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Title: Functional goodness-of-fit tests Authors:  Petr Coupek - Charles University (Czech Republic)
Viktor Dolnik - Charles University (Czech Republic)
Zdenek Hlavka - Charles University (Czech Republic) [presenting]
Daniel Hlubinka - Univerzita Karlova (Czech Republic)
Abstract: The distribution of a Gaussian functional variable (random process) is uniquely determined by the mean function and the covariance operator but it may be characterized also by the so-called characteristic functional (CF). We consider a goodness-of-fit (GOF) test based on a Cramer-von Mises distance between the observed empirical CF and the theoretical CF corresponding to the null hypothesis---in this case, the null hypothesis states the functional observations were generated from a specific family of Gaussian processes. Compared to previously proposed tests of this type, we investigate the functional GOF test also in presence of nuisance parameters, we establish bootstrap consistency, and we discuss the choice of necessary tuning parameters. As an example, we test, e.g., the null hypothesis that the functional observations were generated from an Ornstein-Uhlenbeck process, Vasicek model, or a (fractional) Brownian motion, both with and without unknown parameters, against suitable alternatives. The small sample properties are investigated in a simulation study.