1
$\begingroup$

Let $X$ be a random variable, and let $f$ be a concave function.

Are there any known lower bounds for (or methods of lower bounding) $\mathbb{E}[f(X)]$?

Jensen's inequality only gives an upper bound on the above.

$\endgroup$
  • 2
    $\begingroup$ An obvious lower bound is $\text{E}[f(X)]\geqslant \inf_{x\in S} f(x)$ where $S$ is the support of the random variable $X$. $\endgroup$ – StijnDeVuyst Nov 27 '16 at 13:03
  • 1
    $\begingroup$ That bound given by @Stijn is the best possible in general: since there is no assumption connecting $X$ to $f$, $X$ could have all its probability concentrated in an interval $[\inf f, \inf f+\epsilon)$ for arbitrarily small positive $\epsilon$. $\endgroup$ – whuber Nov 27 '16 at 15:37
2
$\begingroup$

I don't know if this is helpful, but for any function $f$ set $Z \equiv f(X)$. Then by Jensen's Inequality

$$E(Z^2) > [E(Z)]^2 \implies \sqrt {E(Z^2)} > \big |E(Z)\big |$$

$$\implies -\sqrt {E(Z^2)} < E(Z) < \sqrt {E(Z^2)} $$

$$ \implies -\sqrt {E([f(X)]^2)} < E[f(X)] < \sqrt {E([f(X)]^2)} $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.