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( R $^{+}$) m − $^{1}$. We have 

<img src='content_image/306369.jpg'>

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 

<img src='content_image/306368.jpg'>

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 r 

<img src='content_image/306366.jpg'>

