Inference from heteroscedastic functional data

Abstract

Technological advancements have produced an abundance of data sets in which a large number of repeated measurements are observed within a subject or stratum. Many of these data sets are based on a small number of subjects rendering most existing inferential methods unsuitable. This paper develops test procedures based on a novel model for nested heteroscedastic high-dimensional data which we propose. The novelty of the model rests on the fact that the random effects are assumed to be neither uncorrelated nor normal. The model is nonparametric in the sense that it leaves the covariance structure unspecified and applies to both discrete and continuous data. The test procedures developed are useful for evaluating the effects of time as well as their interactions with the crossed factors on the stratum. The asymptotic theory of the test statistics is driven by a large number of measurements per subject and the assumption of nonstationary α-mixing on the error term. Simulation studies and real applications show that the proposed tests are more powerful in detecting effects compared with benchmark methods in data with very limited number of replications.

Keywords:

• functional data analysis
• nonparametric inference
• hypothesis testing
• high dimensional multivariate analysis

AMS Subject Classification :

• 62G10
• 62G20
• 62H15

Acknowledgements

We thank the Editor, Associate Editor and two anonymous referees for their valuable constructive suggestions, which significantly helped improve the structure and presentation of this paper. We also thank Prof. James Higgins, Meghan Buckley and Prof. Gerard Kluitenberg at Kansas State University for sharing the soil water content data with us for illustration of the analysis. We are grateful to Prof. Stephen B. Vardeman from Iowa State University and Prof. Raymond J. Carroll from Texas A&M University for their helpful comments that led to essential improvement of the earlier versions of the manuscript.