Let $f(x) = \log(\cosh(x))$, and define the kernel density:

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

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

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

is decreasing on $[0,\pi/2]$, and:

$$ \frac{\partial}{\partial \phi} \frac{\partial}{\partial \theta}\hspace{2pt} \log p_r(\phi;\theta) \le 0,\quad 0 \le \phi,\theta\le \mbox{$\large\frac{\pi}{2}$}$$

If we define: $$G_1(x,y) = \big(2-\tanh^2(x)-\tanh^2(y)\big)\,\big(f(x)-f(y)\big) - \tanh^2(x) + \tanh^2(y) $$ then define: $$g(x,y) = x\tanh(x) + y\tanh(y) + x^2(1-\tanh^2(x)) + y^2(1-\tanh^2(y)) $$ and: $$G_2(x,y) = x^2\tanh^2(x) - y^2\tanh^2(y) - g(x,y)\big(f(x)-f(y)\big) $$

then the result will be true if, for $x>y>0$:

$$G_1(x,y) > 0 \quad \text{and} \quad G_2(x,y) > 0 $$

$G_1(x,y)$ and $G_2(x,y)$ are actually monotonic in $x$ and $y$. Since it is 0 when $x=y$ it suffices to show that the derivative with respect to $x$ is non-negative for $x,y \ge 0$.

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 variation diminishing property of MLR densities.