AMS subject classifications: 62H99 62E05 15A23 15A52 PACS: 62H10 62E15 15A09 15A52 Keywords: Statistical shape theory Configuration density Elliptically contoured distributions Zonal and invariant polynomials Matrix variate Kotz Pearson Bessel and Jensen-logistic distributions a b s t r a c tThe noncentral configuration density, derived under an elliptical model, generalizes and corrects the Gaussian configuration and some Pearson results. Partition theory is then used to obtain explicit configuration densities associated with matrix variate symmetric Kotz type distributions (including the normal distribution), matrix variate Pearson type VII distributions (including t and Cauchy distributions), the matrix variate symmetric Bessel distribution (including the Laplace distribution) and the matrix variate symmetric Jensenlogistic distribution.

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This paper proposes a validation-comparison method for burned area (BA) products. The technique considers: (1) bootstrapping of scenes for validation-comparison and (2) permutation tests for validation. The research focuses on the tropical regions of Northern Hemisphere South America and Northern Hemisphere Africa and studies the accuracy of the BA products: MCD45, MCD64C5.1, MCD64C6, Fire CCI C4.1, and Fire CCI C5.0. The first and second parts consider methods based on random matrix theory for zone differentiation and multiple ancillary variables such as BA, the number of burned fragments, ecosystem type, land cover, and burned biomass. The first method studies the zone effect using bootstrapping of Riemannian, full Procrustes, and partial Procrustes distances. The second method explores the validation by using distance permutation tests under uncertainty. The results refer to Fire CCI 5.0 with the best BA description, followed by MCD64C6, MCD64C5.1, MCD45, and Fire CCI 4.1. It was also found that biomass, total BA, and the number of fragments affect the BA product accuracy.