Geodesic
On the rank of necessary and sufficient statistics for a~sample of multivariate distributions
Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part V, Tome 108 (1981), pp. 57-71

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The rank of necessary and sufficient statistics is described in terms of functional independency of linear combinations of ratio densities logarithms. 
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     author = {M. S. Ermakov},
     title = {On the rank of necessary and sufficient statistics for a~sample of multivariate distributions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {57--71},
     publisher = {mathdoc},
     volume = {108},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_108_a4/}
}
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M. S. Ermakov. On the rank of necessary and sufficient statistics for a~sample of multivariate distributions. Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part V, Tome 108 (1981), pp. 57-71. http://geodesic.mathdoc.fr/item/ZNSL_1981_108_a4/