Limit theorem associated with Wishart matrices with application to hypothesis testing for common principal components

The Wishart distribution is a classical distribution of random matrices, which arises in the field of multivariate statistical analysis. A lot of studies have investigated its asymptotic properties under the traditional regime of multivariate statistics: $n\to\infty$ with fixed $p$, where $n$ denotes the degree-of-freedom and $p$ is the size of the matrix parameter of the distribution. On the other hand, as observed variables have increased with the development of information technology, statistical methodologies based on the traditional regime do not work. Hence, recently more and more studies considered another asymptotic regime: $n\to\infty$ together with $p\to\infty$. Given this background, we derive a new property of the Wishart distribution when $n$ and $p$ grow simultaneoulsy. Particularly, the asymptotic normality of \[ \frac{1}{p^2\sqrt{n_a n_b n_c n_d}}{\rm tr}\bigl( \mathbf{T}_a(n_a) \mathbf{T}_b(n_b) \mathbf{T}_c(n_c) \mathbf{T}_d(n_d) \bigr) \] is shown under the asymptotic regime $n_a,n_b,n_c,n_d = O(p^{\delta})$, $0<\delta < 1$, where $\mathbf{T}_a(n_a), \mathbf{T}_b(n_b), \mathbf{T}_c(n_c), \mathbf{T}_d(n_d)$ are independent $p$-dimensional Wishart matrices whose degrees-of-freedoms are $n_a, n_b, n_c, n_d$ and matrix parameters are $\mathbf{\Sigma}_a, \mathbf{\Sigma}_b, \mathbf{\Sigma}_c, \mathbf{\Sigma}_d$ which are $p\times p$ positive definite matrices, respectively. As an application of the result, we propose a test procedure for the common principal components hypothesis. For this problem, the proposed test statistic is asymptotically normal under the null hypothesis, and the proposed test statistic diverges to positive infinity in probability under the alternative hypothesis.