Log-concavity of $y\mapsto E[(y+X)^\eta]$

Question

I know the function $y\mapsto(y+x)^\eta$ for $x>0$ and $\eta>0$ is log-concave. Is this true of the expectation over a positive R.V. in the following sense, $$y\mapsto E[(y+X)^\eta]$$ where X is a positive random variable? Any pointers appreciated. Thanks!

Answer 1

If $\eta\leq 1$, the function $y\mapsto(y+x)^\eta$ is concave, so $y\mapsto \mathbb{E}[(y+X)^\eta]$ is concave too, therefore it is log-concave.

For $\eta>1$, it is false in general. I give you a counterexample in the case $\eta=2$ because the computations are easier, though it can be generalized. Consider the discrete random variable $X$ whose law is $\frac13(\delta_1+\delta_2+\delta_x)$, with the parameter $x$ to be chosen later. We have $$ \log\mathbb{E}[(y+X)^2] = \log\left(\frac{(y+1)^2+(y+2)^2+(y+x)^2}3\right). $$ Its second derivative is $$ 2\frac{-3-12x+x^2-(18+6x)y-9y^2}{\bigl(5+x^2+(6+2x)y+3y^2\bigr)^2}, $$ which evaluated at $y=0$ equals $$ 2\frac{-3-12x+x^2}{(5+x^2)^2}. $$ For $x$ sufficiently large, this is positive, disproving the concavity of $\log\mathbb{E}[(y+X)^2]$.

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Edit: it is true if $X$ is log-concave. It follows from my answer here.

Assume that $X$ has a log-concave density $f(x)$. Then the function $F(x,y) = (x+y)^\eta f(x)$ is log-concave, because $\log F(x,y) = \eta(x+y) + \log f(x)$ is the sum of an affine and a concave function. Therefore by the fact that log-concavity is preserved under marginalization, $\mathbb{E}[(y+X)^\eta] = \int F(x,y)\,dx$ is also log-concave.