Computable error bounds for high-dimensional approximations of an LR statistic for additional information in canonical correlation analysis

Let $\lambda$ be the LR criterion for testing an additional information hypothesis on a subvector of $p$-variate random vector ${x}$ and a subvector of $q$-variate random vector ${y}$, based on a sample of size $N=n+1$. Using the fact that the null distribution of $-(2/N)\log \lambda$ can be expressed as a product of two independent $\Lambda$ distributions, we first derive an asymptotic expansion as well as the limiting distribution of the standardized statistic $T$ of $-(2/N)\log \lambda$ under a high-dimensional framework when the sample size and the dimensions are large. Next, we derive computable error bounds for the high-dimensional approximations. Through numerical experiments it is noted that our error bounds are useful in a wide range of $p$, $q$, and $n$.