Linear and Non-Linear Regression Models Assuming a Stable Distribution

Modelos de regresión lineal y no lineal suponiendo una distribución estable

In this paper, we present some computational aspects for a Bayesian
analysis involving stable distributions. It is well known that, in general, there is no closed form for the probability density function of a stable distribution. However, the use of a latent or auxiliary random variable facilitates obtaining any posterior distribution when related to stable distributions. To show the usefulness of the computational aspects, the methodology is applied to linear and non-linear regression models. Posterior summaries of interest are obtained using the OpenBUGS software.

En este trabajo, presentamos algunos aspectos computacionales de análisis bayesiano con distribuciones estables. Es bien sabido que, en general, no hay forma cerrada para la función de densidad de probabilidad de distribuciones estables. Sin embargo, el uso de una variable aleatoria latente facilita obtener la distribución a posteriori. La metodología se aplica a regresión lineal y non lineal utilizando el software OpenBUGS.

Linear and Non-Linear Regression Models Assuminga Stable Distribution Modelos de regressión lineal y no linealsuponiendo una distribución estable JORGE A. ACHCAR¹, SÍLVIA R. C. LOPES²

¹Medical School-USP, Ribeirão Preto, Brazil. Professor. Email: achcar@fmrp.usp.br
²Mathematics Institute-UFRGS, Porto Alegre, Brazil. Professor. Email: silvia.lopes@ufrgs.br

---

Abstract

Key words: Stable Laws, Bayesian Analysis, Mcmc Methods, OpenBUGS Software.

---

Resumen

En este trabajo, presentamos algunos aspectos computacionales de análisis bayesiano con distribuciones estables. Es bien sabido que, en general, no hay forma cerrada para la función de densidad de probabilidad de distribuciones estables. Sin embargo, el uso de una variable aleatoria latente facilita obtener la distribución a posteriori. La metodologia se aplica a regresión lineal y non lineal utilizando el software OpenBUGS.

Palabras clave: leyes estable, análisis bayesiano, métodos MCMC, software OpenBUGS.

---

Texto completo disponible en PDF

---

References

1. Achcar, J., Achcar, A. & Martinez, E. (2013), 'Robust linear regression models: use of a stable distribution for the response data', Open Journal of Statistics 3, 409-416.

2. Achcar, J., Lopes, S., Mazucheli, J. & Linhares, R. (2013), 'A bayesian approach for stable distributions: some computational aspects', Open Journal of Statistics 3, 268-277.

3. Bache, C., Serum, J., Youngs, W. & Lisk, D. (1972), 'Polychlorinated bibhenyl residues: accumulation in cayuga lake trout with age', Science 117, 1192-1193.

4. Bates, D. & Watts, D. (1988), Nonlinear regression analysis and its applications, Wiley, New York.

5. Buckle, D. (1995), 'Bayesian inference for stable distributions', Journal of the American Statistical Association 90, 605-613.

6. Damien, P., Wakefield, J. & Walker, S. (1999), 'Gibbs sampling for bayesian non-conjugate and hierarchical models by using auxiliary variables', Journal of the Royal Statistical Society. Series B 61, 331-344.

7. Draper, N. & Smith, H. (1981), Applied regression analysis, Wiley, New York.

8. Gnedenko, B. & Kolmogorov, A. (1968), Limit distributions for sums of independent random variables, Addison-Wesley, Massachussetts.

9. Ibragimov, I. & Chernin, K. (1959a), 'On the unimodality of stable laws', Teoriya Veroyatnostei i ee Primeneniya 4, 453-456.

10. Ibragimov, I. & Chernin, K. (1959b), 'On the unimodality of stable laws', Teoriya Veroyatnostei i ee Primeneniya 4, 453-456.

11. Johnson, R. & Bhattacharyya, G. (1980), Statistical Principles and Methods, 1 edn, John Wiley, New York.

12. Kanter, M. (1976), 'On the unimodality of stable densities', Annals of Probability 4, 1006-1008.

13. Léevy, P. (1924), 'Théeorie des erreurs la loi de gauss et les lois exceptionelles', Bulletin Society Mathematical 52, 49-85.

14. Lukacs, E. (1970), Characteristic Functions, Hafner Publishing, New York.

15. Nolan, J. (2015), Stable Distributions - Models for Heavy Tailed Data, Birkhäuser, Boston.

16. Ratkowsky, D. (1983), Nonlinear regression modelling: a unified practical approach, Marcel Dekker, Boston.

17. Samorodnitsky, G. & Taqqu, M. (1994), Stable Non-Gaussian Random Processes, Chapman & Hall, New York.

18. Seber, G. & Lee, A. (2003), Linear regression analysis, Wiley, New York.

19. Seber, G. & Wild, C. (1989), Nonlinear regression, Wiley, New York.

20. Skorohod, A. (1961), On a theorem concerning stable distributions, 'Selected Translations in Mathematical Statistics and Probability', Vol. 1, Institute of Mathematical Statistics and American Mathematical Society, , , Providence, Rhode Island.

21. Spiegelhalter, D., Best, N., Carlin, B. & van der Linde, A. (2002), 'Bayesian measures of model complexity and fit', Journal of the Royal Statistical Society. Series B 64(4), 583-639.

22. Spiegelhalter, D., Thomas, A., Best, N. & Lunn, D. (2003), WinBUGS User's Manual, MRC Biostatistics Unit, Cambridge.

23. Tanner, M. & Wong, W. (1987), 'The calculation of posterior distributions by data augmentation', Journal of American Statistical Association 82, 528-550.

---

[Recibido en octubre de 2014. Aceptado en marzo de 2015]

---

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv39n1a08,
    AUTHOR  = {Achcar, Jorge A. and Lopes, SÍlvia R. C.},
    TITLE   = {{Linear and Non-Linear Regression Models Assuminga Stable Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2016},
    volume  = {39},
    number  = {1},
    pages   = {109-128}
}

---