Recovering Low-Dimensional Structure in Estimated Conditional Kendall’s Tau Curves via Functional PCA
S.H.H. Groen (TU Delft - Electrical Engineering, Mathematics and Computer Science)
A.F.F. Derumigny – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
D. Kurowicka – Graduation committee member (TU Delft - Electrical Engineering, Mathematics and Computer Science)
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Abstract
Functional Principal Component Analysis (FPCA) is a powerful method for identifying the dominant modes of variation within a large collection of curves, thereby revealing their underlying low-dimensional structure. However, the existing FPCA framework assumes that the functional data are directly observed. This thesis develops a new theoretical framework for applying FPCA when the functional objects are estimated rather than observed, focusing specifically on estimated conditional Kendall's tau (CKT) curves. The analysis is supported by additional theoretical work on conditional U-statistics, which form the basis for establishing the asymptotic properties of the proposed method. In the theoretical setting where the CKT curves are estimated over the entire domain, we prove the consistency of the associated Hilbert-Schmidt integral operator and under suitable regularity and asymptotic conditions, we show that the spectral decomposition of the estimated operator converges to the true one with an error rate of O(h² + (nhd)-½). We then extend the analysis to the practically relevant setting in which the curves are observed on a finite grid of N points and establish convergence of the spectral properties with a rate of O(N-1/d + h² + (nhd)-½). This illustrates the bias-variance trade-off with respect to the choice of the bandwidth h, as well as the numerical bias introduced by the discretization step, N-1/d.
We support our theoretical results with an empirical study of global financial assets, applying ARMA-GARCH filtering and conditioning on different markets to recover the most dominant modes of variation within a large dataset of CKT curves.
We find that all CKT curves can be approximately summarized by only two degrees of freedom: a constant baseline level, representing the average conditional Kendall's tau, and a parabolic component that contrasts extreme market conditions with more typical periods, capturing the contagion effects.
Ultimately, this framework provides a foundation for future applications, including evaluating the simplifying assumption.