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The advent of polarimetric synthetic aperture radar has spurred a growing interest in statistical models for complex-valued covariance matrices, which is the common representation of multilook polarimetric radar images. In this paper, we respond to an emergent need by proposing statistical tests for the simple and composite goodness-of-fit (GoF) problem for a class of compound matrix distributions. The tests are based on Mellin-kind matrix cumulants. These are derived from a novel characteristic function for positive definite Hermitian random matrices, defined in terms of a matrix-variate Mellin transform instead of the conventional Fouriér transform, and belong to a new framework for statistical analysis of multilook polarimetric radar data recently introduced by the authors. The cumulant-based tests are easy to compute, and the asymptotic sampling distribution of the test statistic is chi-square distributed in the simple hypothesis case. Under the composite hypothesis, the sampling distribution is obtained by Monte Carlo simulations. We evaluate the power of the proposed GoF tests with simulated data. We also use them to assess the fit of several matrix distributions to real data acquired by Radarsat-2 in fine-quad polarization mode.