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Common functional principal components. (English) Zbl 1169.62057

Summary: Functional principal components analysis (FPCA) based on the Karhunen-Loève decomposition has been successfully applied in many applications, mainly for one-sample problems. 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 log-returns of IVs are samples of (smooth) random functions and the methods proposed here study the similarities of their stochastic behavior.
First we present a new method for estimation of functional principal components from discrete noisy data. Next we present the two sample inference for FPCA and develop the two sample theory. We propose bootstrap tests for testing the equality of eigenvalues, eigenfunctions, and mean functions of two functional samples, illustrate the test-properties by simulation studies and apply the method to the IV analysis.

MSC:

62H25 Factor analysis and principal components; correspondence analysis
62G08 Nonparametric regression and quantile regression
62G09 Nonparametric statistical resampling methods
62P05 Applications of statistics to actuarial sciences and financial mathematics

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