The Annals of Statistics

Common functional principal components

Michal Benko, Wolfgang Härdle, and Alois Kneip

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Functional principal component analysis (FPCA) based on the Karhunen–Loève decomposition has been successfully applied in many applications, 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 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 study and apply the method to the IV analysis.

Article information

Ann. Statist., Volume 37, Number 1 (2009), 1-34.

First available in Project Euclid: 16 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis 62G08: Nonparametric regression
Secondary: 62P05: Applications to actuarial sciences and financial mathematics

Functional principal components nonparametric regression bootstrap two sample problem


Benko, Michal; Härdle, Wolfgang; Kneip, Alois. Common functional principal components. Ann. Statist. 37 (2009), no. 1, 1--34. doi:10.1214/07-AOS516.

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