  We discuss the problem how to determine the quality of a nonlinear system
with respect to a measurement task. Due to amplification, filtering,
quantization and internal noise sources physical measurement equipment in
general exhibits a nonlinear and random input-to-output behaviour. This usually
makes it impossible to accurately describe the underlying statistical system
model. When the individual operations are all known and deterministic, one can
resort to approximations of the input-to-output function. The problem becomes
challenging when the processing chain is not exactly known or contains
nonlinear random effects. Then one has to approximate the output distribution
in an empirical way. Here we show that by measuring the first two sample
moments of an arbitrary set of output transformations in a calibrated setup,
the output distribution of the actual system can be approximated by an
equivalent exponential family distribution. This method has the property that
the resulting approximation of the statistical system model is guaranteed to be
pessimistic in an estimation theoretic sense. We show this by proving that an
equivalent exponential family distribution in general exhibits a lower Fisher
information measure than the original system model. With various examples and a
model matching step we demonstrate how this estimation theoretic aspect can be
exploited in practice in order to obtain a conservative measurement-driven
quality assessment method for nonlinear measurement systems.
