English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Gaussian errors are sometimes inappropriate in a multivariate linearregression setting because, for example, the data contain outliers. In suchsituations, it is often assumed that the error density is a scale mixture ofmultivariate normal densities that takes the form $f(\varepsilon) =\int_0^\infty |\Sigma|^{-\frac{1}{2}} u^{\frac{d}{2}} \, \phi_d \big(\Sigma^{-\frac{1}{2}} \sqrt{u} \, \varepsilon \big) \, h(u) \, du$, where $d$is the dimension of the response, $\phi_d(\cdot)$ is the standard $d$-variatenormal density, $\Sigma$ is an unknown $d \times d$ positive definite scalematrix, and $h(\cdot)$ is some fixed mixing density. Combining this alternativeregression model with a default prior on the unknown parameters results in ahighly intractable posterior density. Fortunately, there is a simple dataaugmentation (DA) algorithm and a corresponding Haar PX-DA algorithm that canbe used to explore this posterior. This paper provides conditions (on $h$) forgeometric ergodicity of the Markov chains underlying these Markov chain MonteCarlo (MCMC) algorithms. These results are extremely important from a practicalstandpoint because geometric ergodicity guarantees the existence of the centrallimit theorems that form the basis of all the standard methods of calculatingvalid asymptotic standard errors for MCMC-based estimators. The main result isthat, if $h$ converges to 0 at the origin at an appropriate rate, and$\int_0^\infty u^{\frac{d}{2}} \, h(u) \, du < \infty$, then the DA and HaarPX-DA Markov chains are both geometrically ergodic. This result is quitefar-reaching. For example, it implies the geometric ergodicity of the DA andHaar PX-DA Markov chains whenever $h$ is generalized inverse Gaussian,log-normal, inverted gamma (with shape parameter larger than $d/2$), orFr\'{e}chet (with shape parameter larger than $d/2$).

Convergence Analysis of the Data Augmentation Algorithm for Bayesian Linear Regression with Non-Gaussian Errors