  This paper investigates a statistical procedure for testing the equality of
two independently estimated covariance matrices when the number of potentially
dependent data vectors is large and proportional to the size of the vectors,
that is, the number of variables. Inspired by the spike models used in random
matrix theory, we concentrate on the largest eigenvalues of the matrices in
order to determine significant differences. To avoid false rejections we must
guard against residual spikes and need a sufficiently precise description of
the properties of the largest eigenvalues under the null hypothesis. In this
paper, we extend arXiv:2002.12741 for perturbation of order $1$ and
arXiv:2002.12703 studying simpler statistic. The residual spike introduce in
the first paper is investigated and leads to a statistic that results in a good
test of equality of two populations. Simulations show that this new test does
not rely on some hypotheses that were necessary for the proofs and in the
second paper.
