Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series A, Pt. 1, 10-27

ON A TWO-PARAMETER FAMILY OF MULTIVARIATE UNIMODAL PROBABILITY MEASURES

By

EMILE BERTIN, *Universiteit Utrecht, Utrecht*

And

RADU THEODORESCU, *Universit¢
e Laval, Qu¢
ebec*

*
*

SUMMARY. The family of all random vectors **X **which are distributed as **U Z**, where the random variable **U** is .beta distributed and the random vector **Z** is independent of **U**, is examined within a general framework Bertain and Theodorescu( 1984) containing, e.g., versions of the theorem of Khinchin (1983) and the theorem of L¢
evy-Shepp (1962). This family leads to a new type of multivariate unimodality, providing a natural unifying background for several existing notions of unimodality.

*AMS (1991) subject classification.* 60E05, 60E10, 60E15.

*Key words and phrases*. Unimodality, generalized unimodality, theorem of Khinchin, theorem of Levy-shepp.