On the maximal multivariate spacing extension and convexity tests

The notion of multivariate spacings was introduced and studied by Deheuvels, P. (1983) for data uniformly distributed on the unit cube. Later on, Janson, S. (1987) extended the results to bounded sets, and obtained a very fine result, namely, he derived the exact asymptotic distribution of the maximal spacing. These results have been very useful in many statistical applications. We extend Janson's result to the case where the data are generated from a positive, bounded support Lipchitz continuous density function, and develop a convexity test for the support of a distribution.