2022/06/27 by Fraiman, Ricardo, Moreno, Leonardo, Ransford, Thomas
#60B11 (primary) #60E10 #62H15 (secondary) #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2206.13612
According to a well-known theorem of Cramér and Wold, if P and Q are two Borel probability measures on ℝd whose projections PL,QL onto each line L in ℝd satisfy PL=QL, then P=Q. Our main result is that, if P and Q are both elliptical distributions, then, to show that P=Q, it suffices merely to check that PL=QL for a certain set of (d2+d)/2 lines L. Moreover (d2+d)/2 is optimal. The class of elliptical distributions contains the Gaussian distributions as well as many other multivariate distributions of interest. Our theorem contrasts with other variants of the Cramér-Wold theorem, in that no assumption is made about the finiteness of moments of P and Q. We use our results to derive a statistical test for equality of elliptical distributions, and carry out a small simulation study of the test, comparing it with other tests from the literature. We also give an application to learning (binary classification), again illustrated with a small simulation