Testing serial independence with functional data
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Testing serial independence with functional data Zdenek ˇ Hlávka1
· Marie Hušková1
· Simos G. Meintanis2,3
Received: 27 January 2020 / Accepted: 29 August 2020 © Sociedad de Estadística e Investigación Operativa 2020
Abstract We consider tests of serial independence for a sequence of functional observations. The new methods are formulated as L2-type criteria based on empirical characteristic functions and are convenient from the computational point of view. We derive asymptotic normality of the proposed test statistics for both discretely and continuously observed functions. In a Monte Carlo study, we show that the new test is sensitive with respect to functional GARCH alternatives, investigate the choice of necessary tuning parameters, and demonstrate that critical values obtained by resampling lead to a test with good performance in any setup, whereas the asymptotic critical values may be recommended only for a sufficiently fine discretization grid. Finite-sample comparison with a distance (auto)covariance test criterion is also included, and the article concludes with application on a real data set. Keywords Functional data · Serial independence · Empirical characteristic function · Testing Mathematics Subject Classification 62R10 · 62G10 · 62H15
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Zdenˇek Hlávka [email protected] Marie Hušková [email protected] Simos G. Meintanis [email protected]
1
Department of Statistics, Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic
2
Department of Economics, National and Kapodistrian University of Athens, Athens, Greece
3
Unit for Business Mathematics and Informatics, North-West University, Potchefstroom, South Africa
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Z. Hlávka et al.
1 Introduction Functional data analysis (FDA) is developing at an ever-increasing pace due to the current availability of large data sets and the rapid development of computer technology. In this connection, it is necessary to develop new procedures for analyzing functional data coming from various areas, e.g., medicine, finance, econometrics, meteorology. This leads to new methods and algorithms and the need to study their theoretical properties, their performance on simulated data as well as their potential for real data applications. In certain situations, methods from the finite-dimensional setting can be adjusted, but in most cases new procedures have to be developed. There are several books and a number of survey papers containing the latest achievements as well as formulations of new open problems. Concerning books, a book by Ramsay and Silverman (2005) focusing on independent functional observations has become something of a classic, Ferraty and Vieu (2006) deal with the nonparametric approach for functional data, while Horváth and Kokoszka (2012) give high level introduction to the mathematical framework of functional data analysis covering both independent and dependent functional observations with a number of applications. Zhang (2013) focuses on ANOVA in a functional setup, and quite recently, a book by Kokos
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