribution. The fundamental properties of this family are examined in the context of a general case and three specific examples. Additionally, general solutions for estimating the parameters of any sine-cosine weighted circular distribution using maximum likelihood are provided. A likelihood-ratio test is performed to check the symmetry of the data. Lastly, two examples are presented that illustrate how the proposed model may be utilized to analyze two real-world case studies with asymmetric datasets.References
Abe, T. and Pewsey, A. (2011). Sine-skewed circular distributions. Statistical Papers, 52: 683-707.
Abe, T., Pewsey, A. and Shimizu, K. (2013).Extending circular distributions through transformation of argument. Annals of the Institute of Statistical Mathematics, 65: 833858.
Abramowitz, M. and Stegun, I.A. (Eds.) (1972). Handbook of Mathematical Functions, Dover, New York.
Agostinelli, C. (2007). Robust estimation for circular data. Computational Statistics and Data Analysis, 51(12): 5867-5875.
Ameijeiras-Alonso, J. and Ley, C. (2020). Sine-skewed toroidal distributions and their application in protein bioinformatics. Biostatistics, DOI: 10.1093/biostatistics/kxaa039.
Azzalini, A. (1985). A class of distribution which includes the normal ones. Scandinavian Journal of Statistics, 12: 171178.
Azzalini, A. (2005). The skew-normal distribution and related multivariate families. Scandinavian Journal of Statistics, 32: 159188.
Batschelet, E. (1981). Circular Statistics in Biology, Academic Press, London.
Fisher, R.A. (1934). The effects of methods of ascertainment upon the estimation of frequencies. Annals of Eugenic, 6: 13-25.
Gatto, R. and Jammalamadaka, S.R. (2007). The generalized von Mises distribution. Stat Method, 4(93): 341-353.
Ghalanos, A. and S. Theussl (2015). Rsolnp: General Non-linear Optimization Using Augmented Lagrange Multiplier Method. R package version 1.
Jammalamadaka, S.R. and Kozubowski, T.J. (2003). A new family of circular models: The wrapped Laplace distributions. Advances and Applications in Statistics, 3(1): 77-103.
Jammalamadaka, S.R. and SenGupta, A. (2001). Topics in Circular Statistics, World Scientific, Singapore.
Jones, M. C. and Pewsey, A. (2005). A family of symmetric distributions on the circle. Journal of the American Statistical Association, 100: 14221428.
Kato, S. and Jones, M. C. (2010). A family of distributions on the circle with links to, and applications arising from, Moebius transformation. Journal of the American Statistical Association, 105: 249262.
Kato, S. and Jones, M. C. (2013). An extended family of circular distributions related to wrapped Cauchy distributions via Brownian motion. Bernoulli, 19: 154171.
Kato, S. and Jones, M. C. (2015). A tractable and interpretable four-parameter family of unimodal distributions on the circle. Biometrika, 102: 181190.
Maksimov, V.M. (1967). Necessary and sufficient statistics for the family of shifts of probability distributions on continuous bicompact groups. Theory of Probability and its Applications, 12: 307-321.
Mardia, K.V. (1972). Statistics of Directional Data, Academic Press, New York.
Mardia, K.V. and Jupp, P.E. (1999). Directional Statistics, Wiley, Chichester.
Patil, G. and Ord, J. (1976). On size-biased sampling and related form-invariant weighted distributions. Sankhya: The Indian Journal of Statistics, Series B, 38: 48-61.
Patil, G. and Rao, C., (1978). Weighted distributions and size-biased sampling with applications to wildlife populations and human families. Biometrics, 34: 179-189.
Pewsey, A., (2000). The wrapped skew-normal distribution on the circle. Communications in Statistics - Theory and Methods, 29: 2459-2472.
Pewsey, A., (2008). The wrapped stable family of distributions as a flexible model for circular data. Computational Statistics and Data Analysis, 52, 1516-1523.
Spurr, B.D. and Koutbeiy, M.A., (1991). A comparison of various methods of estimating the parameters in mixtures of von Mises distributions. Communications in Statistics -Simulation and Computation, 20, 725741.
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