This question is based on the question which I asked before:
For which pair of probability density functions, $L=f_1(x)/f_0(x)$ is increasing?
however I didnt get any answer other than the Gaussian density and I have the conjecture that perhaps only the Gaussian density provides increasing likelihood ratio.
We know that $y\in(-\infty,\infty)$ for $f_0(y)$ and $f_1(y)$ and both densities are non zero on the domain where they are defined. $L(x)=\frac{f_1(x)}{f_0(x)}$ should be increasing.
I will be very happy to see that my conjecture is incorrect.
Thank you very much in advance.
EDIT: The densities should not be in a partial function. For example:
$\frac{1}{\mu^+-\mu^-}e^{\frac{-x}{\mu^+}}\quad for\quad x\geq 0$ $\frac{1}{\mu^+-\mu^-}e^{\frac{-x}{\mu^-}}\quad for\quad x< 0$
for $\mu^+>0$ and $\mu^-<0$ is a valid density with increasing $L(x)$