In this paper, we study the minimax estimation of the Bochner integral
ΞΌkβ(P):=β«Xβk(β ,x)dP(x), also called as the kernel
mean embedding, based on random samples drawn i.i.d.~from P, where
k:XΓXβR is a positive definite
kernel. Various estimators (including the empirical estimator),
ΞΈ^nβ of ΞΌkβ(P) are studied in the literature wherein all of
them satisfy βΞΈ^nββΞΌkβ(P)βHkββ=OPβ(nβ1/2) with
Hkβ being the reproducing kernel Hilbert space induced by k. The
main contribution of the paper is in showing that the above mentioned rate of
nβ1/2 is minimax in β₯β β₯Hkββ and
β₯β β₯L2(Rd)β-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 Rd. 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)