Student-t model

Description

Linear regression with Student-t distributed residuals is also called robust regression.

Definition

For continuous unbounded outcome \(y\) and predictors \(x\), the model is:

\[ \begin{align} y_i &\sim \text{StudentT}(\nu, \eta_i, \sigma) \\ \eta_i &= \alpha + \beta \cdot x_i \end{align} \]

Parameters needing priors:

• \(\alpha\) (intercept)
• \(\beta\) (predictor weights)
• \(\sigma\) (resdiual scale)
• \(\nu\) (degrees of freedom)

Prior for \(\alpha\)

Prior for \(\beta\)

Prior for \(\nu\)

Weakly informative gamma prior

A gamma prior that has increasing density from zero to ~30 was analysed and suggested by Juárez and Steel (2010).

\[ \nu \sim \text{gamma}(2, 0.1) \]

Penalized complexity prior

Simpson et al. (2017)

See also

• Distribution explorer
• PreliZ
• Stan Functions Reference
• bambi example

References

Juárez, Miguel A., and Mark F. J. Steel. 2010. “Model-Based Clustering of Non-Gaussian Panel Data Based on Skew- t Distributions.” Journal of Business & Economic Statistics 28 (1): 52–66. https://doi.org/10.1198/jbes.2009.07145.

Simpson, Daniel, Håvard Rue, Andrea Riebler, Thiago G. Martins, and Sigrunn H. Sørbye. 2017. “Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors.” Statistical Science 32 (1): 1–28. https://doi.org/10.1214/16-STS576.