  We propose and investigate a new estimation method for the parameters of
models consisting of smooth density functions on the positive half axis. The
procedure is based on a recently introduced characterization result for the
respective probability distributions, and is to be classified as a minimum
distance estimator, incorporating as a distance function the $L^q$-norm.
Throughout, we deal rigorously with issues of existence and measurability of
these implicitly defined estimators. Moreover, we provide consistency results
in a common asymptotic setting, and compare our new method with classical
estimators for the exponential-, the Rayleigh-, and the Burr Type XII
distribution in Monte Carlo simulation studies. We also assess the performance
of different estimators for non-normalized models in the context of an
exponential-polynomial family.
