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Title: Hypotheses testing of functional principal components Authors:  Zening Song - Tsinghua University (China) [presenting]
Lijian Yang - Tsinghua University (China)
Yuanyuan Zhang - Soochow University (China)
Abstract: A procedure is proposed to test the hypothesis that the standardized functional principle components (FPCs) of a functional data are equal to a given set of orthonormal basis (e.g., the Fourier basis). Based on B-spline estimators of individual trajectories, a chi-square type statistic is constructed and shown to be oracally efficient in the sense that its limiting distribution is the same as an infeasible statistic if all unobserved trajectories were known by ``oracle''. The limiting distribution is shown to be an infinite Gaussian quadratic form, and a finite sample estimator of its quantile is shown to be consistent. A test statistic is proposed based on the chi-square type statistic and approximate quantile of the Gaussian quadratic form, which is shown to be asymptotically correct. Simulation studies are conducted to illustrate the finite performance of the proposed testing procedure. For an EEG (ElectroEncephalogram) data, the proposed procedure has confirmed an interesting discovery that the centered EEG data is generated from a small set of standard Fourier basis.