(Asking a final time in a new question because the previous version had insufficient conditions, as pointed out in the answer there.)

Define the densities:

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

where $f(x)=g(x^2)$, and the following hold: $g'(x)>0$, $g''(x)>0$, $g'''(x)> 0$, $g'''(x)$ is non-decreasing on $(0,\infty)$, and $f'(0^+) < \infty$. Remarkably, the area of these densities is independent of $\theta$ whenever $g$ is increasing and concave or convex (without requiring the monotonicity of $g''(x)$), which can be shown using an integral representation. For $f(x)=|x|$, we have $\int_0^{\pi/2} p(\phi;\theta,r)d\phi=\frac{2}{3}r$.

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

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

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

Examples of functions are $f(x) = |x|^p$ for $p>2$ and $f(x)=\cosh(x)$.

This result is important to prove uniqueness of stable optima in the unmixing and deconvolution of linear mixtures of independent random variables with strongly sub- and super-gaussian densities, using the Karlin-Rubin theorem. The result can be proved for $f(x)=x^4$ by simplifying the derivative expression. This corresponds to using kurtosis as the the cost function, and the uniqueness result is already known in this case. For $f(x) = |x|$, the likelihood ratio is non-increasing, constant around $\phi=0$ and $\phi=\pi/2$.

(Asking a final time in a new question because the previous version had insufficient conditions, as pointed out in the answer there.)

Define the densities:

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

where $f(x)=g(x^2)$, with $g'(x)>0$, $g''(x)>0$, $g'''(x)> 0$, and $g'''(x)$ monotonic on $(0,\infty)$. Remarkably, the area of these densities is independent of $\theta$ whenever $g$ is increasing and convex, which can be shown using an integral representation.

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

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

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

Examples of functions are $f(x) = |x|^p$ for $p>2$ and $f(x)=\cosh(x)$.

This result is important to prove uniqueness of stable optima in the unmixing and deconvolution of linear mixtures of independent random variables with strongly sub- and super-gaussian densities, using the Karlin-Rubin theorem. The result can be proved for $f(x)=x^4$ by simplifying the derivative expression. This corresponds to using kurtosis as the the cost function, and the uniqueness result is already known in this case. For $f(x) = |x|$, the likelihood ratio is non-increasing, constant around $\phi=0$ and $\phi=\pi/2$.