Properties of design-based functional principal components analysis

By Hervé Cardot, Mohamed Chaouch, Camelia Goga, Catherine Labruère. 2008

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Abstract This work aims at performing functional principal components analysis (FPCA) with Horvitz–Thompson estimators when the observations are curves collected with survey sampling techniques. One important motivation for this study is that FPCA is a dimension reduction tool which is the first step to develop model-assisted approaches that can take auxiliary information into account. FPCA relies on the estimation of the eigenelements of the covariance operator which can be seen as nonlinear functionals. Adapting to our functional context the linearization technique based on the influence function developed by Deville [1999. Variance estimation for complex statistics and estimators: linearization and residual techniques. Survey Methodology 25, 193–203], we prove that these estimators are asymptotically design unbiased and consistent. Under mild assumptions, asymptotic variances are derived for the FPCA’ estimators and consistent estimators of them are proposed. Our approach is illustrated with a simulation study and we check the good properties of the proposed estimators of the eigenelements as well as their variance estimators obtained with the linearization approach.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e8502b3e05bf5918ee91ecf7ef796f9fcab6e5c5

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In Statistics and Probability

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Horvitz--Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling (2009, methodology): “Such an estimation procedure relies, as noted in Cardot et al. (2010), on a parsimonious representation of the trajectories in order to reduce the dimension of the data.”