Login | Register

On Some Circular Distributions Induced by Inverse Stereographic Projection

Title:

On Some Circular Distributions Induced by Inverse Stereographic Projection

Chaubey, Yogendra P. ORCID: https://orcid.org/0000-0002-0234-1429 and Karmaker, Shamal Chandra (2018) On Some Circular Distributions Induced by Inverse Stereographic Projection. Technical Report. Concordia University. Department of Mathematics and Statistics, Montreal, Quebec.

[img]
Preview
Text (application/pdf)
report2-18.pdf
Available under License Spectrum Terms of Access.
214kB

Abstract

In earlier studies of circular data, the corresponding probability distributions considered were mostly assumed to be symmetric. However, the assumption of symmetry may not be meaningful for some data. Thus there has been increased interest, more recently, in developing skewed circular distributions. In this article we introduce three skewed circular models based on inverse stereographic projection, originally introduced by Minh and Farnum (2003), by considering three different versions of skewed-t considered in the literature, namely skewed-t by Azzalini (1985), two-piece skewed-t (Fern´andez and Steel, 1998) and skewedt by Jones and Faddy (2003). Shape properties of the resulting circular distributions along with estimation of parameters using maximum likelihood are also discussed in this article. Further, real data sets are used to illustrate the application of the new models. It is found that Azzalini and Jones-Faddy skewed-t versions are good competitors, however, the Jones-Faddy version is computationally more tractable.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Monograph (Technical Report)
Authors:Chaubey, Yogendra P. and Karmaker, Shamal Chandra
Series Name:Department of Mathematics & Statistics. Technical Report No. 2/18
Corporate Authors:Concordia University. Department of Mathematics and Statistics
Institution:Concordia University
Date:26 April 2018
Funders:
  • Natural Sciences and Engineering Research Council
Keywords:Circular data; Skewed-t distribution; Inverse stereographic projection.
ID Code:983833
Deposited By: YOGENDRA CHAUBEY
Deposited On:07 May 2018 13:26
Last Modified:07 May 2018 13:29

References:

[1] Abe, T. and Pewsey, A. (2011). Symmetric circular models through duplication and cosine perturbation. Computational Statistics & Data Analysis, 55(12):3271–3282.
[2] Abe, T., Shimizu, K., and Pewsey, A. (2010). Symmetric unimodal models for directional data motivated by inverse stereographic projection. Journal of the Japan Statistical Society, 40(1):045–061.
[3] Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12:171–178.
[4] Azzalini, A. and Capitanio, A. (2003). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65(2):367–389.
[5] Branco, M. D. and Dey, D. K. (2001). A general class of multivariate skew-elliptical distributions. Journal of Multivariate Analysis, 79(1):99–113.
[6] Bruderer, B. and Jenni, L. (1990). Migration across the alps. In Bird Migration, pages 60–77. Springer.
[7] Cartwright, D. E. (1963). The use of directional spectra in studying the out- put of a wave recorder on a moving ship. In Ocean Wave Spectra, pages 203–218. Englewood Cliffs, NJ: Prentice-Hall.
[8] Fern´andez, C. and Steel, M. F. (1998). On Bayesian modeling of fat tails and skewness. Journal of the American Statistical Association, 93(441):359–371.
[9] Ferreira, J. T. and Steel, M. F. (2007). A new class of skewed multivariate distributions with applications to regression analysis. Statistica Sinica, pages 505–529.
[10] Fisher, N. (1993). Statistical Analysis of Circula Data. Cambridge University Press, London.
[11] Genton, M. G. (2004). Skew-elliptical distributions and their applications: a journey beyond normality. CRC Press.
[12] Holzmann, H., Munk, A., Suster, M., and Zucchini, W. (2006). Hidden markov models for circular and linear-circular time series. Environmental and Ecological Statistics, 13(3):325–347.
[13] Jammalamadaka, S. R. and SenGupta, A. (2001). Topics in Circular Statistics. World Scientific, Singapore.
[14] Johnson, N. L., Kotz, S., and Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2, 2nd Edition. John Wiley & Sons, New York.
[15] Jones, M. C. (2006). A note on rescalings, reparametrizations and classes of distributions. Journal of Statistical Planning and Inference, 136(10):3730–3733.
[16] Jones, M. C. and Faddy, M. J. (2003). A skew extension of the t−distribution, with applications. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65(1):159–174.
[17] Jones, M. C. and Pewsey, A. (2012a). Inverse batschelet distributions for circular data. Biometrics, 68(1):183–193.
[18] Jones, M. C. and Pewsey, A. (2012b). Inverse batschelet distributions for circular data. Biometrics, 68(1):183–193.
[19] Kato, S. and Jones, M. (2010). A family of distributions on the circle with links to, and applications arising from, Mobius transformation. ¨ J. Amer. Statist. Assoc., 105:249–262.
[20] Kato, S. and Jones, M. (2015). A tractable and interpretable four-parameter family of unimodal distributions on the circle. Biometrika, 102(1):181–190.
[21] Ma, Y. and Genton, M. G. (2004). Flexible class of skew-symmetric distributions. Scandinavian Journal of Statistics, 31(3):459–468.
[22] Mardia, K. (1972). Statistics of Directional Data. Academic Press, New York.
[23] Mardia, K. V. and Jupp, P. E. (2009). Directional Statistics. John Wiley & Sons.
All items in Spectrum are protected by copyright, with all rights reserved. The use of items is governed by Spectrum's terms of access.

Repository Staff Only: item control page

Back to top Back to top