Let $Z \sim \mathcal{N}(\mu,1)$ and optimize the following problem. For a given $p>0$ \begin{align} \min_{a\in \mathbb{R} }E \left[|Z-a|^p\right] \end{align}
The goal is to find optimal $a$.
Things I tried
W.l.o.g we can transform the problem over $N \sim \mathcal{N}(0,1)$ since
\begin{align} \min_{a\in \mathbb{R} }E \left[|Z-a|^p\right]=\ \min_{a\in \mathbb{R} }E \left[|N+\mu-a|^p\right]=\min_{b\in \mathbb{R} }E \left[|N-b|^p\right]. \end{align}
Moreover, we know that $E[|N|^p]=\frac{2^{p/2} \Gamma \left( \frac{p+1}{2}\right)}{\sqrt{p}} $ and $E[N^p]=0$ for $p$=odd.
So, the problem becomes kind of manageable for $p$=even since
\begin{align} \min_{b\in \mathbb{R} }E \left[|N-b|^p\right]=\min_{b\in \mathbb{R} }E \left[(N-b)^p\right]=\min_{b\in \mathbb{R}} \sum_{n=0}^p{p \choose n}E[N^{p-n}]a^n \end{align}
and we basically have to find a minimum of a polynomial of order $p$. However, getting a close form solution for large $p$ is not that easy. (At least for me).
My main difficulty is how to approach a general $p>0$?
At first, I thought that the minim was going to be $a=E[N]$ but I don't think this is true.
Thanks for your help in advance.