# A Cramér--Wold theorem for elliptical distributions

Dia | 2023-03-31 10:30:00-03:00 |

Hora | 2023-03-31 10:30:00-03:00 |

Lugar | Facultad de Ciencias Económicas y Administración (entrada por Lauro Muller). |

### A Cramér--Wold theorem for elliptical distributions

#### Ricardo Fraiman (Udelar)

According to a well-known theorem of Cramér and Wold,

if P and Q are two Borel probability measures on R^d whose projections P_L,Q_L onto each line L in R^d satisfy P_L=Q_L, 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 P_L=Q_L for a certain set of (d^2+d)/2 lines L.

Moreover (d^2+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 Cram\'er--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.