Define the family of densities:

$$p(\phi;\theta) = \Big(f\big(\hspace{-1pt}\cos(\phi-\theta)\big) - f\big(\hspace{-1pt}\cos(\phi+\theta)\big)\Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)}, \quad 0 \le \phi,\theta\le \pi/2$$

where $f(x)=g(x^2)$ with $g$ non-negative, increasing, and convex or concave, on $[0,\infty)$. (Interestingly these densities indeed have the same measure for all $\theta$ whenever $g$ is concave or convex, which can be shown by an integral representation.)

Show that $p(\phi;\theta)$ has a monotone likelihood ratio (increasing for concave $g$, decreasing for convex $g$). I.e., for $0\le\theta_1 < \theta_2\le\pi/2$:

$$ h(\phi) = \frac{f\big(\hspace{-1pt}\cos(\phi-\theta_2)\big) - f\big(\hspace{-1pt}\cos(\phi+\theta_2)\big)}{f\big(\hspace{-1pt}\cos(\phi-\theta_1)\big) - f\big(\hspace{-1pt}\cos(\phi+\theta_1)\big)}$$

is monotonic on $[0,\pi/2]$.

Examples of functions are $f(x) = |x|^p$, $1\le p<2$ or $p>2$. (For $p=2$, $p(\phi;\theta) = \sin^2(2\phi)$.) Also the function $f(x) = \log( \cosh(x))$ is concave in $x^2$ (i.e. $\log(\cosh(\sqrt{x}))$ is concave on $[0,\infty)$), and twice differentiable (unlike $|x|^p$, $p<2$).

This is important to prove uniqueness of blind source separation and deconvolution optima with strongly sub- and super-gaussian sources.