 Solution

Let $a={\pi }^2({\epsilon }^{-1}-1)$. From the supplied second moment of a mixture likelihood ratio, dropping a nonpositive term givesThe last step uses the assumed separation. It has a real square root only when $16{\nu }^{2}d\ge 1$; otherwise no parameters satisfy it. The chi-squared testing lower bound follows from the Cauchy-Schwarz inequality: $\Vert P_1-P_0{\Vert }_{\mathrm{TV}}\le \frac{1}{2}\sqrt{{\chi }^2(P_1\Vert P_0)}\le 2\nu$. Every statistical hypothesis testing rule has sum of its Type I error and Type II error at least $1-\Vert P_1-P_0{\Vert }_{\mathrm{TV}}$. Its larger error is at least half the sum, hence