Nearest neighbors estimation for long memory functional data
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Nearest neighbors estimation for long memory functional data Lihong Wang1 Accepted: 15 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper, we consider the asymptotic properties of the nearest neighbors estimation for long memory functional data. Under some regularity assumptions, we investigate the asymptotic normality and the uniform consistency of the nearest neighbors estimators for the nonparametric regression models when the explanatory variable and the errors are of long memory and the explanatory variable takes values in some abstract functional space. The finite sample performance of the proposed estimator is discussed through simulation studies. Keywords Asymptotic normality · Functional data · Long memory · Nearest neighbors estimation · Uniform consistency Mathematics Subject Classification 62M10
1 Introduction As indicated in Ferraty and Vieu (2006), functional data are often seen in many fields of applied sciences ranging from environmetrics, chemometrics, biometrics to econometrics. Statistical analysis for functional data has been receiving increasing attention in the recent years, see, e.g. Horváth and Kokoszka (2012), Bongiorno et al. (2014), Cuevas (2014), Hsing and Eubank (2015), and Goia and Vieu (2016). Among all the methods explored to deal with the observations with complex functional structure, nonparametric statistical methods has been discussed extensively. For example, Gasser et al. (1998) considered density and mode estimations for the explanatory variable taking values in a normed vector space. Ferraty and Vieu (2004)
This work was supported by National Natural Science Foundation of China (NSFC) Grants 11671194 and 11501287.
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Lihong Wang [email protected] Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
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and Masry (2005) studied the asymptotic properties of the Nadaraya–Watson kernel type estimator for weak dependent functional data. Chen and Zhang (2009) established the strong consistency and the asymptotic normality of the nonparametric M-estimator for mixing functional data. Chagny and Roche (2016) proposed an adaptive kernel estimator based on a data-driven bandwidth selection rule for independent functional data and derived the convergence rates of the estimator. Both Messaci et al. (2015) and Al-Awadhi et al. (2018) considered the local polynomial modelling of the conditional quantile for functional data. Kara-Zaitri et al. (2017a) presented some functional uniform in bandwidth consistency results for various kernel estimators in the i.i.d. setting. Nearest neighbors estimation methods have become an important tool for nonparametric regression in infinite dimensional problems, because the local features of the data are more heterogeneous when the dimension increases and the nearest neighbors technique can provide location adaptive estimators through a locally adaptive smoothing parameter (see, e.g. Kudraszowa and Vieu (2013) and Kara-Zaitri et al. (2017b)). Recently, Burb
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