An alternative to the inverted gamma for the variances to modelling outliers and structural breaks in dynamic models. (English) Zbl 1319.62031

Fúquene, Jairo; Pérez, María-Eglée; Pericchi, Luis R.

Braz. J. Probab. Stat. 28, No. 2, 288-299 (2014).

Summary: We propose a new wide class of hypergeometric heavy tailed priors that is given as the convolution of a Student-\(t\) density for the location parameter and a Scaled Beta 2 prior for the squared scale parameter. These priors may have heavier tails than Student-\(t\) priors, and the variances have a sensible behaviour both at the origin and at the tail, making it suitable for objective analysis. Since the representation of our proposal is a scale mixture, it is suitable to detect sudden changes in the model. Finally, we propose a Gibbs sampler using this new family of priors for modelling outliers and structural breaks in Bayesian dynamic linear models. We demonstrate in a published example, that our proposal is more suitable than the Inverted Gamma’s assumption for the variances, which makes very hard to detect structural changes.

MSC:

62E15	Exact distribution theory in statistics
62E20	Asymptotic distribution theory in statistics
62F15	Bayesian inference

Keywords:

Bayesian inference; robust priors; scaled beta 2 distribution; Student \(t\) distribution; dynamic linear models; change point detection; inverted-gamma distribution

Software:

dlm

References:

[1]	Abramowitz, M. and Stegun, I., eds. (1970). Handbook of Mathematical Functions. Applied Mathematics Series 55 . Washington, DC: National Bureau of Standards; Reprinted by Dover Publications, New York. · Zbl 0171.38503
[2]	Andrade, J. and O’Hagan, A. (2005). Bayesian robustness modelling using regularly varying distributions. Bayesian Analysis 1 , 169-188. · Zbl 1331.62175
[3]	Carvalho, C. M., Polson, N. G. and Scott, J. G. (2010). The horseshoe estimator for sparse signals. Biometrika 97 , 465-480. · Zbl 1406.62021
[4]	Fruwirth-Schnatter, S. (1994). Data augmentation and dynamic linear models. Journal of Time Series Analysis 15 , 183-202. · Zbl 0815.62065 · doi:10.1111/j.1467-9892.1994.tb00184.x
[5]	Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian Analysis 1 , 515-534. · Zbl 1331.62139
[6]	Gradshteyn, I. and Ryzhink, I. (1965). Table of Integrals, Series, and Products , 4th ed. New York: Academic Press.
[7]	Pericchi, L. R. (2010). Discussion of Polson, N., and Scott, J. In Bayesian Statistics 9 . Oxford: Oxford Univ. Press.
[8]	Pericchi, L. R. and Pérez, M. E. (2009). The case for a fully robust hierarchical Bayesian analysis of clinical trials. Technical report, Centro de Bioestadística y Bioinformática, Univ. Puerto Rico, Rio Piedras Campus. Available at .
[9]	Pericchi, L. R. and Pérez, M. E. (2010). Limiting the shrinkage for the exceptional by objective robust Bayesian analysis. Technical report, Centro de Bioestadística y Bioinformática, Univ. Puerto Rico, Rio Piedras Campus. Available at .
[10]	Petris, G. (2010). An R package for dynamic linear models. Journal of Statistical Software 36 , 1-16.
[11]	Petris, G., Petrone, S. and Campagnoli, P. (2010). Dynamic Linear Models with R . New York: Springer. · Zbl 1176.62088
[12]	Prado, R. and West, M. (2010). Times Series: Modelling, Computation, and Inference . Boca Raton, FL: Chapman & Hall. · Zbl 1245.62105
[13]	Richter, W.-D. (2007). Generalized spherical and simplicial coordinates. Journal of Mathematical Analysis and Applications 336 , 1187-1202. · Zbl 1143.60017 · doi:10.1016/j.jmaa.2007.03.047
[14]	West, M. (1984). Outliers models and prior distributions in Bayesian linear regression. Journal of the Royal Statistical Society, Ser. B 46 , 431-439. · Zbl 0567.62022
[15]	West, M. and Harrison, P. J. (1997). Bayesian Forecasting and Dynamic Models . New York: Springer. · Zbl 0871.62026