On intersection volumes of confidence hyper-ellipsoids and two geometric Monte Carlo methods

Abstract The intersection or the overlap region of two n-dimensional ellipsoids plays an important role in statistical decision making in a number of applications. For instance, the intersection volume of two n-dimensional ellipsoids has been employed to define dissimilarity measures in time series clustering (see [M. Bakoben, T. Bellotti and N. M. Adams, Improving clustering performance by incorporating uncertainty, Pattern Recognit. Lett. 77 2016, 28–34]). Formulas for the intersection volumes of two n-dimensional ellipsoids are not known. In this article, we first derive exact formulas to determine the intersection volume of two hyper-ellipsoids satisfying a certain condition. Then we adapt and extend two geometric type Monte Carlo methods that in principle allow us to compute the intersection volume of any two generalized convex hyper-ellipsoids. Using the exact formulas, we evaluate the performance of the two Monte Carlo methods. Our numerical experiments show that sufficiently accurate estimates can be obtained for a reasonably wide range of n, and that the sample-mean method is more efficient. Finally, we develop an elementary fast Monte Carlo method to determine, with high probability, if two n-ellipsoids are separated or overlap.