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Setup
=====


Let $X \sim \text{Binomial}(p, n)$, and $r \ge 1$.


Question
========


What is a good upper-bound for $\mathbb E[|X-np|^r]$ ?


Solution for small $r$
======================


* If $r=2$, then $\mathbb E[|X-np|^2] = np(1-p)$.
* If $1 \le r < 2$, then Jensen's inequality gives
$$
\mathbb E[|X-np|^r] =(\mathbb E[|X-np|^r])^{2/r})^{(r/2)} \le (\mathbb E[|X-np|^2])^{r/2} = (np(1-p))^{(r/2)}.
$$


Notes
=====


I'm ultimately interested in bounding (via McDiamid's inequality) the $\ell\_r$-distance between a distribution $P$ and it's empirical version $\hat{P}\_n$.



Good upper-bound for $\mathbb E[|X-np|^r]$ where $X \sim \text{Binomial}(n,p)$ and $r \ge 1$
By the main result of the paper [Exact Rosenthal-type bounds](https://projecteuclid.org/euclid.aop/1441792292), we have 
$$E|X-np|^r\le c^r E|\Pi\_\lambda-\lambda|^r
$$
for real $r\in(2,\infty)\setminus(3,4)\setminus(4,5)$, where real $c>0$ and $\lambda>0$ are defined by the conditions 
$$c^r\lambda=n(q^rp+p^rq)\quad \text{and}\quad c^2\lambda=npq;
$$
$q:=1-p$; and $\Pi\_\lambda$ is a Poisson random variable with parameter $\lambda$. 


Other results on Rosenthal-type bounds can be found e.g. in [this paper](https://epubs.siam.org/doi/pdf/10.1137/S0040585X97T987417) or its [arXiv version](https://arxiv.org/pdf/1212.1912), and in references therein. 


**Added:** In a comment, the OP stated that it may be assumed that $1\le r\le 2$. This simplifies the matter a great deal. Indeed, in this case we have 
\begin{equation}
 E|X-np|^r\le\min((npq)^{r/2},2npq(q^{r-1}+p^{r-1}))\ll s^r\wedge s^2,\tag{1}
\end{equation}
where $(npq)^{r/2}$ is the bound the OP obtained by using Jensen's inequality; and $2npq(q^{r-1}+p^{r-1})$ is a bound immediately obtained by using the von Bahr--Esseen inequality -- see e.g. [this paper](https://projecteuclid.org/euclid.afa/1435763999) or [its arXiv version](https://arxiv.org/pdf/1008.5350); $s:=\sqrt{npq}$. For positive expressions $e\_1$ and $e\_2$, we write $e\_1\ll e\_2$ or, equivalently, $e\_2\gg e\_1$ if $e\_1\le C e\_2$ for some universal positive real constant $C$. 


The upper bound on $E|X-np|^r$ in (1) is optimal up to a universal constant factor: for $r\in[1,2)$
\begin{equation}
 E|X-np|^r\gg s^r\wedge s^2.\tag{2}
\end{equation}
This lower bound on $E|X-np|^r$ is obtained by using the log-convexity of $m\_t:=E|X-np|^t$ in $t>0$ and an *upper* Rosenthal-type bound -- as follows: By (say) Theorem 1.5 in the already cited paper [Exact Rosenthal-type bounds](https://projecteuclid.org/euclid.aop/1441792292), we have $m\_3\ll s^3\wedge s^2$. Now using the log-convexity of $m\_t:=E|X-np|^t$ in $t>0$ or, equivalently, the Hölder inequality, we have 
\begin{equation\*}
 m\_2\le m\_r^{1/a}m\_3^{1-1/a},
\end{equation\*}
where $a:=3-r$. Hence,
\begin{equation\*}
 E|X-np|^r=m\_r\ge m\_2^{3-r}m\_3^{r-2}\gg s^{6-2r}(s^{3r-6}\wedge s^{2r-4})
=s^r\wedge s^2. 
\end{equation\*}
Thus, (2) indeed holds. 


