Some asymptotic theory for Silverman's smoothed functional principal components in an abstract Hilbert space

Unlike classical principal component analysis (PCA) for multivariate data, one needs to smooth or regularize when estimating functional principal components. Silverman's method for smoothed functional principal components has nice theoretical and practical properties. Some theoretical properties of Silverman's method were obtained using tools in the L2 and the Sobolev spaces. This paper proposes an approach, in a general manner, to study the asymptotic properties of Silverman's method in an abstract Hilbert space. This is achieved by exploiting the perturbation results of the eigenvalues and the corresponding eigenvectors of a covariance operator. Consistency and asymptotic distributions of the estimators are derived under mild conditions. First we restrict our attention to the first smoothed functional principal component and then extend the same method for the first K smoothed functional principal components.