Wrapped Skew Laplace Distribution on Integers:A New Probability Model for Circular Data

ABSTRACT

In this paper we propose a new family of circular distributions, obtained by wrapping discrete skew Laplace distribution on Z = 0, ±1, ±2, around a unit circle. In contrast with many wrapped distributions, here closed form expressions exist for the probability density function, the distribution function and the characteristic function. The properties of this new family of distribution are studied.

In this paper we propose a new family of circular distributions, obtained by wrapping discrete skew Laplace distribution on Z = 0, ±1, ±2, around a unit circle. In contrast with many wrapped distributions, here closed form expressions exist for the probability density function, the distribution function and the characteristic function. The properties of this new family of distribution are studied.

Cite this paper

K. Jayakumar and S. Jacob, "Wrapped Skew Laplace Distribution on Integers:A New Probability Model for Circular Data,"*Open Journal of Statistics*, Vol. 2 No. 1, 2012, pp. 106-114. doi: 10.4236/ojs.2012.21011.

K. Jayakumar and S. Jacob, "Wrapped Skew Laplace Distribution on Integers:A New Probability Model for Circular Data,"

References

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[2] K. V. Mardia, “Statistics of Directional Data,” Academic Press, London, 1972.

[3] R. A. Fisher, “Dispersion on a Sphere,” Proceedings of the Royal Society of London A, Vol. 217, No. 1130, 1953, pp. 295-305. doi:10.1098/rspa.1953.0064

[4] J. A. Greenwood and D. Durand, “The Distribution of Length and Components of the Sum of n Random Unit Vectors,” Annals of Mathematical Statistics, Vol. 26, No. 2, 1955, pp. 233-246. doi:10.1214/aoms/1177728540

[5] G. S. Watson and E. J. William, “On the Construction of Significance Tests on the Circle and the Sphere,” Biometrika, Vol. 43, 1956, pp. 344-352.

[6] A. Pewsey, “A Wrapped Skew—Normal Distribution on the Circle,” Communications in Statistics: Theory and Methods, Vol. 29, No. 11, 2000, pp. 2459-2472. doi:10.1080/03610920008832616

[7] S. R. Jammalamadaka and T. J. Kozubowski, “A Wrapped Exponential Circular Model,” Proceedings of the Andhra Pradesh Academy of Sciences, Vol. 5, 2001, pp. 43-56.

[8] S. R. Jammalamadaka and T. J. Kozubowski, “A New Family of Circular Models: The Wrapped Laplace Distributions,” Advances and Applications in Statistics, Vol. 3, No. 1, 2003, pp. 77-103.

[9] S. R. Jammalamadaka and A. Sengupta, “Topics in Circular Statistics,” World Scientific, Singapore, 2001. doi:10.1142/9789812779267

[10] P. E. Jupp and K. V. Mardia, “A Unified View of the Theory of Directional Statistics, 1975-1988,” International Statistical Review, Vol. 57, No. 3, 1989, pp. 261- 294. doi:10.2307/1403799

[11] S. Inusah and T. J. Kozubowski, “A Discrete Analogue of the Laplace Distribution,” Journal of Statistical Planning and Inference, Vol. 136, No. 3, 2006, pp. 1090-1102. doi:10.1016/j.jspi.2004.08.014

[12] A. W. Kemp, “Characterization of a Discrete Normal Distribution,” Journal of Statistical Planning and Inference, Vol. 63, No. 2, 1997, pp. 223-229. doi:10.1016/S0378-3758(97)00020-7

[13] P. J. Szablowski, “Discrete Normal Distribution and Its Relationship with Jacobi Theta Functions,” Statistics and Probability Letters, Vol. 52, No. 3, 2001, pp. 289-299. doi:10.1016/S0167-7152(00)00223-6

[14] W. Feller, “An Introduction to Probability Theory and Its Applications,” John Wiley & Sons, New York, 1957.

[1] K. V. Mardia and P. E. Jupp, “Directional Statistics,” 2nd Edition, Wiley, New York, 2001.

[2] K. V. Mardia, “Statistics of Directional Data,” Academic Press, London, 1972.

[3] R. A. Fisher, “Dispersion on a Sphere,” Proceedings of the Royal Society of London A, Vol. 217, No. 1130, 1953, pp. 295-305. doi:10.1098/rspa.1953.0064

[4] J. A. Greenwood and D. Durand, “The Distribution of Length and Components of the Sum of n Random Unit Vectors,” Annals of Mathematical Statistics, Vol. 26, No. 2, 1955, pp. 233-246. doi:10.1214/aoms/1177728540

[5] G. S. Watson and E. J. William, “On the Construction of Significance Tests on the Circle and the Sphere,” Biometrika, Vol. 43, 1956, pp. 344-352.

[6] A. Pewsey, “A Wrapped Skew—Normal Distribution on the Circle,” Communications in Statistics: Theory and Methods, Vol. 29, No. 11, 2000, pp. 2459-2472. doi:10.1080/03610920008832616

[7] S. R. Jammalamadaka and T. J. Kozubowski, “A Wrapped Exponential Circular Model,” Proceedings of the Andhra Pradesh Academy of Sciences, Vol. 5, 2001, pp. 43-56.

[8] S. R. Jammalamadaka and T. J. Kozubowski, “A New Family of Circular Models: The Wrapped Laplace Distributions,” Advances and Applications in Statistics, Vol. 3, No. 1, 2003, pp. 77-103.

[9] S. R. Jammalamadaka and A. Sengupta, “Topics in Circular Statistics,” World Scientific, Singapore, 2001. doi:10.1142/9789812779267

[10] P. E. Jupp and K. V. Mardia, “A Unified View of the Theory of Directional Statistics, 1975-1988,” International Statistical Review, Vol. 57, No. 3, 1989, pp. 261- 294. doi:10.2307/1403799

[11] S. Inusah and T. J. Kozubowski, “A Discrete Analogue of the Laplace Distribution,” Journal of Statistical Planning and Inference, Vol. 136, No. 3, 2006, pp. 1090-1102. doi:10.1016/j.jspi.2004.08.014

[12] A. W. Kemp, “Characterization of a Discrete Normal Distribution,” Journal of Statistical Planning and Inference, Vol. 63, No. 2, 1997, pp. 223-229. doi:10.1016/S0378-3758(97)00020-7

[13] P. J. Szablowski, “Discrete Normal Distribution and Its Relationship with Jacobi Theta Functions,” Statistics and Probability Letters, Vol. 52, No. 3, 2001, pp. 289-299. doi:10.1016/S0167-7152(00)00223-6

[14] W. Feller, “An Introduction to Probability Theory and Its Applications,” John Wiley & Sons, New York, 1957.