Supplement to “estimation in Functional Regression for General Exponential Families” by Winston

In this supplemental article, we introduce some useful results in spectral theory and perturbation theory. Some of the results are well-established. We briefly review them for the purpose of easy reference. For example, the results for eigenvalues have become quite standard for decades (see, e.g. Dunford and Schwartz, 1988, Chapter VII.6). We derive a bound for the perturbation of eigenprojections (Lemma 15) which plays a key role in the slope function estimation problem. This bound is closely related to Proposition 2 in Cardot, Mas and Sarda (2007), which was tailored to solve the prediction problem at a random design. However, the two results are different. A comparison between their result and our bound in Lemma 4 is discussed later following Lemma 4. We could not find the same (or stronger) bound explicitly in the existing perturbation literature. The spectral theory and the perturbation theory in Hilbert spaces have been serving as powerful tools that allow statisticians to tackle the statistical approximation problems in an elegant way. From Lemma 1 to Lemma 3 we review the wellestablished perturbation-theoretic results for eigenvalues and eigenvectors of positive and self-adjoint compact operators respectively. Our main contribution of this piece is to extend the perturbation result for eigenprojections, obtained by Tyler (1981, Lemma 4.1), from the matrix case to the general operator case. Our perturbation result for eigenprojections will be introduced in Lemma 4. Suppose T is a positive and self-adjoint compact operator in a Hilbert space H. According to the spectral theory for positive and self-adjoint compact operators (see e.g. Birman and Solomjak, 1987, Page 209), the operator T has a sequence of decreasing nonnegative eigenvalues {θi} and a sequence of corresponding eigenvectors {ei}. That is, Tei = θiei with θ1 ≥ θ2 ≥ · · · ≥ 0. Furthermore, T has the spectral decomposition