Geodesic
Laplace expansion for Bartlett--Nanda--Pillai's test statistic and its error bound
Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 4, pp. 705-718

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We construct asymptotic expansions for the distribution function of the Bartlett–Nanda–Pillai statistic under the condition that the null linear hypothesis is valid in a multivariate linear model. Computable estimates of the accuracy of approximation are obtained via the Laplace approximation method, which is generalized to integrals for matrix-valued functions.
Keywords: Bartlett–Nanda–Pillai statistics, nonasymptotic analysis, Laplace approximation method. 
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     author = {H. Wakaki and V. V. Ulyanov},
     title = {Laplace expansion for {Bartlett--Nanda--Pillai's} test statistic and its error bound},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2023_68_4_a2/}
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H. Wakaki; V. V. Ulyanov. Laplace expansion for Bartlett--Nanda--Pillai's test statistic and its error bound. Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 4, pp. 705-718. http://geodesic.mathdoc.fr/item/TVP_2023_68_4_a2/