A Note on Exponential Inequalities in Hilbert Spaces for Spatial Processes with Applications to the Functional Kernel Regression Model


In this manuscript, we present exponential inequalities for spatial lattice processes which take values in a separable Hilbert space and satisfy certain dependence conditions. We consider two types of dependence: spatial data under \(\alpha\)-mixing conditions and spatial data which satisfy a weak dependence condition introduced by Dedecker and Prieur (Prob Theory Relat Fields 132(2):203–236, 2005). We demonstrate their usefulness in the functional kernel regression model of Ferraty and Vieu (Nonparametr Stat 16(1–2):111–125, 2004), where we study uniform consistency properties of the estimated regression operator on increasing subsets of the underlying function space.

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  1. 1.

    Álvarez-Liébana J, Ruiz-Medina M (2019) Prediction of air pollutants pm 10 by arbx (1) processes. Stoch Environ Res Risk Assess 33(10):1721–1736

    Article  Google Scholar 

  2. 2.

    Álvarez-Liébana J, Bosq D, Ruiz-Medina MD (2017) Asymptotic properties of a component-wise arh (1) plug-in predictor. J Multivar Anal 155:12–34

    MathSciNet  Article  Google Scholar 

  3. 3.

    Andrews DW (1984) Non-strong mixing autoregressive processes. J Appl Probab 21(4):930–934

    MathSciNet  Article  Google Scholar 

  4. 4.

    Antoniadis A, Sapatinas T (2003) Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes. J Multivar Anal 87(1):133–158

    MathSciNet  Article  Google Scholar 

  5. 5.

    Besse PC, Cardot H, Stephenson DB (2000) Autoregressive forecasting of some functional climatic variations. Scand J Stat 27(4):673–687

    Article  Google Scholar 

  6. 6.

    Bosq D (2000) Linear processes in function spaces: theory and applications, vol 149. Springer Science & Business Media, Berlin

    Google Scholar 

  7. 7.

    Bradley RC (2005) Basic properties of strong mixing conditions a survey and some open questions. Probab Surv 2(2):107–144

    MathSciNet  Article  Google Scholar 

  8. 8.

    Carbon M, Hallin M, Tran LT (1996) Kernel density estimation for random fields: the \(L^1\) theory. J Nonparametr Stat 6(2–3):157–170

    Article  Google Scholar 

  9. 9.

    Carbon M, Francq C, Tran LT (2007) Kernel regression estimation for random fields. J Stat Plan Inference 137(3):778–798

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chiou J-M, Müller H-G, Wang J-L (2004) Functional response models. Stat Sin 14:675–693

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Collomb G (1977) Estimation non paramétrique de la régression par la méthode du noyau: propriété de convergence asymptotiquememt normale indépendante. Annales scientifiques de l’Université de Clermont. Mathématiques 65(15):24–46

    MATH  Google Scholar 

  12. 12.

    Cressie N (1993) Statistics for spatial data. Wiley series in probability and mathematical statistics: applied probability and statistics. Wiley, New York

    Google Scholar 

  13. 13.

    Cuevas A (2014) A partial overview of the theory of statistics with functional data. J Stat Plan Inference 147:1–23

    MathSciNet  Article  Google Scholar 

  14. 14.

    Cuevas A, Febrero M, Fraiman R (2002) Linear functional regression: the case of fixed design and functional response. Can J Stat 30(2):285–300

    MathSciNet  Article  Google Scholar 

  15. 15.

    Dedecker J, Doukhan P (2003) A new covariance inequality and applications. Stoch Process Appl 106(1):63–80

    MathSciNet  Article  Google Scholar 

  16. 16.

    Dedecker J, Prieur C (2005) New dependence coefficients. Examples and applications to statistics. Probab Theory Relat Fields 132(2):203–236

    MathSciNet  Article  Google Scholar 

  17. 17.

    Delsol L (2009) Advances on asymptotic normality in non-parametric functional time series analysis. Statistics 43(1):13–33

    MathSciNet  Article  Google Scholar 

  18. 18.

    Ferraty F, Vieu P (2002) The functional nonparametric model and application to spectrometric data. Comput Stat 17(4):545–564

    MathSciNet  Article  Google Scholar 

  19. 19.

    Ferraty F, Vieu P (2004) Nonparametric models for functional data, with application in regression, time series prediction and curve discrimination. Nonparametr Stat 16(1–2):111–125

    MathSciNet  Article  Google Scholar 

  20. 20.

    Ferraty F, Laksaci A, Vieu P (2006) Estimating some characteristics of the conditional distribution in nonparametric functional models. Stat Inference Stoch Process 9(1):47–76

    MathSciNet  Article  Google Scholar 

  21. 21.

    Ferraty F, Mas A, Vieu P (2007) Nonparametric regression on functional data: inference and practical aspects. Aust N Z J Stat 49(3):267–286

    MathSciNet  Article  Google Scholar 

  22. 22.

    Ferraty F, Van Keilegom I, Vieu P (2012) Regression when both response and predictor are functions. J Multivar Anal 109:10–28

    MathSciNet  Article  Google Scholar 

  23. 23.

    García-Portugués E, Álvarez-Liébana J, Álvarez-Pérez G, González-Manteiga W (2019) A goodness-of-fit test for the functional linear model with functional response. arXiv preprint arXiv:1909.07686

  24. 24.

    Guyon X (1995) Random fields on a network: modeling, statistics, and applications. Springer Science & Business Media, Berlin

    Google Scholar 

  25. 25.

    Hallin M, Lu Z, Tran LT (2004) Local linear spatial regression. Ann Stat 32(6):2469–2500

    MathSciNet  Article  Google Scholar 

  26. 26.

    Hörmann S, Kokoszka P (2010) Weakly dependent functional data. Ann Stat 38(3):1845–1884

    MathSciNet  Article  Google Scholar 

  27. 27.

    Ibragimov IA (1962) Some limit theorems for stationary processes. Theory Probab Appl 7(4):349–382

    MathSciNet  Article  Google Scholar 

  28. 28.

    Imaizumi M, Kato K (2018) PCA-based estimation for functional linear regression with functional responses. J Multivar Anal 163:15–36

    MathSciNet  Article  Google Scholar 

  29. 29.

    Krebs JTN (2018) Orthogonal series estimates on strong spatial mixing data. J Stat Plan Inference 193:15–41

    MathSciNet  Article  Google Scholar 

  30. 30.

    Laib N, Louani D (2010) Nonparametric kernel regression estimation for functional stationary Ergodic data: asymptotic properties. J Multivar Anal 101(10):2266–2281

    MathSciNet  Article  Google Scholar 

  31. 31.

    Li L (2016) Nonparametric regression on random fields with random design using wavelet method. Stat Inference Stoch Process 19(1):51–69

    MathSciNet  Article  Google Scholar 

  32. 32.

    Maume-Deschamps V (2006) Exponential inequalities and functional estimations for weak dependent data: applications to dynamical systems. Stoch Dyn 6(04):535–560

    MathSciNet  Article  Google Scholar 

  33. 33.

    Merlevède F, Peligrad M, Rio E (2009) Bernstein inequality and moderate deviations under strong mixing conditions, volume 5 of Collections, pages 273–292. Institute of Mathematical Statistics, Beachwood, Ohio, USA

  34. 34.

    Politis DN, Romano JP (1994) Limit theorems for weakly dependent Hilbert space valued random variables with application to the stationary bootstrap. Stat Sin 4:461–476

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Ramsay JO, Silverman B (1997) Functional data analysis. Springer, Berlin

    Google Scholar 

  36. 36.

    Rosenblatt M (1956) A central limit theorem and a strong mixing condition. Proc Natl Acad Sci 42(1):43–47

    MathSciNet  Article  Google Scholar 

  37. 37.

    Slaoui Y (2019) Wild bootstrap bandwidth selection of recursive nonparametric relative regression for independent functional data. J Multivar Anal 173:494–511

    MathSciNet  Article  Google Scholar 

  38. 38.

    Slaoui Y (2020) Recursive nonparametric regression estimation for independent functional data. Stat Sin 30(1):417–37

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Tran LT (1990) Kernel density estimation on random fields. J Multivar Anal 34(1):37–53

    MathSciNet  Article  Google Scholar 

  40. 40.

    Valenzuela-Domínguez E, Krebs JTN, Franke JE (2017) A Bernstein inequality for spatial lattice processes. arXiv preprint arXiv:1702.02023

  41. 41.

    van der vaart A, Wellner J (2013) Weak convergence and empirical processes: with applications to statistics. Springer series in statistics. Springer, New York

    Google Scholar 

  42. 42.

    Yao F, Müller H-G, Wang J-L (2005) Functional linear regression analysis for longitudinal data. Ann Stat 33:2873–2903

    MathSciNet  Article  Google Scholar 

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This research was supported by the Deutsche Forschungsgemeinschaft (DFG), grant number KR 4977/1-1.

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Correspondence to Johannes Krebs.

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Lemma 6.1

( [27]) Let \(Z_1,\ldots ,Z_n\) be real-valued non-negative random variables each a.s. bounded. Set \(\alpha \,{:}{=}\,\sup _{s\in \{1,\ldots ,n\} } \alpha \left( \sigma ( Z_i: i \le k), \sigma ( Z_i: i > k) \right)\). Then \(\left| {\mathbb {E}}\left[ \, \prod _{i=1}^n Z_i \, \right] - \prod _{i=1}^n {\mathbb {E}}\left[ \, Z_i \, \right] \right| \le (n-1) \, \alpha \, \prod _{i=1}^n \left\Vert Z_i \right\Vert _{\infty }\).

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Krebs, J. A Note on Exponential Inequalities in Hilbert Spaces for Spatial Processes with Applications to the Functional Kernel Regression Model. J Stat Theory Pract 15, 20 (2021). https://doi.org/10.1007/s42519-020-00147-y

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  • Exponential inequalities
  • Functional data analysis
  • Functional kernel regression
  • Nonlinear regression operator
  • Nonparametric curve estimation
  • Spatial lattice processes
  • Strong mixing
  • Weak dependence measures

Mathematics Subject Classification

  • Primary: 62G08
  • 62M40
  • Secondary: 37A25
  • 62G20