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.
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3$\begingroup$ 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 $X$. $\endgroup$– StijnDeVuystNov 27, 2016 at 13:03
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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, 2016 at 15:37
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2 Answers
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I don't know if this is helpful, but for any function $f$ set $Z \equiv f(X)$. Then by Jensen's Inequality
$$E(Z^2) > [E(Z)]^2 \implies \sqrt {E(Z^2)} > \big |E(Z)\big |$$
$$\implies -\sqrt {E(Z^2)} < E(Z) < \sqrt {E(Z^2)} $$
$$ \implies -\sqrt {E([f(X)]^2)} < E[f(X)] < \sqrt {E([f(X)]^2)} $$
answered Nov 27, 2016 at 15:42
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A lower bound for E[f(X)]? (for a concave function f) is any upper bound on the expectation E[f(X)] of a convex function f, e.g., see Edmundson-Madansky type inequalities: 1 - Edmundson, H.P. (1956): Bounds on the expectation of a convex function of a random variable. Technical report, The Rand Corporation Paper 982, Santa Monica, California; and, 2 - Madansky, A. (1959): Bounds on the expectation of a convex function of a multivariate random variable. Ann. Math. Stat. 30, 743–746; or, Dula's type bounds: 3 - Dul´a, J.H. (1992): An upper bound on the expectation of simplicial functions of multivariate random variables. Math. Program. 55, 69–80
