Robust Linear Regression Models: Use of a Stable Distribution for the Response Data

ABSTRACT

In this paper, we study some robustness aspects of linear regression models of the presence of outliers or discordant observations considering the use of stable distributions for the response in place of the usual normality assumption. It is well known that, in general, there is no closed form for the probability density function of stable distributions. However, under a Bayesian approach, the use of a latent or auxiliary random variable gives some simplification to obtain any posterior distribution when related to stable distributions. To show the usefulness of the computational aspects, the methodology is applied to two examples: one is related to a standard linear regression model with an explanatory variable and the other is related to a simulated data set assuming a 2^{3} factorial experiment. Posterior summaries of interest are obtained using MCMC (Markov Chain Monte Carlo) methods and the OpenBugs software.

KEYWORDS

Stable Distribution; Bayesian Analysis; Linear Regression Models; MCMC Methods; OpenBugs Software

Stable Distribution; Bayesian Analysis; Linear Regression Models; MCMC Methods; OpenBugs Software

Cite this paper

J. Achcar, A. Achcar and E. Martinez, "Robust Linear Regression Models: Use of a Stable Distribution for the Response Data,"*Open Journal of Statistics*, Vol. 3 No. 6, 2013, pp. 409-416. doi: 10.4236/ojs.2013.36048.

J. Achcar, A. Achcar and E. Martinez, "Robust Linear Regression Models: Use of a Stable Distribution for the Response Data,"

References

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http://dx.doi.org/10.1080/01621459.1995.10476553

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http://dx.doi.org/10.1214/aop/1176995944

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http://dx.doi.org/10.1111/1467-9868.00179

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http://dx.doi.org/10.1080/01621459.1987.10478458

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http://dx.doi.org/10.4236/ojs.2013.34031

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http://dx.doi.org/10.1007/978-1-4757-4145-2

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[1] D. J. Buckle, “Bayesian Inference for Stable Distributions,” Journal of the American Statistical Association, Vol. 90, 1995, pp. 605-613.

http://dx.doi.org/10.1080/01621459.1995.10476553

[2] P. Lévy, “Théorie des Erreurs la loi de Gauss et les Lois Exceptionelles,” Bulletin Society Mathematical, Vol. 52, 1924, pp. 49-85.

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

[4] J. P. Nolan, “Stable Distributions—Models for Heavy Tailed Data,” Birkhauser, Boston, 2009. In Progress, Chapter 1. academic2.american.edu/~jpnolan

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

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

[7] I. A. Ibragimov and K. E. Cernin, “On the Unimodality of Stable Laws,” Teoriya Veroyatnostei i ee Primeneniya, Vol. 4, No. 1, 1959, pp. 453-456.

[8] M. Kanter, “On the Unimodality of Stable Densities,” Annals of Probability, Vol. 4, No. 6, 1976, pp. 1006-1008.

http://dx.doi.org/10.1214/aop/1176995944

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

[10] G. G. Roussas, “Measure-Theoretic Probability,” Elsevier, San Diego, 2005.

[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.

http://dx.doi.org/10.1111/1467-9868.00179

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

http://dx.doi.org/10.1080/01621459.1987.10478458

[13] J. A. Achcar, S. R. C. 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.

http://dx.doi.org/10.4236/ojs.2013.34031

[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.

http://dx.doi.org/10.1007/978-1-4757-4145-2

[16] N. R. Draper and H. Smith, “Applied Regression Analysis,” Wiley Series in Probability and Mathematical Statistics, 1981.

[17] G. A. F. Seber and A. J. Lee, “Linear Regression Analysis,” 2nd Edition, Wiley Series in Probability and Mathematical Statistics, 2003.

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