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Summary The paper is concerned with testing normality in samples of curves and error curves estimated from functional regression models. We propose a general paradigm based on the application of multivariate normality tests to vectors of functional principal components scores. We examine finite sample performance of a number of such tests and select the best performing tests. We apply them to several extensively used functional data sets and determine which can be treated as normal, possibly after a suitable transformation. We also offer practical guidance on software implementations of all tests we study and develop large sample justification for tests based on sample skewness and kurtosis of functional principal component scores. 

Spatio-temporal data indexed by sampling locations and sampling time points are encountered in many scientific disciplines such as climatology, environ- mental sciences, and public health. Here, we propose a novel spatio-temporal expanding distance (STED) asymptotic framework for studying the proper- ties of statistical inference for nonstationary spatio-temporal models. In particular, to model spatio-temporal dependence, we develop a new class of locally stationary spatio-temporal covariance functions. The STED asymp- totic framework has a fixed spatio-temporal domain for spatio-temporal pro- cesses that are globally nonstationary in a rescaled fixed domain and locally stationary in a distance expanding domain. The utility of STED is illus- trated by establishing the asymptotic properties of the maximum likelihood estimation for a general class of spatio-temporal covariance functions. A simulation study suggests sound finite-sample properties and the method is applied to a sea-surface temperature dataset.