Geodesic
On the hypothesis of multisample sphericity
Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part VI, Tome 136 (1984), pp. 153-161 
A. M. Matai. On the hypothesis of multisample sphericity. Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part VI, Tome 136 (1984), pp. 153-161. http://geodesic.mathdoc.fr/item/ZNSL_1984_136_a10/
@article{ZNSL_1984_136_a10,
     author = {A. M. Matai},
     title = {On the hypothesis of multisample sphericity},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {153--161},
     year = {1984},
     volume = {136},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_136_a10/}
}
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Thd hypothesis of multisample sphericity appears in different areas such as repeated measures designs, checking the validity of $F$-ratios in analysis of variance problems and testing circularity. This article deals with the modified likelihood ratio test for testing multisample sphericity, null and non-null moments as well as the null and non-null distributions of the test statistic. Some approximations are also considered.