Common functional principal components

TL;DR

This paper extends functional principal component analysis to compare two samples, with applications to implied volatility functions, introducing new estimation methods, inference procedures, and bootstrap tests for equality of key functional features.

Contribution

It introduces a novel approach for estimating common FPCA components from noisy data and develops two-sample inference methods, including bootstrap tests, for comparing functional data.

Findings

New estimation method for discrete noisy functional data

Bootstrap tests effectively compare eigenvalues, eigenfunctions, and means

Application to implied volatility functions demonstrates practical utility

Abstract

Functional principal component analysis (FPCA) based on the Karhunen--Lo\`{e}ve 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.