We have where ˜ γ depends only on n$_{p}$ 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; O ( p ) is the set of all p × p orthogonal matrices; and (d Q ) is the invariant measure on the orthogonal group O ( p ), normalized to make the total measure unity. Formula (1) is a special case of the density given in James (1964, p.483) for n$_{p}$ ≥ p, and follows from Theorems 2 and 6 in Uhlig (1994) for n$_{p}$ < p . Let x = x$_{1}$ + ... + x$_{m}$ and let y$_{i}$ = x$_{i}$/x. Note that the Jacobian of the coordinate change from ( x$_{1}$, ..., x$_{m}$ ) to ( y$_{1}$$^{, ..., y$_{m}$}$− $_{1}$, x ) is x m − $^{1}$. Changing variables in (1) and integrating x out, we obtain where Y = diag ( y$_{1}$, ..., y$_{m}$, 0 , ..., 0) is a ( p × p ) diagonal matrix. Consider the likelihood ratios L$_{p}$ ( h ; λ$_{p}$ ) = f$_{λp}$ ( λ$_{p}$ ; h ) /f$_{λp}$ ( λ$_{p}$ ; 0) and L$_{p}$ ( h ; µ ) = f$_{µp}$ ( µ$_{p}$ ; h ) /f$_{µp}$ ( µ$_{p}$ ; 0 ) . Formulae (1) and (2) imply the following proposition. Proposition 1 Let O ( p ) be the set of all p × p orthogonal matrices. Denote by (d Q ) the invariant measure on the orthogonal group O ( p ) normalized to make the total measure unity. Put Λ$_{p}$ = diag ( λ$_{p}$$_{1}$, ..., λ$_{pp}$ ) , S$_{p}$ = λ$_{p}$$_{1}$ + ... + λ$_{pp}$, and let D$_{p}$ be the p × p diagonal matrix diag ( 1 2 c$_{p}$ h$_{1}$ 1+ $_{h$_{1}$}$, ..., 1 2 c$_{p}$ h$_{r}$ 1+ $_{h$_{r}$}$, 0 , ..., 0 ) , where c$_{p}$ = p/n$_{p}$ . f$_{µp}$ (˜ y ; h ) = ˜ γ ∏ m i =1 $_{y}$| $^{p}$−n p | − 1 2 i ∏ m i<j ( y$_{i}$ − y$_{j}$ ) ∏ r j $_{=1}$(1 + h$_{j}$ ) n$_{p}$/ 2 ∫ ∞ 0 x n p p 2 − 1 ∫ O ( p ) $_{e}$− n p 2 x tr(Π Q $^{′}$Y Q $^{)}$(d Q ) d x, (2) f$_{λp}$ (˜ x ; h ) = ˜ γ ∏ m i =1 $_{x}$| $^{p}$−n p | − 1 2 i ∏ m i<j ( x$_{i}$ − x$_{j}$ ) ∏ r j $_{=1}$(1 + h$_{j}$ ) n$_{p}$/ 2 ∫ O ( p ) $_{e}$− n p 2 tr(Π Q $^{′}$X Q $^{)}$(d Q ) , (1) 7