Calibrated multivariate distributions for improved conditional prediction

The specification of multivariate prediction regions, having coverage probability closed to the target nominal value, is a challenging problem both from the theoretical and the practical point of view. In this paper we define a well-calibrated multivariate predictive distribution giving suitable conditional prediction intervals with the desired overall coverage accuracy. This distribution is the extension in the multivariate setting of a calibrated predictive distribution defined for the univariate case and it is found on the idea of calibrating prediction regions for improving the coverage probability. This solution is asymptotically equivalent to that one based on asymptotic calculations and, whenever its explicit computation is not feasible, an approximation based on a simple bootstrap simulation procedure is readily available. Moreover, we state a simple, simulation-based, procedure for computing the associated improved conditional prediction limits.