  Let $f$ be analytic on $[0,1]$ with $|f^{(k)}(1/2)|\leq A\alpha^kk!$ for some
constant $A$ and $\alpha<2$. We show that the median estimate of
$\mu=\int_0^1f(x)\,\mathrm{d}x$ under random linear scrambling with $n=2^m$
points converges at the rate $O(n^{-c\log(n)})$ for any $c<
3\log(2)/\pi^2\approx 0.21$. We also get a super-polynomial convergence rate
for the sample median of $2k-1$ random linearly scrambled estimates, when
$k=\Omega(m)$. When $f$ has a $p$'th derivative that satisfies a
$\lambda$-H\"older condition then the median-of-means has error $O(
n^{-(p+\lambda)+\epsilon})$ for any $\epsilon>0$, if $k\to\infty$ as
$m\to\infty$.
