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Parameter estimation of probability density functions is one of the major steps in the area of statistical image and signal processing. In this paper we explore several properties and limitations of the recently proposed method of logarithmic cumulants (MoLC) parameter estimation approach which is an alternative to the classical maximum likelihood (ML) and method of moments (MoM) approaches. We derive the general sufficient condition for a strong consistency of the MoLC estimates which represents an important asymptotic property of any statistical estimator. This result enables the demonstration of the strong consistency of MoLC estimates for a selection of widely used distribution families originating from (but not restricted to) synthetic aperture radar image processing. We then derive the analytical conditions of applicability of MoLC to samples for the distribution families in our selection. Finally, we conduct various synthetic and real data experiments to assess the comparative properties, applicability and small sample performance of MoLC notably for the generalized gamma and K families of distributions. Supervised image classification experiments are considered for medical ultrasound and remote-sensing SAR imagery. The obtained results suggest that MoLC is a feasible and computationally fast yet not universally applicable alternative to MoM. MoLC becomes especially useful when the direct ML approach turns out to be unfeasible.