( R $^{+}$) m − $^{1}$. We have where ˜ γ depends only on n and p ; Π = diag ( (1 + h$_{1}$ ) − 1 , ..., (1 + h$_{r}$ ) − 1 , 1 , ..., 1 ) ; X = diag ( x$_{1}$, ..., x$_{m}$, 0 , ..., 0) is a ( p × p ) diagonal matrix, so that there are no zeros along the diagonal if m = p ; U ( p ) is the set of all p × p unitary matrices; and (d G ) is the invariant measure on the unitary group U ( p ) normalized to make the total measure unity. Formula (2) is a special case of the densities given in James (1964, p.489) for n ≥ p and in Ratnarajah and Vaillancourt (2005) for n < p . Let s = x$_{1}$ + ... + x$_{m}$ and let y$_{j}$ = x$_{j}$/s. Note that the Jacobian of the coordinate change from ( x$_{1}$, ..., x$_{m}$ ) to ( y$_{1}$, ..., y$_{m}$$_{−}$$_{1}$, s ) equals s m − $^{1}$. Changing variables in (2) and integrating s out, we obtain where Y = diag ( y$_{1}$, ..., y$_{m}$$_{−}$$_{1}$, 0 , ..., 0) is a ( p × p ) diagonal matrix. Consider the likelihood ratios: L ( h ; λ ) = p$_{λ}$ ( λ ; h ) /p$_{λ}$ ( λ ; 0) and L ( h ; µ ) = p$_{µ}$ ( µ ; h ) /p$_{µ}$ ( µ ; 0). Formulae (2) and (3) imply the following Proposition. Proposition 1 Let U ( p ) be the set of all p × p unitary matrices. Denote by (d G ) the invariant measure on the unitary group U ( p ) normalized to make the total measure unity. Further, let Λ = diag ( λ$_{1}$, ..., λ$_{p}$ ) and M = diag ( µ$_{1}$, ..., µ$_{p}$ ) . Then L ( h ; λ ) = r ∏ i =1 (1 + h$_{i}$ ) − n ∫ U ( p ) e − n tr ( (Π − I ) G Λ G − 1 ) (d G ) and (4) L ( h ; µ ) = ∏ r i $_{=1}$(1 + h$_{i}$ ) − n n np Γ ( np ) ∫ ∞ 0 s np − $^{1}$e − ns ∫ U ( p ) e − ns tr ( (Π − I ) GMG − 1 ) (d G ) d s. (5) 8 p$_{µ}$ ( y ; h ) = ˜ γ ∏ m i =1 y | p − n | i ∏ m i<j ( y$_{i}$ − y$_{j}$ ) 2 ∏ r i $_{=1}$(1 + h$_{i}$ ) n ∫ ∞ 0 s np − 1 ∫ U ( p ) e − ns tr ( Π G Y G − 1 ) (d G ) d s, (3) p$_{λ}$ ( x ; h ) = ˜ γ ∏ m i =1 x | p − n | i ∏ m i<j ( x$_{i}$ − x$_{j}$ ) 2 ∏ r i $_{=1}$(1 + h$_{i}$ ) n ∫ U ( p ) e − n tr ( Π G X G − 1 ) (d G ) , (2)