  Kronecker product covariance structure provides an efficient way to modeling
the inter-correlations of matrix-variate data. In this paper, we propose
testing statistics for Kronecker product covariance matrix based on linear
spectral statistics of renormalized sample covariance matrices. Central limit
theorem is proved for the linear spectral statistics with explicit formulas for
mean and covariance functions, which fills the gap in the literature. We then
theoretically justify that the proposed testing statistics have well-controlled
sizes and strong powers. To facilitate practical usefulness, we further propose
a bootstrap resampling algorithm to approximate the limiting distributions of
associated linear spectral statistics. Consistency of the bootstrap procedure
is guaranteed under mild conditions. A more general model which allows the
existence of noises will also be discussed. In the simulations, the empirical
sizes of the proposed testing procedure and its bootstrapped version are close
to corresponding theoretical values, while the powers converge to one quickly
as the dimension and sample size grow.
