Hypothesis testing for two-sample functional/longitudinal data

Abstract

During recent two decades, functional data commonly arise from many scientific fields such as transportation flow, climatology, neurological science and human mortality among others. The corresponding data recorded may be in the form of curves, shapes, images and functions that may be correlated, multivariate, or both. The intrinsic infinite dimensionality of functional data poses challenges in the development of theory, methodology and computation for functional data analysis. Tests of significance are essential statistical problems and are challenging for functional data due to the demands coming from real world applications. Motivated by requirements in real-world data analysis, we have focused on two topics of study. 1) Multivariate functional data have received considerable attention. It is natural to validate whether two mean surfaces are homogeneous but existing work is few. 2) In existing literature, most testing methods were designed for validity of dense and regular functional data samples, whereas in practice, functional samples may be sparse and irregular or even partly dense. In such functional data setting, there is rare work for testing equality of covariance functions or mean curves. To address these problems, we aim to two targets: 1) We propose novel sequential and parallel projection testing procedures that can detect the difference in mean surfaces powerfully. Furthermore, we apply the idea to present testing statistics for test of equality of mean curves for two functional data samples irrespective of the data type. Furthermore, the other related work takes auxiliary information into consideration. We propose a new functional regression model to characterize the conditional mean of functional response given covariates. 2) We derive a novel test procedure for test of equality of covariance functions that can deal with any functional data type, even irregular or sparse data. In addition, by using the stringing technique, once a high-dimensional data can map into functional data, we excogitate a testing procedure for comparison of covariance matrices under the high-dimensional data setting. Our method outperforms the existing testing methods in high-dimensional data testing procedures. Almost all work mentioned above include asymptotic theory and rigorous theorem proof, intensive numerical experiments and real-world data analysis.