Nonparametric integral statistics ωkn = nk/2 ∫∞−∞ [Sn(x) − F(x)]k dF(x): Main properties and applications

The main characteristics of nonparametric statistics, ωkn, which can be represented as integrals of the kth degree of the empirical process, are considered. An algebraic form of these statistics, convenient for practical usage, is derived. Tables of percentage points for k = 1,2, …, 5 and small empirical sample sizes n = 1,2, …, 10 are obtained. Goodness-of-fit criteria based on ωkn statistics are constructed, and their main properties are studied. A method for classifying multidimensional data based on these criteria, which was successfully applied in several experiments, is described.