Smallest Pareto confidence regions and applications

Available joint confidence sets for the parameters of the Pareto model are not the regions with minimum area. In order to determine the smallest joint confidence region among all those which are based on the minimal sufficient statistic, a computational procedure is proposed which is applicable even when some of the smallest and largest observations have been discarded or censored; i.e., both single (right or left) and double censoring are allowed. The smallest Pareto region is determined by using iterative linear interpolation, as well as numerical integration and optimization methods. A few iterations are often enough to achieve the optimal solution. The reduction in area of the smallest confidence regions with respect to the existing sets is substantial in most situations, and enormous in some cases. Applications of the present approach include uses in estimation and hypothesis testing. In particular, it permits to construct confidence intervals for functions of the Pareto parameters, as well as pointwise and simultaneous confidence bands for the Pareto distribution function. Data sets concerning component lifetimes, fire claims and business failures are studied for illustrative and comparative purposes.