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Algorithms are derived for detecting targets in cluttered backgrounds, where the background is modeled as a product of univariate distributions independently fit to each of the principal component projections. Thus, fatter-than-Gaussian tails are fit to the data, with a different fatness parameter for each principal component. Comparisons are made to elliptically-contoured distributions (which, unlike these product distributions, are isotropic in the whitened space), including the multivariate t and the Gaussian. Numerical experiments are performed on hyperspectral data from the SHARE 2012 exercise, with target detection performance evaluated on both actual and simulated targets. Both direct and residual data are considered, with the residual data obtained from local background subtraction – these residual data are found to exhibit not only lower variance, but qualitatively different tail statistics. More direct target …