Let's have a function $f(x, \theta)$, and some probability distribution on $x$. Let's say I have found $\theta^* = \operatorname{argmax}(f(E[x], \theta) $, and $f$ is concave in $x$. I would like to prove, that $E_x[f(\theta^*, x)] \ge E_x[f(\theta, x)] $, or in other words, for concave function $f$, is it true, that $E_x[f(x)]$ is maximized, when $f(E_x[x])$ is maximized.
I've been through Jensen's inequality, but I haven't been able to find a prove for this. I know that the distribution of $x$ is Rayleigh distribution.
Thanks a million !