 Solution

Let $Q=P_1-P_0$ and $R=I-P_1$. For nested subspaces these are orthogonal projections onto ${\Theta }_1\cap {\Theta }_0^{\perp }$ and ${\Theta }_1^{\perp }$, of ranks $d_1-d_0$ and $d-d_1$. Under the null, both annihilate $\mu$. In an orthonormal basis adapted to ${\Theta }_0$, these two spaces and their complement, the coordinates of $X-\mu$ are independent $N(0,{\sigma }^2)$. Consequentlyand the two variables are independent. Equivalently the joint probability density of the two unscaled sums $a,b>0$ factors into gamma densities of respective shapes $(d-d_1)/2,(d_1-d_0)/2$ and common scale $2{\sigma }^2$.

The nuisance variance cancels inReject for a value above the upper $\alpha$ critical quantile to obtain a level-$\alpha$ test. Under a genuine alternative the numerator is noncentral chi-square with noncentrality $\Vert Q\mu {\Vert }^2/{\sigma }^2$, while the denominator retains its central law, explaining the upper-tail choice.