Fréchet-differentiation of functions of operators with application to functional data analysis
It is well-known that the sample covariance operator converges in distribution in the Hilbert space of Hilbert-Schmidt operators, and that this result entails the asymptotic distribution of simple eigenvalues and corresponding eigenvectors. Several estimators and test statistics for the analysis of functional data require the asymptotic distribution of eigenvalues and eigenvectors of certain functions of sample covariance operators, and it turns out that the asymptotic distribution of such a function of the sample covariance operator is a prerequisite. To obtain such a result, the main result in this dissertation is the determination of the Fr´echet-derivative of an analytic functions of a bounded operator, tangentially to the space of all bounded operators, and an ensuing delta-method to solve this problem. In the Hermitian case, moreover, some results on perturbation of an isolated eigenvalue, its eigenprojection, and its eigenvector if the eigenvalue is simple, are also included. The results are applied to obtain the asymptotic distribution of a test statistic for testing the equality of two covariance operators.