A Bayesian Approach for Stable Distributions: Some Computational Aspects

Affiliation(s)

Medical School, USP, Ribeir?o Preto, Brazil.

Mathematical Institute, UFRGS, Porto Alegre, Brazil.

Departament of Statistics, UEM, Maringá, Brazil.

Medical School, USP, Ribeir?o Preto, Brazil.

Mathematical Institute, UFRGS, Porto Alegre, Brazil.

Departament of Statistics, UEM, Maringá, Brazil.

ABSTRACT

In this work, we study some computational aspects for the Bayesian analysis involving stable distributions. It is well known that, in general, there is no closed form for the probability density function of stable distributions. However, the use of a latent or auxiliary random variable facilitates to obtain any posterior distribution when being related to stable distributions. To show the usefulness of the computational aspects, the methodology is applied to two examples: one is related to daily price returns of Abbey National shares, considered in [1], and the other is the length distribution analysis of coding and non-coding regions in a Homo sapiens chromosome DNA sequence, considered in [2]. Posterior summaries of interest are obtained using the OpenBUGS software.

In this work, we study some computational aspects for the Bayesian analysis involving stable distributions. It is well known that, in general, there is no closed form for the probability density function of stable distributions. However, the use of a latent or auxiliary random variable facilitates to obtain any posterior distribution when being related to stable distributions. To show the usefulness of the computational aspects, the methodology is applied to two examples: one is related to daily price returns of Abbey National shares, considered in [1], and the other is the length distribution analysis of coding and non-coding regions in a Homo sapiens chromosome DNA sequence, considered in [2]. Posterior summaries of interest are obtained using the OpenBUGS software.

Cite this paper

J. Achcar, S. Lopes, J. Mazucheli and R. Linhares, "A Bayesian Approach for Stable Distributions: Some Computational Aspects,"*Open Journal of Statistics*, Vol. 3 No. 4, 2013, pp. 268-277. doi: 10.4236/ojs.2013.34031.

J. Achcar, S. Lopes, J. Mazucheli and R. Linhares, "A Bayesian Approach for Stable Distributions: Some Computational Aspects,"

References

[1] D. J. Buckle, “Bayesian Inference for Stable Distributions,” Journal of the American Statistical Association, Vol. 90, No. 430, 1995, pp. 605-613. doi:10.1080/01621459.1995.10476553

[2] N. Crato, R. R. Linhares and S. R. C. Lopes, “α-stable Laws for Noncoding Regions in DNA Sequences,” Journal of Applied Statistics, Vol. 38, No. 2, 2011, pp. 267271. doi:10.1080/02664760903406447

[3] P. Lévy, “Théorie des Erreurs la loi de Gauss et les lois exceptionelles”, Bulletin de la Société Mathématique de France, Vol. 52, No. 1, 1924, pp. 49-85.

[4] E. Lukacs, “Characteristic Functions,” Hafner Publishing, New York, 1970.

[5] J. P. Nolan, “Stable Distributions—Models for Heavy Tailed Data,” Birkhauser, Boston, 2009.

[6] B. V. Gnedenko and A. N. Kolmogorov, “Limit Distributions for Sums of Independent Random Variables,” Addison-Wesley, Massachusetts, 1968.

[7] A. V. Skorohod, “On a Theorem Concerning Stable Distributions,” In: Institute of Mathematical Statistics, Ed., Selected Translations in Mathematical Statistics and Probability, Vol. 1, 1961, pp.169-170.

[8] I. A. Ibragimov and K. E. Cernin, “On the Unimodality of Stable Laws,” Teoriya Veroyatnostei i ee Primeneniya, Vol. 4, 1959, pp. 453-456.

[9] M. Kanter, “On the Unimodality of Stable Densities,” Annals of Probability, Vol. 4, No. 6, 1976, pp.1006-1008. doi:10.1214/aop/1176995944

[10] W. Feller, “An Introduction to Probability Theory and Its Applications,” Vol. II. John Wiley, New York, 1971.

[11] P. Damien, J. Wakefield and S. Walker, “Gibbs Sampling for Bayesian Non-Conjugate and Hierarchical Models by Using Auxiliary Variables,” Journal of the Royal Statistical Society, Series B, Vol. 61, No. 2, 1999, pp. 331-344. doi:10.1111/1467-9868.00179

[12] M. A. Tanner and W. H. Wong, “The Calculation of Posterior Distributions by Data Augmen-Tation,” Journal of American Statistical Association, Vol. 82, No. 398, 1987, pp. 528-550. doi:10.1080/01621459.1987.10478458

[13] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,” National Bureau of Standards Applied Mathematics Series, Vol. 65, Dover Publications, Washington DC, 1964.

[14] S. Chib and E. Greenberger, “Understanding the Metropolis-Hastings Algorithm,” The American Statistician, Vol. 49, No. 4, 1995, pp. 327-335.

[15] C. P. Robert and G. Casella, “Monte Carlo Statistical Methods,” 2nd Edition, Springer-Verlag, New York, 2004. doi:10.1007/978-1-4757-4145-2

[16] D. J. Spiegelhalter, A. Thomas, N. G. Best and D. Lunn, “WinBUGS User’s Manual,” MRC Biostatistics Unit, Cambridge, 2003.

[1] D. J. Buckle, “Bayesian Inference for Stable Distributions,” Journal of the American Statistical Association, Vol. 90, No. 430, 1995, pp. 605-613. doi:10.1080/01621459.1995.10476553

[2] N. Crato, R. R. Linhares and S. R. C. Lopes, “α-stable Laws for Noncoding Regions in DNA Sequences,” Journal of Applied Statistics, Vol. 38, No. 2, 2011, pp. 267271. doi:10.1080/02664760903406447

[3] P. Lévy, “Théorie des Erreurs la loi de Gauss et les lois exceptionelles”, Bulletin de la Société Mathématique de France, Vol. 52, No. 1, 1924, pp. 49-85.

[4] E. Lukacs, “Characteristic Functions,” Hafner Publishing, New York, 1970.

[5] J. P. Nolan, “Stable Distributions—Models for Heavy Tailed Data,” Birkhauser, Boston, 2009.

[6] B. V. Gnedenko and A. N. Kolmogorov, “Limit Distributions for Sums of Independent Random Variables,” Addison-Wesley, Massachusetts, 1968.

[7] A. V. Skorohod, “On a Theorem Concerning Stable Distributions,” In: Institute of Mathematical Statistics, Ed., Selected Translations in Mathematical Statistics and Probability, Vol. 1, 1961, pp.169-170.

[8] I. A. Ibragimov and K. E. Cernin, “On the Unimodality of Stable Laws,” Teoriya Veroyatnostei i ee Primeneniya, Vol. 4, 1959, pp. 453-456.

[9] M. Kanter, “On the Unimodality of Stable Densities,” Annals of Probability, Vol. 4, No. 6, 1976, pp.1006-1008. doi:10.1214/aop/1176995944

[10] W. Feller, “An Introduction to Probability Theory and Its Applications,” Vol. II. John Wiley, New York, 1971.

[11] P. Damien, J. Wakefield and S. Walker, “Gibbs Sampling for Bayesian Non-Conjugate and Hierarchical Models by Using Auxiliary Variables,” Journal of the Royal Statistical Society, Series B, Vol. 61, No. 2, 1999, pp. 331-344. doi:10.1111/1467-9868.00179

[12] M. A. Tanner and W. H. Wong, “The Calculation of Posterior Distributions by Data Augmen-Tation,” Journal of American Statistical Association, Vol. 82, No. 398, 1987, pp. 528-550. doi:10.1080/01621459.1987.10478458

[13] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,” National Bureau of Standards Applied Mathematics Series, Vol. 65, Dover Publications, Washington DC, 1964.

[14] S. Chib and E. Greenberger, “Understanding the Metropolis-Hastings Algorithm,” The American Statistician, Vol. 49, No. 4, 1995, pp. 327-335.

[15] C. P. Robert and G. Casella, “Monte Carlo Statistical Methods,” 2nd Edition, Springer-Verlag, New York, 2004. doi:10.1007/978-1-4757-4145-2

[16] D. J. Spiegelhalter, A. Thomas, N. G. Best and D. Lunn, “WinBUGS User’s Manual,” MRC Biostatistics Unit, Cambridge, 2003.