Proof of the conjectures of H. Uhlig on the singular multivariate beta and the Jacobian of a certain matrix transformation. (English) Zbl 0881.62058

Díaz-García, José A.; Gutiérrez Jáimez, Ramón

Ann. Stat. 25, No. 5, 2018-2023 (1997).

Summary: H. Uhlig [ibid. 22, No. 1, 395-405 (1994; Zbl 0795.62052)] proposes two conjectures. The first concerns the Jacobian of the transformation \(Y=B \times B'\) where \(B\) is the matrix \(m \times m\) and \(m\) and \(X\), \(Y\) belong to the class of positive semidefinite matrices of the order of \(m \times m\) of rank \(n<m\), \(S^+_{m,n}\). The second is concerned with the singular multivariate beta distribution.
This article seeks to prove the two conjectures. The latter result is then extended to the case of the singular multivariate \(F\) distribution, and the respective density functions are located for the nonzero positive eigenvalues of the singular Beta and \(F\) matrices.

MSC:

62H05	Characterization and structure theory for multivariate probability distributions; copulas
62H10	Multivariate distribution of statistics
15A99	Basic linear algebra

Keywords:

Wishart distribution; singular matrix distributions; beta distribution

Citations:

Zbl 0795.62052