Some Asymptotics for Localized Principal Components and Curves

The asymptotic behavior of localized principal components applying kernels as weights is investigated. In particular, we show that the first-order approximation of the first localized principal component at any given point only depends on the bandwidth parameter(s) and the density at that point. This result is extended to the context of local principal curves, where the characteristics of the points at which the curve stops at the edges are identified. This is used to provide a method which allows the curve to proceed beyond its natural endpoint if desired.