  That data follow a Gompertz distribution is a widely used assumption in
diverse fields of applied sciences, e.g., in biology or when analysing survival
times. Since misspecified models may lead to false conclusions, assessing the
fit of the data to an underlying model is of central importance. We propose a
new family of characterisation-based weighted $L^2$-type tests of fit to the
family of Gompertz distributions, hence tests for the composite hypothesis when
the parameters are unknown. The characterisation is motivated by distributional
transforms connected to Stein's method of distributional approximation. We
provide the limit null distribution of the test statistics in a Hilbert space
setting and, since the limit distribution depends on the unknown parameters, we
propose a parametric bootstrap procedure. Consistency of the testing procedure
is shown. An extensive simulation study as well as applications to real data
examples show practical benefits of the procedures: the first data set we
analyse consists of lifetimes of fruitflies, the second has been synthetically
generated from life tables for women born in Germany in 1948.
