Nonasymptotic analysis of Bartlett–Nanda–Pillai statistic for high-dimensional data

The author gets the computable error bounds for normal approximation of Bartlett–Nanda–Pillai statistic when dimensionality grows proportionally to the sample size. This result enables one to get more precise calculations of the $p$-values in applications of multivariate analysis. In practice, more and more often, analysts encounter situations when the number of factors is large and comparable with the sample size. The examples include signal processing. The proof is essentially based on the normality of the distribution of the elements of the matrices under consideration with the Wishart distribution. For random variables that are the matrix traces of the product and squares of matrices with the normalized Wishart distribution, convenient upper bounds for $1-F$ are found where $F$ is the distribution function of the corresponding matrix trace. Applying the properties of inverse matrices and positive semidefinite matrices, the Bartlett–Nanda–Pillai statistic is bounded from above by a combination of the above-mentioned matrix traces.