Common Functional Principal Components

Functional principal component analysis (FPCA) based on theKarhunen-Lo`eve decomposition has been successfully applied in manyapplications, mainly for one sample problems. In this paper we consider common functional principal components for two sample problems. Our research is motivated not only by the theoretical challenge of this data situation but also by the actual question of dynamics of implied volatility (IV) functions. For different maturities the logreturns of IVs are samples of (smooth) random functions and themethods proposed here study the similarities of their stochastic behavior. Firstly we present a new method for estimation of functionalprincipal components from discrete noisy data. Next we present thetwo sample inference for FPCA and develop two sample theory. Wepropose bootstrap tests for testing the equality of eigenvalues, eigenfunctions, and mean functions of two functional samples, illustratethe test-properties by simulation study and apply the method to theIV analysis.