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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 $L(x)$

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    @GautamShenoy: I need to make some experiments with the densities that I obtain. A$f$ter the likelihood ratio is inverted it should be inserted into some complicated equations and I search some solutions $f$or t$h$em. The example which I gave has two step $f$unctions and when I finally invert the likelihood from that partial function mathematica can not solve the complicated equations. However I guess it means differentiability, althouhgh i didnt check.2012-10-29

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Here is a family of solutions:

Fix any density function $f_0$. For any positive increasing function $g$ integrable with respect to the measure $f_0(x)\,\mathrm dx$, consider the density function $ f_1:x\mapsto f_0(x)g(x)/\mathbb E(g(X_0)), $ where $X_0$ denotes any random variable with density $f_0$. In other words, $\mathbb E(g(X_0))$ is the integral of $f_0g$ and $f_1$ is the density of any random variable $X_1$ such that, for every bounded measurable test function $\varphi$, $ \mathbb E(\varphi(X_1))=\frac{\mathbb E(\varphi(X_0)g(X_0))}{\mathbb E(g(X_0))}. $ Then, the pair $(f_0,f_1)$ is a solution.

Naturally, this is barely more than a reformulation of the question, however it shows the diversity of the solutions $f_1$, for any given density function $f_0$.

Assume for concreteness that $f_0:x\mapsto\mathrm e^{-x^2/2}/\sqrt{2\pi}$ is the standard normal density. Then, as already noted, for every $\mu\geqslant0$, the function $g:x\mapsto\mathrm e^{\mu x}$ is admissible. This yields for $f_1$ the normal density with mean $\mu$ and variance $1$.

But many more examples exist, for example $g$ might be $x\mapsto\mathrm e^{cx\cdot|x|^a}$ for some $c\gt0$ and $|a|\lt1$, or the logistic $x\mapsto1/(1+a\mathrm e^{-cx})$ for some $c\gt0$ and $a\gt0$, or the trigonometric inverse $g=\frac\pi2+\arctan$, or the error function $\Phi$, or some shifted versions of these, or, more generally, any positive power of any cumulative distribution function, or...

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    I am sorry perhaps I was not clear. In my comment I was asking for a probable *simple* case such that my equations would not explode in mathematica. If I get one of your example, I am very much afraid that $\mathbb{E}(g(X_0))$ would look like horrible.2012-10-29