Projection pursuit based tests of normality with functional data

Abstract

Methods for validating the assumption of normality of functional data have been only lightly developed to date, with existing methods based primarily on summarizing the data by their projections into random or principal component subspaces, and applying multivariate normality tests to the vectors of scores defining these projections. While this is effective in some cases, we show with both real and synthetic data examples some pitfalls of this approach, including their sensitivity to the basis used to smooth the raw data. We propose a new normality test for functional data based on a projection pursuit that overcomes some of these challenges. Asymptotic theory is developed for the proposed statistics, and we develop several new computational tools needed to implement the high-dimensional projection pursuit. As a by-product of evaluating the test statistic, our method furnishes a way of decomposing functional data into its approximately Gaussian and non-Gaussian components, which is useful for the purpose of data visualization and subsequent analyses. A simulation study and analysis of three data sets demonstrate the complimentary advantages of the proposed test to those currently available in the literature.