  In a recent contribution, we introduced and applied a detector-independent
method to uncover nonclassicality. Here, we extend those techniques and give
more details on the performed analysis. We derive a general theory of the
positive-operator-valued measure that describes multiplexing layouts with
arbitrary detectors. From the resulting quantum version of a multinomial
statistics, we infer nonclassicality probes based on a matrix of normally
ordered moments. We discuss these criteria and apply the theory to our data
which are measured with superconducting transition-edge sensors. Our experiment
produces heralded multi-photon states from a parametric down-conversion light
source. We show that the known notions of sub-Poisson and sub-binomial light
can be deduced from our general approach, and we establish the concept of
sub-multinomial light, which is shown to outperform the former two concepts of
nonclassicality for our data.
