On Some Nonparametric Tests of Dispersion

Let X and Y be random variables with distribution functions F and G and medians ζand η, respectively. X will be said to be more dispersed about the median than Y if F(x+ζ) – F(–x+ζ) G (x+η) – G (–x+η)for all x > O, with sharp inequality holding for some positive x. Asymptotically nonparametric tests for the hypothesis that F (x+ζ) = G (x+η) against alternatives of the above type (for ζ and η unknown), provided F(x+ζ) (but not necessarily G (x+η)) is symmetric and Fand G have bounded densities, are constructed from a subclass Of SUKHATME'S modified generalized U - statistics 12]. One simple test of this subclass turns out to be the MANN-WHITNEY test as performed on the |X_i–ζ| and | Y_j–η| where ζ and η are the corresponding sample medians. Consistency and PITMAN efficiencies of some of the tests proposed are discussed under two families of alternatives in which the Y's differ from the X's through departure from symmetry as well as by being more (less) dispersed: a model of “heterogeneous response” and the wellknown gross error model. The usual scale alternatives are also studied and the relationship of our tests to those previously given in the literature is pointed out.