Suppose $\epsilon \sim N(0,\sigma^2)$, $a\in(0,1)$ and $x>0$ are constant. Is there any way of estimaing the following:
$$(\frac{x+\epsilon}{x})^a$$
Although $\mathbb{E}[\frac{x+\epsilon}{x}]=1$, I think generally $\mathbb{E}[(\frac{x+\epsilon}{x})^a]$ is not.
Can we say $median[(\frac{x+\epsilon}{x})^a]=1$? If not, is there any other approach to get it close to or 'make' it $1$? If still not, is there any extra restriction needed so that we can have something close to $1$?