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We study quasi-Monte Carlo (QMC) integration over the multi-dimensional unit
cube in several weighted function spaces with different smoothness classes. We
consider approximating the integral of a function by the median of several
integral estimates under independent and random choices of the underlying QMC
point sets (either linearly scrambled digital nets or infinite-precision
polynomial lattice point sets). Even though our approach does not require any
information on the smoothness and weights of a target function space as an
input, we can prove a probabilistic upper bound on the worst-case error for the
respective weighted function space, where the failure probability converges to
0 exponentially fast as the number of estimates increases. Our obtained rates
of convergence are nearly optimal for function spaces with finite smoothness,
and we can attain a dimension-independent super-polynomial convergence for a
class of infinitely differentiable functions. This implies that our
median-based QMC rule is universal in the sense that it does not need to be
adjusted to the smoothness and the weights of the function spaces and yet
exhibits the nearly optimal rate of convergence. Numerical experiments support
our theoretical results.