Comparing the marginal densities of two strictly stationary linear processes
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Comparing the marginal densities of two strictly stationary linear processes Paul Doukhan1 · Ieva Grublyte˙ 2 · Denys Pommeret3 · Laurence Reboul4 Received: 3 July 2018 / Revised: 5 July 2019 © The Institute of Statistical Mathematics, Tokyo 2019
Abstract In this paper, we adapt a data-driven smooth test to the comparison of the marginal distributions of two independent, short or long memory, strictly stationary linear sequences. Some illustrations are shown to evaluate the performances of our test. Keywords Linear processes · Local Whittle estimator · Long memory · Schwarz’s rule · Smooth test · Strictly stationary process
1 Introduction Long-range dependent (LRD) strictly stationary processes, empirically observed by a slowly decaying autocovariance function, is a topic of active research in probability theory (see e.g., Robinson 2003; Surgailis 2000; Beran et al. 2013) but also in applications (Hsing 2000; Taqqu 1975). The importance of these processes in many fields, such as Econometrics, Finance, Hydrology and other physical sciences, is abundantly demonstrated [see Doukhan et al. (2003), Baillie (1996) and the references therein]. Long-memory linear processes are an important class of such processes. Some of their theoretical properties have been studied in e.g., Giraitis et al. (2012), Surgailis
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10463019-00730-6) contains supplementary material, which is available to authorized users.
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Denys Pommeret [email protected]
1
Department of Mathematics, University Cergy-Pontoise, AGM-UMR 8080, 2 Bd. Adolphe Chauvin, 95000 Cergy-Pontoise, France
2
Institute of Mathematics and Informatics of Vilnius University, 4 Akademijos, 08663 Vilnius, Lithuania
3
ISFA, LSAF, and Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Univ. Lyon 1, 163 Avenue de Luminy, 13009 Marseille, France
4
CNRS, Centrale Marseille, I2M, Aix Marseille Univ, 163 Avenue de Luminy, 13009 Marseille, France
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P. Doukhan et al.
(2000) and Ho and Hsing (1999). Classical examples are fractional autoregressiveintegrated moving average (ARFIMA) time series (see Dittmann and Granger 2002; Hosking 1981), extensively used in Econometrics. Nonlinear transformations of such processes allow to construct ordinary statistics of linear processes, such as the empirical variance or the empirical process. These models received much attention in recent years (see Abadir et al. (2014); Giraitis and Surgailis (1999); Dittmann and Granger (2002); Ho and Hsing (1997); Wu (2006, 2002); Sang and Sang (2016); Giraitis et al. (2012); Giraitis (1985); Ho (2000)). One of the interesting problems highlighted by these previous works is to make nonparametric inference on the marginal distribution of such processes. In this paper, we propose a nonparametric test of comparison for the marginal densities of strictly stationary linear processes. More precisely, consider the linear processes X = (X t )t∈Z and Y = (Yt )t∈Z such that αt− j j and Yt = βt− j e j
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