Convergence rate of principal component analysis with local-linear smoother for functional data under a unified weighing scheme

The unified weighing scheme for the local-linear smoother in analysing functional data can deal with data that are dense, sparse or of neither type. In this paper, we focus on the convergence rate of functional principal component analysis using this method. Almost sure asymptotic consistency and rates of convergence for the estimators of eigenvalues and eigenfunctions have been established. We also provide the convergence rate of the variance estimation of the measurement error. Based on the results, the number of observations within each curve can be of any rate relative to the sample size, which is consistent with the earlier conclusions about the asymptotic properties of the mean and covariance estimators.