Why MLR (monotone likelihood ratio) implies stochastic increasing? the following argument holds:
for $\theta_1&lt;\theta_2$, $\dfrac{f(x\mid\theta_2)}{f(x\mid\theta_1)}$ is increasing in $x$. Then, $F(x\mid\theta_2)\leq F(x\mid\theta_1)$ for all $x$.

Intuitively, this seems to be natural but I could not find a way.
Would you let me know this or have some hints?
Let's suppose all is one-dimensional, and denote $g(x) = \dfrac{f(x\mid\theta_2)}{f(x\mid\theta_1)}$, then $g$ is increasing.  

Define $t = \sup\{x: g(x) \leq 1\}$, then obviously for all $x \leq t$, $F(x\mid\theta_2)\leq F(x\mid\theta_1)$ since on $(-\infty,t)$ $f(x\mid\theta_2) \leq f(x\mid\theta_1)$ and $F(x\mid\theta_i) = \int_\infty^x f(y\mid\theta_i)dy, i = 1,2$

And for $x &gt; t$, remark that $F(x\mid\theta_i) = 1 - \int^\infty_x f(y\mid\theta_i)dy, i = 1,2$ and on $(t, +\infty)$ we have $f(x\mid\theta_2) \geq f(x\mid\theta_1)$
Sometimes it is good learning about doubts. This relation is well known from the beginning of mathematical statistics. In the classic book of Lehmann "Testing Statistical Hypothesis" you can find this implication and more.