Properties of Design-Based Functional Principal Components Analysis

TL;DR

This paper develops a survey sampling-based FPCA method using Horvitz-Thompson estimators, proving their asymptotic properties and demonstrating effectiveness through simulations.

Contribution

It introduces a novel FPCA approach with survey sampling, adapting influence function linearization for covariance operator eigenanalysis.

Findings

Estimators are asymptotically design unbiased and consistent.

Variance estimators are effectively derived using linearization.

Simulation confirms good properties of the proposed estimators.

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), 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.