Perturbations of Operators with Application to Testing Equality of Covariance Operators.

The generalization of multivariate statistical procedures to infinite dimension naturally requires extra theoretical work. In this dissertation, we will focus on testing the equality of covariance operators. We derive a procedure from the Union Intersection principle in conjunction with a Likelihood Ratio test. This procedure leads to a statistic which is the largest eigenvalue of a product of operators. We generalize this procedure by using a test statistic that is based on the first $m\in N$ largest eigenvalues. Perturbation theory of operators and functional calculus of covariance operators are extensively used to derieve the required asymptotics. It is shown that the power of the test is improved with inclusion of more eigenvalues. We perform simulations to corroborate the testing procedure, using samples from two Gaussian distributions.