Adaptive Polar Sampling (APS) algorithms are proposed for Bayesian analysis of models with
nonelliptical, possibly, multimodal posterior distributions. A location-scale transformation
and a transformation to polar coordinates are used. After the transformation to polar
coordinates, a Metropolis-Hastings method or, alternatively, an importance sampling
method is applied to sample directions and, conditionally on these, distances are
generated by inverting the cumulative distribution function. A sequential procedure is
applied to update the initial location and scaling matrix in order to sample directions
in an efficient way. Tested on a set of canonical mixture models that feature multimodality,
strong correlation, and skewness, the APS algorithms compare favourably with the standard
Metropolis-Hastings and importance samplers in terms of flexibility and robustness. APS is
applied to several econometric and statistical examples. The empirical results for a
regression model with scale contamination, an ARMA-GARCH-Student t model with near
cancellation of roots and heavy tails, a mixture model for economic growth, and a
nonlinear threshold model for industrial production growth confirm the practical
flexibility and robustness of APS
