Geodesic
The testing of composite hypotheses with nuisance parameters in multidimensional case
Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 2, pp. 291-302

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In the paper the class of generalized $C(\alpha)$-tests for multidimensional parameter is introduced. We found the subclass of these tests, which is the complete class for examinated problem. 
@article{TVP_1980_25_2_a4,
     author = {A. V. Bern\v{s}hteǐn},
     title = {The testing of composite hypotheses with nuisance parameters in multidimensional case},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {291--302},
     publisher = {mathdoc},
     volume = {25},
     number = {2},
     year = {1980},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1980_25_2_a4/}
}
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A. V. Bernšhteǐn. The testing of composite hypotheses with nuisance parameters in multidimensional case. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 2, pp. 291-302. http://geodesic.mathdoc.fr/item/TVP_1980_25_2_a4/