  In this paper, we study the minimax estimation of the Bochner integral
$$\mu_k(P):=\int_{\mathcal{X}} k(\cdot,x)\,dP(x),$$ also called as the kernel
mean embedding, based on random samples drawn i.i.d.~from $P$, where
$k:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$ is a positive definite
kernel. Various estimators (including the empirical estimator),
$\hat{\theta}_n$ of $\mu_k(P)$ are studied in the literature wherein all of
them satisfy $\bigl\|
\hat{\theta}_n-\mu_k(P)\bigr\|_{\mathcal{H}_k}=O_P(n^{-1/2})$ with
$\mathcal{H}_k$ being the reproducing kernel Hilbert space induced by $k$. The
main contribution of the paper is in showing that the above mentioned rate of
$n^{-1/2}$ is minimax in $\|\cdot\|_{\mathcal{H}_k}$ and
$\|\cdot\|_{L^2(\mathbb{R}^d)}$-norms over the class of discrete measures and
the class of measures that has an infinitely differentiable density, with $k$
being a continuous translation-invariant kernel on $\mathbb{R}^d$. The
interesting aspect of this result is that the minimax rate is independent of
the smoothness of the kernel and the density of $P$ (if it exists). This result
has practical consequences in statistical applications as the mean embedding
has been widely employed in non-parametric hypothesis testing, density
estimation, causal inference and feature selection, through its relation to
energy distance (and distance covariance).
