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A General Approach for Obtaining Wrapped Circular Distributions via Mixtures

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Abstract

We show that the operations of mixing and wrapping linear distributions around a unit circle commute, and can produce a wide variety of circular models. In particular, we show that many wrapped circular models studied in the literature can be obtained as scale mixtures of just the wrapped Gaussian and the wrapped exponential distributions, and inherit many properties from these two basic models. We also point out how this general approach can produce flexible asymmetric circular models, the need for which has been noted by many authors.

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References

  • Agiomyrgiannakis, Y. and Stylianou, Y. (2009). Wrapped Gaussian mixture models for modeling and high-rate quantization of phase data of speech. IEEE Trans. Audio, Speech, Language Process. 17, 775–786.

    Article  Google Scholar 

  • Andrews, D. F. and Mallows, C. L. (1974). Scale mixtures of normal distributions. J. Roy. Statist. Soc. Ser. B 36, 99–102.

    MathSciNet  MATH  Google Scholar 

  • Arellano-Valle, R. B., Gómez, H. W. and Quintana, F. A. (2005). Statistical inference for a general class of asymmetric distributions. J. Statist. Plann. Inference 128, 427–443.

    Article  MathSciNet  MATH  Google Scholar 

  • Ayebo, A. and Kozubowski, T. J. (2003). An asymmetric generalization of Gaussian and Laplace laws. J. Probab. Statist. Sci. 1, 187–210.

    Google Scholar 

  • Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178.

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen, O, Kent, J. and Sorensen, H. (1982). Normal variance-mean mixtures and z distributions. Internat. Statist. Rev. 50, 145–159.

    Article  MathSciNet  MATH  Google Scholar 

  • Batschelet, E. (1981). Circular Statistics in Biology. Academic, London.

    MATH  Google Scholar 

  • Branco, M. D. and Dey, D. K. (2001). A general class of multivariate skew-elliptical distributions. J. Multivariate Anal. 79, 99–113.

    Article  MathSciNet  MATH  Google Scholar 

  • Coelho, C. A. (2011). The wrapped gamma distribution and wrapped sums and linear combinations of independent gamma and Laplace distributions. J. Stat. Theory Pract. 1, 1–29.

    Article  MathSciNet  MATH  Google Scholar 

  • Fernandez, C. and Steel, M. F. J. (1998). On Bayesian modelling of fat tails and skewness. J. Amer. Statist. Assoc. 93, 359–371.

    MathSciNet  MATH  Google Scholar 

  • Fishe, N. I. (1993). Statistical Analysis of Circular Data. Cambridge University Press, Cambridge.

    Book  MATH  Google Scholar 

  • Gatto, R. and Jammalamadaka, S. (2003). Inference for wrapped α-stable circular models. Sankhya 65, 333–355.

    MathSciNet  MATH  Google Scholar 

  • Ghosh, M., Choi, K. P. and Li, J. (2010) A commentary on the logistic distribution. In The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, (K. Alladi, J. R. Klauder and C. R. Rao, eds.). pp. 351–357, Springer, New York.

  • Gleser, L. J. (1987). The gamma distribution as a mixture of exponential distributions. Technical Report No 87-28. Department of Statistics, Purdue University.

  • Gneiting, T. (1997). Normal scale mixtures and dual probability densities. J. Statist. Comput. Sim. 59, 375–384.

    Article  MATH  Google Scholar 

  • Gradshteyn, I. S. and Ryzhik, I. M. (1994). Table of Integrals, Series and Products, 5th Edn. Academic, San Diego.

    MATH  Google Scholar 

  • Jammalamadaka, S. and Kozubowski, T. J. (2001). A wrapped exponential circular model. Proc. Andhra Pradesh Academy Sci. 5, 43–56.

    Google Scholar 

  • Jammalamadaka, S. and Kozubowski, T. J. (2003). A new family of circular models: The wrapped Laplace distributions. Adv. Appl. Statist. 3, 77–103.

    MathSciNet  MATH  Google Scholar 

  • Jammalamadaka, S. and Kozubowski, T. J. (2004). New families of wrapped distributions for modeling skew circular data. Comm. Statist. Theory Methods 33, 2059–2074.

    Article  MathSciNet  MATH  Google Scholar 

  • Jammalamadaka, S. and SenGupta, A. (2001). Topics in Circular Statistics. World Scientific, Singapore.

    Book  Google Scholar 

  • Jewell, N. P. (1982). Mixtures of exponential distributions. Ann. Statist. 10, 479–484.

    Article  MathSciNet  MATH  Google Scholar 

  • Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, 1, 2nd Edn. Wiley, New York.

  • Johnson, N. L., Kotz S. and Balakrishnan, N. (1995). Continuous Univariate Distributions, 2, 2nd Edn. Wiley, New York.

  • Kato, S. and Shimizu, K. (2004). A further study of t-distributions on spheres. Technical Report School of Fundamental Science and Technology. Keio University, Yokohama.

  • Kim, H. -M. and Genton, M. G. (2011). Characteristic functions of scale mixtures of multivariate skew-normal distributions. J. Multivariate Anal. 102, 1105–1117.

    Article  MathSciNet  MATH  Google Scholar 

  • Kotz, S., Kozubowski, T.J. and Podgórski K. (2001). The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering and Finance. Birkhaüser, Boston.

    Book  MATH  Google Scholar 

  • Mardia, K. V. (1972). Statistics of Directional Data. Academic Press, London.

    MATH  Google Scholar 

  • Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics. Wiley, Chichester.

    MATH  Google Scholar 

  • Mudholkar, G. S. and Hutson, A. D. (2000). The epsilon-skew-normal distribution for analyzing near-normal data. J. Statist. Plann. Inference 83, 291–309.

    Article  MathSciNet  MATH  Google Scholar 

  • Pewsey, A. (2000). The wrapped skew-normal distribution on the circle. Comm. Statist. Theory Methods 29, 2459–2472.

    Article  MathSciNet  MATH  Google Scholar 

  • Pewsey, A. (2008). The wrapped stable family of distributions as a flexible model for circular data. Comput. Statist. Data Anal. 52, 1516–1523.

    Article  MathSciNet  MATH  Google Scholar 

  • Pewsey, A., Lewis, T. and Jones, M. C. (2007). The wrapped t family of circular distributions. Aust. N. Z. J. Statist. 49, 79–91.

    Article  MathSciNet  MATH  Google Scholar 

  • Pewsey, A., Neuhäuser, M. and Ruxton, G. D. (2013). Circular Statistics in R. Oxford University Press, Oxford.

    MATH  Google Scholar 

  • Rao, A. V. D., Sarma, I. R. and Girija, S. V. S. (2007). On wrapped version of some life testing models. Comm. Statist. Theory Methods 36, 2027–2035.

    Article  MathSciNet  MATH  Google Scholar 

  • Reed, W. J. (2007). Brownian-Laplace motion and its application in financial modeling. Comm. Statist. Theory Methods 36, 473–484.

    Article  MathSciNet  MATH  Google Scholar 

  • Reed, W. J. and Pewsey, A. (2009). Two nested families of skew-symmetric circular distributions. Test 18, 516–528.

    Article  MathSciNet  MATH  Google Scholar 

  • Sarma, I. R., Rao, A. V. D. and Girija, S. V. S. (2011). On characteristic functions of the wrapped lognormal and the wrapped Weibull distributions. J. Statist. Comput. Sim. 81, 579–589.

    Article  MathSciNet  MATH  Google Scholar 

  • Stefanski, L. A. (1991). A normal scale mixture representation of the logistic distribution. Statist. Probab. Lett. 11, 69–70.

    Article  MathSciNet  MATH  Google Scholar 

  • Steutel, F. W. and van Harn, K. (2004). Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, New York.

  • Umbach, D. and Jammalamadaka, S. (2009). Building asymmetry into circular distributions. Statist. Probab. Lett. 79, 659–663.

    Article  MathSciNet  MATH  Google Scholar 

  • West, M. (1984). Outlier models and prior distributions in Bayesian linear regression. J. Roy. Statist. Soc. Ser. B 46, 431–439.

    MathSciNet  MATH  Google Scholar 

  • West, M. (1987). On scale mixtures of normal distributions. Biometrika 74, 646–648.

    Article  MathSciNet  MATH  Google Scholar 

  • Wong, R. (1989). Asymptotic Approximations of Integrals. Academic, Boston.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics & Applied Probability, University of California, Santa Barbara, CA, 93106, USA

    S. Rao Jammalamadaka

  2. Department of Mathematics & Statistics, University of Nevada, Reno, NV, 89557, USA

    Tomasz J. Kozubowski

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  1. S. Rao Jammalamadaka
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  2. Tomasz J. Kozubowski
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Correspondence to Tomasz J. Kozubowski.

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Jammalamadaka, S.R., Kozubowski, T.J. A General Approach for Obtaining Wrapped Circular Distributions via Mixtures. Sankhya A 79, 133–157 (2017). https://doi.org/10.1007/s13171-017-0096-4

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  • DOI: https://doi.org/10.1007/s13171-017-0096-4

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