Robust nonparametric methods for functional data

This thesis presents three novel statistical methods for the robust analysis of functional data and theoretical insights into two pre-existing nonparametricregression methods, namely regression with penalized and smoothing splines. Beginning from Chapter 2, we review the functional linear regression paradigmand propose a two step-estimation procedure that combines robust functional principal components and robust linear regression. Moreover, we propose a transformation that reduces the curvature of the estimators and can be advantageous in many settings. For these estimators we prove Fisher-consistency and consistency for finite-dimensional processes under mild regularity conditions. Through the study of their influence function we are able to formally show that the proposed estimators are much less vulnerable to outlying observations thanclassical estimators based on generalized least-squares procedures. A simulation study shows that in finite-samples the proposed estimators have reasonableefficiency, protect against outlying observations, produce smooth estimates and perform well in comparison to existing approaches.In Chapter 3 we switch focus to robust nonparametric regression and in particular the popular penalized-spline estimator. The classical objective function isleast-squares based and therefore highly susceptible to model deviations and atypical observations. To remedy these deficiencies several authors haveproposed penalized-splines with a more resistant objective function but the theoretical properties of these general (M-type) penalized spline estimators hadnot been fully understood. We show in this chapter that M-type penalized spline estimators achieve the same rates of convergence as their least-squarescounterparts, even with auxiliary scale estimation. We illustrate the benefits of M-type penalized splines in a Monte-Carlo study and two real-data examples,which contain atypical observations.In Chapter 4 we demonstrate that robust penalized spline estimation can also find fertile ground in functional linear regression. Under a balance condition on the design matrix, these estimators exhibit the same asymptotic properties as the corresponding least-squares estimators, while being considerably more reliable in the presence of outliers. The proposed methods easily generalize to functional models that include additional functional predictors, scalar covariates or nonparametric components, thus providing a wide framework of estimation.The finite-sample performance of the proposed family of estimators is illustrated on a Monte-Carlo study as well as a real data set, where it is found that the proposed estimators can combine high efficiency with protection against outliers, and produce smooth estimates that compare favourably with existingapproaches, robust and non-robust alike.Chapter 5 is a theoretical contribution to the study of smoothing spline estimators with general objective functions. Such estimators have been popularin numerical analysis for a long time but they were only introduced in Statistics by Grace Wahba around the mid 70s. We provide a general treatment of thesmoothing spline problem and assuming only the convexity of the objective function, show that the least-squares (super-)convergence rates can be extended to M-type estimators whose asymptotic properties have not been hitherto described. We further show that auxiliary scale estimates may be handled under significantly weaker assumptions than those found in the literature and we establish optimal rates of convergence for the derivatives, which have not been obtained outside the least-squares framework. A simulation study and areal-data example illustrate the competitive performance of non-smooth M-type splines in relation to the least-squares spline on regular data and their superiorperformance on data that contain anomalies.Finally, Chapter 6 describes the first theoretically-optimal robust estimator for the mean function of discretely-sampled functional data. The proposed method is based on M-type smoothing spline estimation with repeated measurements and is suitable for densely observed trajectories as well as for sparsely observed trajectories that are subject to measurement error. Our analysis clearly delineates the role of the sampling frequency in the determination of the asymptotic properties of the M-type smoothing spline estimators: for commonly observed trajectories, the sampling frequency dominates the error when it is small but ceases to be important when it is large. On the other hand, for independently observed trajectories the sampling frequency plays a more limited role as the asymptotic error is jointly determined by the sampling frequency and the sample size. We illustrate the excellent performance of the proposed family of estimators relative to existing methods in a Monte-Carlo study and a real-data example that contains outlying observations.

Jaar van publicatie:2020

Toegankelijkheid:Open