A note about calibrated prediction regions and distributions

In this work we address the problem of the construction of prediction regions and distribution functions, with particular regard to the multidimensional setting. Firstly,\ud we define a simple procedure for calculating the predictive distribution function which\ud gives improved prediction limits. Secondly, with a multivariate generalization of a\ud result presented in Ueki and Fueda(2007), we propose a method for correcting\ud estimative prediction regions, to reduce their coverage error to the third-order\ud accuracy. The improved prediction regions and the associated distribution functions\ud are easy to calculate using a suitable bootstrap procedure. Examples of application are\ud included, showing the good performance of the proposed method, even if we consider\ud an approximated model for prediction purposes