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Functional Uniform-in-Bandwidth Moderate Deviation Principle for the Local Empirical Processes Involving Functional Data

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Abstract

Our research employs general empirical process methods to investigate and establish moderate deviation principles for kernel-type function estimators that rely on an infinite-dimensional covariate, subject to mild regularity conditions. In doing so, we introduce a valuable moderate deviation principle for a function-indexed process, utilizing intricate exponential contiguity arguments. The primary objective of this paper is to contribute to the existing literature on functional data analysis by establishing functional moderate deviation principles for both Nadaraya–Watson and conditional distribution processes. These principles serve as fundamental tools for analyzing and understanding the behavior of these processes in the context of functional data analysis. By extending the scope of moderate deviation principles to the realm of functional data analysis, we enhance our understanding of the statistical properties and limitations of kernel-type function estimators when dealing with infinite-dimensional covariates. Our findings provide valuable insights and contribute to the advancement of statistical methodology in functional data analysis.

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Notes

  1. Let us first recall the concept of large and moderate deviations. A sequence \(\left\{Z_{n},n\geq 1\right\}\) of \(\mathbb{R}\)-valued random variables is said to satisfy a large deviation principle (LDP) with speed \(v_{n}\) and rate function \(I(\cdot)\) if for any closed set \(F\subset\mathbb{R}\),

    $$\limsup_{n\rightarrow\infty}v_{n}^{-1}\log\left(\mathbb{P}\left(Z_{n}\in F\right)\right)\leq-\inf_{x\in F}I(x)$$

    and any open set \(G\subset\mathbb{R}\),

    $$\liminf_{n\rightarrow\infty}v_{n}^{-1}\log\left(\mathbb{P}\left(Z_{n}\in G\right)\right)\geq-\inf_{x\in G}I(x).$$

    Let \(a_{n}\) be a nonrandom sequence that goes to infinity, if there exists function \(c(n)\), and \(\left(a_{n}\left(Z_{n}-c(n)\right)\right)\) satisfies an LDP, then \(Z_{n}\) is said to satisfy a moderate deviation principles (MDP). Roughly speaking, the MDP for \(Z_{n}\) is the LDP for \(\left(a_{n}\left(Z_{n}-c(n)\right)\right)\).

  2. A semi-metric (sometimes called pseudo-metric) \(d(\cdot,\cdot)\) is a metric which allows \(d(x_{1},x_{2})=0\) for some \(x_{1}\neq x_{2}\).

  3. Given two functions \(l\) and \(u\), the interval \([l,u]\) represents the set of all functions \(f\) such that \(l\leq f\leq u\). An \(\varepsilon\)-bracket is defined as \([l,u]\) with \(||u-l||<\varepsilon\). The bracketing number \(N_{[]}(\mathcal{F},||\cdot||,\varepsilon)\) corresponds to the minimum number of \(\varepsilon\)-brackets required to encompass the class \(\mathcal{F}\). The entropy with bracketing is expressed as the logarithm of the bracketing number. It’s important to note that, in the definition of the bracketing number, the upper and lower bounds \(u\) and \(l\) of the brackets need not be part of \(\mathcal{F}\) itself, but they are assumed to have finite norms, refer to Definition 2.1.6 in [131].

REFERENCES

  1. I. M. Almanjahie, S. Bouzebda, Z. Chikr Elmezouar, and A. Laksaci, ‘‘The functional \(k\)NN estimator of the conditional expectile: Uniform consistency in number of neighbors,’’ Stat. Risk Model. 38 (3–4), 47–63 (2022).

    Article  MathSciNet  Google Scholar 

  2. I. M. Almanjahie, S. Bouzebda, Z. Kaid, and A. Laksaci, ‘‘Nonparametric estimation of expectile regression in functional dependent data,’’ J. Nonparametr. Stat. 34 (1), 250–281 (2022).

    Article  MathSciNet  Google Scholar 

  3. I. M. Almanjahie, S. Bouzebda, Z. Kaid, and A. Laksaci, ‘‘The local linear functional \(k\)NN estimator of the conditional expectile: Uniform consistency in number of neighbors,’’ Metrika 34 (1), 1–29 (2024).

    Google Scholar 

  4. G. Aneiros, R. Cao, R. Fraiman, and P. Vieu, ‘‘Editorial for the special issue on functional data analysis and related topics,’’ J. Multivariate Anal. 170, 1–2 (2019).

    Article  MathSciNet  Google Scholar 

  5. G. Aneiros-Pérez and P. Vieu, ‘‘Nonparametric time series prediction: A semi-functional partial linear modeling,’’ J. Multivariate Anal. 99 (5), 834–857 (2008).

    Article  MathSciNet  Google Scholar 

  6. M. A. Arcones, ‘‘The large deviation principle for stochastic processes. I,’’ Teor. Veroyatnost. i Primenen. 47 (4), 727–746 (2002).

    Article  Google Scholar 

  7. M. A. Arcones, ‘‘The large deviation principle for stochastic processes. II,’’ Teor. Veroyatnost. i Primenen. 48 (1), 122–150 (2003).

    Article  MathSciNet  Google Scholar 

  8. M. A. Arcones, Moderate Deviations of Empirical Processes. In Stochastic Inequalities and Applications, Vol. 56 of Progr. Probab. (Birkhäuser, Basel, 2003), pp. 189–212.

    Google Scholar 

  9. R. R. Bahadur, Some limit theorems in statistics, No. 4: Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (Philadelphia, PA, 1971).

    Book  Google Scholar 

  10. R. R. Bahadur and S. L. Zabell, ‘‘Large deviations of the sample mean in general vector spaces,’’ Ann. Probab. 7 (4), 587–621 (1979).

    Article  MathSciNet  Google Scholar 

  11. N. Berrahou, ‘‘Principe de grandes déviations uniforme pour l’estimateur de la densité par la méthode des delta-suites,’’ C. R. Math. Acad. Sci. Paris 343 (9), 595–600 (2006).

    Article  MathSciNet  Google Scholar 

  12. N. Berrahou, ‘‘Large deviations probabilities for a symmetry test statistic based on delta-sequence density estimation,’’ Statist. Probab. Lett. 78 (3), 238–248 (2008).

    Article  MathSciNet  Google Scholar 

  13. V. I. Bogachev, Gaussian Measures, Vol. 62: Mathematical Surveys and Monographs. American Mathematical Society (Providence, RI, 1998).

  14. D. Bosq, Linear Processes in Function Spaces, Vol. 149: Lecture Notes in Statistics (Springer-Verlag, New York. Theory and Applications, 2000).

  15. S. Bouzebda, ‘‘On the strong approximation of bootstrapped empirical copula processes with applications,’’ Math. Methods Statist. 21 (3), 153–188 (2012).

    Article  MathSciNet  Google Scholar 

  16. S. Bouzebda, ‘‘General tests of conditional independence based on empirical processes indexed by functions,’’ Jpn. J. Stat. Data Sci. 6 (1), 115–177 (2023).

    Article  MathSciNet  Google Scholar 

  17. S. Bouzebda, ‘‘On the weak convergence and the uniform-in-bandwidth consistency of the general conditional \(U\)-processes based on the copula representation: Multivariate setting,’’ Hacet. J. Math. Stat. 52 (5), 1303–1348 (2023).

    Article  MathSciNet  Google Scholar 

  18. S. Bouzebda and M. Chaouch, ‘‘Uniform limit theorems for a class of conditional \(Z\)-estimators when covariates are functions,’’ J. Multivariate Anal. 189 (104872), 21 (2022).

    Article  Google Scholar 

  19. S. Bouzebda and M. Cherfi, ‘‘General bootstrap for dual \(\phi\)-divergence estimates,’’ J. Probab. Stat., Art. ID 834107, 33 (2012).

    Google Scholar 

  20. S. Bouzebda and S. Didi, ‘‘Some results about kernel estimators for function derivatives based on stationary and ergodic continuous time processes with applications,’’ Comm. Statist. Theory Methods 51 (12), 3886–3933 (2022).

    Article  MathSciNet  Google Scholar 

  21. S. Bouzebda and I. Elhattab, ‘‘Uniform in bandwidth consistency of the kernel-type estimator of the Shannon’s entropy,’’ C. R. Math. Acad. Sci. Paris 348 (5–6), 317–321 (2010).

    Article  MathSciNet  Google Scholar 

  22. S. Bouzebda and I. Elhattab, ‘‘Uniform-in-bandwidth consistency for kernel-type estimators of Shannon’s entropy,’’ Electron. J. Stat. 5, 440–459 (2011).

    Article  MathSciNet  Google Scholar 

  23. S. Bouzebda and N. Limnios, ‘‘The uniform CLT for the empirical estimator of countable state space semi-Markov kernels indexed by functions with applications,’’ J. Nonparametr. Stat. 34 (4), 758–788 (2022).

    Article  MathSciNet  Google Scholar 

  24. S. Bouzebda and B. Nemouchi, ‘‘Weak-convergence of empirical conditional processes and conditional \(U\)-processes involving functional mixing data,’’ Stat. Inference Stoch. Process. 26 (1), 33–88 (2023).

    Article  MathSciNet  Google Scholar 

  25. S. Bouzebda and A. Nezzal, ‘‘Uniform consistency and uniform in number of neighbors consistency for nonparametric regression estimates and conditional \(U\)-statistics involving functional data,’’ Jpn. J. Stat. Data Sci. 5 (2), 431–533 (2022).

    Article  MathSciNet  Google Scholar 

  26. S. Bouzebda and A. Nezzal, ‘‘Asymptotic properties of conditional \(U\)-statistics using delta sequences,’’ Comm. Statist. Theory Methods, 1–56 (2024). https://doi.org/10.1080/03610926.2023.2179887

  27. S. Bouzebda and A. Nezzal, ‘‘Uniform in number of neighbors consistency and weak convergence of \(k\)NN empirical conditional processes and \(k\)NN conditional \(U\)-processes involving functional mixing data,’’ AIMS Math. 9 (2), 4427–4550 (2024).

    Article  MathSciNet  Google Scholar 

  28. S. Bouzebda and I. Soukarieh, ‘‘Non-parametric conditional \(U\)-processes for locally stationary functional random fields under stochastic sampling design,’’ Mathematics 11 (1), 1–70 (2023).

    Google Scholar 

  29. S. Bouzebda and N. Taachouche, On the variable bandwidth kernel estimation of conditional \(U\)-statistics at optimal rates in sup-norm. Phys. A 625 (129000), 72 (2023).

    Article  Google Scholar 

  30. S. Bouzebda and N. Taachouche, ‘‘Rates of the strong uniform consistency for the kernel-type regression function estimators with general kernels on manifolds,’’ Math. Methods Statist. 32 (1), 27–80 (2023).

    Article  MathSciNet  Google Scholar 

  31. S. Bouzebda and N. Taachouche, ‘‘Rates of the strong uniform consistency with rates for conditional \(U\)-statistics estimators with general kernels on manifolds,’’ Math. Methods Statist. 33 (1), 1–55 (2024).

    Google Scholar 

  32. S. Bouzebda and T. Zari, ‘‘Strong approximation of multidimensional \(\mathbb{P}\)\(\mathbb{P}\) plots processes by Gaussian processes with applications to statistical tests,’’ Math. Methods Statist. 23 (3), 210–238 (2014).

    Article  MathSciNet  Google Scholar 

  33. S. Bouzebda, I. Elhattab, and C. T. Seck, ‘‘Uniform in bandwidth consistency of nonparametric regression based on copula representation,’’ Statist. Probab. Lett. 137, 173–182 (2018).

    Article  MathSciNet  Google Scholar 

  34. S. Bouzebda, I. Elhattab, and B. Nemouchi, ‘‘On the uniform-in-bandwidth consistency of the general conditional \(U\)-statistics based on the copula representation,’’ J. Nonparametr. Stat. 33 (2), 321–358 (2021).

    Article  MathSciNet  Google Scholar 

  35. S. Bouzebda, M. Chaouch, and S. Didi Biha, ‘‘Asymptotics for function derivatives estimators based on stationary and ergodic discrete time processes,’’ Ann. Inst. Statist. Math. 74 (4), 737–771 (2022).

    Article  MathSciNet  Google Scholar 

  36. S. Bouzebda, I. Elhattab, and A. A. Ferfache, ‘‘General \(M\)-estimator processes and their \(m\) out of \(n\) bootstrap with functional nuisance parameters,’’ Methodol. Comput. Appl. Probab. 24 (4), 2961–3005 (2022b).

    Article  MathSciNet  Google Scholar 

  37. S. Bouzebda, A. Laksaci, and M. Mohammedi, ‘‘Single index regression model for functional quasi-associated time series data,’’ REVSTAT 20 (5), 605–631 (2022c).

    MathSciNet  Google Scholar 

  38. S. Bouzebda, A. Laksaci, and M. Mohammedi, ‘‘The \(k\)-nearest neighbors method in single index regression model for functional quasi-associated time series data,’’ Rev. Mat. Complut. 36 (2), 361–391 (2023).

    Article  MathSciNet  Google Scholar 

  39. J. E. Chacón and T. Duong, Multivariate Kernel Smoothing and Its Applications, Vol. 160: Monographs on Statistics and Applied Probability (CRC Press, Boca Raton, FL, 2018).

  40. D. Chen, P. Hall, and H.-G. Müller, ‘‘Single and multiple index functional regression models with nonparametric link,’’ Ann. Statist. 39 (3), 1720–1747 (2011).

    Article  MathSciNet  Google Scholar 

  41. M. Cherfi, ‘‘Large deviations theorems in nonparametric regression on functional data,’’ C. R. Math. Acad. Sci. Paris 349 (9–10), 583–585 (2011).

    Article  MathSciNet  Google Scholar 

  42. H. Chernoff, ‘‘A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations,’’ Ann. Math. Statistics 23, 493–507 (1952).

    Article  MathSciNet  Google Scholar 

  43. K. Chokri and S. Bouzebda, ‘‘Uniform-in-bandwidth consistency results in the partially linear additive model components estimation,’’ Comm. Statist. Theory Methods, 1–42 (2023).

  44. J. A. Clarkson and C. R. Adams, ‘‘On definitions of bounded variation for functions of two variables,’’ Trans. Amer. Math. Soc. 35 (4), 824–854 (1933).

    Article  MathSciNet  Google Scholar 

  45. S. Dabo-Niang and A. Laksaci, ‘‘Nonparametric quantile regression estimation for functional dependent data,’’ Comm. Statist. Theory Methods 41 (7), 1254–1268 (2012).

    Article  MathSciNet  Google Scholar 

  46. P. Deheuvels, ‘‘One bootstrap suffices to generate sharp uniform bounds in functional estimation,’’ Kybernetika (Prague) 47 (6), 855–865 (2011).

    MathSciNet  Google Scholar 

  47. P. Deheuvels, ‘‘Uniform-in-bandwidth functional limit laws for multivariate empirical processes,’’ in: High dimensional probability VIII–The Oaxaca volume, Vol. 74 of Progr. Probab. (Birkhäuser/Springer, Cham, 2019), pp. 201–239.

    Google Scholar 

  48. P. Deheuvels and D. M. Mason, ‘‘General asymptotic confidence bands based on kernel-type function estimators,’’ Stat. Inference Stoch. Process. 7 (3), 225–277 (2004).

    Article  MathSciNet  Google Scholar 

  49. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Vol. 38: Applications of Mathematics, 2nd ed. (Springer-Verlag, New York, 1998).

    Google Scholar 

  50. J.-D. Deuschel and D. W. Stroock, Large Deviations, Vol. 137: Pure and Applied Mathematics (Academic Press, Inc., Boston, MA, 1989).

  51. L. Devroye and L. Györfi, Nonparametric Density Estimation. Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics (John Wiley and Sons, Inc., New York, The \(L_{1}\) View, 1985).

  52. L. Devroye and G. Lugosi, Combinatorial Methods in Density Estimation, Springer Series in Statistics (Springer-Verlag, New York, 2001).

  53. J. Dony and U. Einmahl, ‘‘Uniform in bandwidth consistency of kernel regression estimators at a fixed point,’’ in: High dimensional probability V: The Luminy volume, Vol. 5: Inst. Math. Stat. (IMS) Collect. Inst. Math. Statist (Beachwood, OH, 2009), p. 308–325.

  54. L. Douge, ‘‘Théorèmes limites pour des variables quasi-associées hilbertiennes,’’ Ann. I.S.U.P. 54 (1–2), 51–60 (2010).

    MathSciNet  Google Scholar 

  55. R. M. Dudley, Uniform Central Limit Theorems, Vol. 63: Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 1999).

  56. N. Dunford and J. T. Schwartz, Linear Operators, I: General Theory, Pure and Applied Mathematics, Vol. 7 (Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London. With the assistance of W. G. Bade and R. G. Bartle, 1958).

  57. P. P. B. Eggermont and V. N. LaRiccia, Maximum Penalized Likelihood Estimation, Vol. II (Springer Series in Statistics. Springer, Dordrecht, 2009).

  58. U. Einmahl and D. M. Mason, ‘‘An empirical process approach to the uniform consistency of kernel-type function estimators,’’ J. Theoret. Probab. 13 (1), 1–37 (2000).

    Article  MathSciNet  Google Scholar 

  59. U. Einmahl and D. M. Mason, ‘‘Uniform in bandwidth consistency of kernel-type function estimators,’’ Ann. Statist. 33 (3), 1380–1403 (2005).

    Article  MathSciNet  Google Scholar 

  60. R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics, Classics in Mathematics (Springer-Verlag, Berlin, Reprint of the 1985 original, 2006).

  61. F. Ferraty and P. Vieu, ‘‘Dimension fractale et estimation de la régression dans des espaces vectoriels semi-normés,’’ C. R. Acad. Sci. Paris Sér. I Math. 330 (2), 139–142 (2000).

    Article  MathSciNet  Google Scholar 

  62. F. Ferraty and P. Vieu, Nonparametric Functional Data Analysis, Springer Series in Statistics (Springer, New York, Theory and Practice, 2006).

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

    Article  MathSciNet  Google Scholar 

  64. F. Ferraty, A. Mas, and P. Vieu, ‘‘Nonparametric regression on functional data: Inference and practical aspects,’’ Australian and New Zealand Journal of Statistics 49, 267–286 (2007).

    Article  MathSciNet  Google Scholar 

  65. F. Ferraty, A. Laksaci, A. Tadj, and P. Vieu, ‘‘Rate of uniform consistency for nonparametric estimates with functional variables,’’ J. Statist. Plann. Inference 140 (2), 335–352 (2010).

    Article  MathSciNet  Google Scholar 

  66. J. C. Fu, ‘‘Large sample point estimation: A large deviation theory approach,’’ Ann. Statist. 10 (3), 762–771 (1982).

    Article  MathSciNet  Google Scholar 

  67. F. Gao and X. Zhao, ‘‘Delta method in large deviations and moderate deviations for estimators,’’ Ann. Statist. 39 (2), 1211–1240 (2011).

    Article  MathSciNet  Google Scholar 

  68. T. Gasser, P. Hall, and B. Presnell, ‘‘Nonparametric estimation of the mode of a distribution of random curves,’’ J. R. Stat. Soc. Ser. B Stat. Methodol. 60 (4), 681–691 (1998).

    Article  Google Scholar 

  69. I. Gijbels, M. Omelka, and N. Veraverbeke, ‘‘Multivariate and functional covariates and conditional copulas,’’ Electron. J. Stat. 6, 1273–1306 (2012).

    Article  MathSciNet  Google Scholar 

  70. R. D. Gill, ‘‘Non- and semi-parametric maximum likelihood estimators and the von Mises method. I,’’ Scand. J. Statist. 16 (2), 97–128. With a discussion by J. A. Wellner and J. Præstgaard and a reply by the author (1989).

  71. E. Giné and A. Guillou, ‘‘On consistency of kernel density estimators for randomly censored data: Rates holding uniformly over adaptive intervals,’’ Ann. Inst. H. Poincaré Probab. Statist. 37 (4), 503–522 (2001).

    Google Scholar 

  72. E. Giné and R. Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, New York, 2016).

  73. E. Giné, V. Koltchinskii, and J. Zinn, ‘‘Weighted uniform consistency of kernel density estimators,’’ Ann. Probab. 32 (3B), 2570–2605 (2004).

    Article  MathSciNet  Google Scholar 

  74. A. Goia and P. Vieu, ‘‘A partitioned single functional index model,’’ Comput. Statist. 30 (3), 673–692 (2015).

    Article  MathSciNet  Google Scholar 

  75. P. Groeneboom, J. Oosterhoff, and F. H. Ruymgaart, ‘‘Large deviation theorems for empirical probability measures,’’ Ann. Probab. 7 (4), 553–586 (1979).

    Article  MathSciNet  Google Scholar 

  76. L. Györfi, M. Kohler, A. Krzyżak, and H. Walk, A Distribution-Free Theory of Nonparametric Regression, Springer Series in Statistics (Springer-Verlag, New York, 2002).

  77. P. Hall, ‘‘Asymptotic properties of integrated square error and cross-validation for kernel estimation of a regression function,’’ Z. Wahrsch. Verw. Gebiete 67 (2), 175–196 (1984).

    Article  MathSciNet  Google Scholar 

  78. W. Härdle, Applied Nonparametric Regression, Vol. 19: Econometric Society Monographs (Cambridge University Press, Cambridge, 1990).

  79. W. Härdle and J. S. Marron, ‘‘Optimal bandwidth selection in nonparametric regression function estimation,’’ Ann. Statist. 13 (4), 1465–1481 (1985).

    Article  MathSciNet  Google Scholar 

  80. G. H. Hardy, ‘‘On double fourier series and especially those which represent the double zeta-function with real and incommensurable parameters,’’ Quart. J. Math 37 (1), 53–79 (1905).

    Google Scholar 

  81. E. W. Hobson, The theory of functions of a real variable and the theory of Fourier’s series, Vol. II (Dover Publications, Inc., New York, N.Y., 1958).

    Book  Google Scholar 

  82. L. Horváth and P. Kokoszka, Inference for Functional Data with Applications (Springer Series in Statistics. Springer, New York, 2012).

  83. P. J. Huber, ‘‘Robust estimation of a location parameter,’’ Ann. Math. Statist. 35, 73–101 (1964).

    Article  MathSciNet  Google Scholar 

  84. W. C. M. Kallenberg, ‘‘Chernoff efficiency and deficiency,’’ Ann. Statist. 10 (2), 583–594 (1982).

    Article  MathSciNet  Google Scholar 

  85. W. C. M. Kallenberg, ‘‘Intermediate efficiency, theory and examples,’’ Ann. Statist. 11 (1), 170–182 (1983a).

    Article  MathSciNet  Google Scholar 

  86. W. C. M. Kallenberg, ‘‘On moderate deviation theory in estimation,’’ Ann. Statist. 11 (2), 498–504 (1983b).

    Article  MathSciNet  Google Scholar 

  87. L.-Z. Kara, A. Laksaci, M. Rachdi, and P. Vieu, ‘‘Data-driven \(k\)NN estimation in nonparametric functional data analysis,’’ J. Multivariate Anal. 153, 176–188 (2017).

    Article  MathSciNet  Google Scholar 

  88. A. D. M. Kester and W. C. M. Kallenberg, ‘‘Large deviations of estimators,’’ Ann. Statist. 14 (2), 648–664 (1986).

    Article  MathSciNet  Google Scholar 

  89. M. R. Kosorok, Introduction to Empirical Processes and Semiparametric Inference (Springer Series in Statistics, Springer, New York, 2008).

  90. M. Krause, ‘‘Über Mittelwertsätze im Gebiete der Doppelsummen und Doppelintegrale,’’ Leipz. Ber. 55, 239–263 (1903).

    Google Scholar 

  91. W. V. Li and Q.-M. Shao, Gaussian Processes: Inequalities, Small Ball Probabilities, and Applications, in: Stochastic Processes: Theory and Methods, Vol. 19: Handbook of Statist. (North-Holland, Amsterdam, 2001), p. 533–597.

  92. H. Lian, ‘‘Functional partial linear model,’’ J. Nonparametr. Stat. 23 (1), 115–128 (2011).

    Article  MathSciNet  Google Scholar 

  93. N. Ling and P. Vieu, ‘‘Nonparametric modelling for functional data: Selected survey and tracks for future,’’ Statistics 52 (4), 934–949 (2018).

    Article  MathSciNet  Google Scholar 

  94. Q. Liu and S. Zhao, ‘‘Pointwise and uniform moderate deviations for nonparametric regression function estimator on functional data,’’ Statist. Probab. Lett. 83 (5), 1372–1381 (2013).

    Article  MathSciNet  Google Scholar 

  95. D. Louani and S. M. Ould Maouloud, ‘‘Large deviation results for the nonparametric regression function estimator on functional data,’’ Math. Methods Statist. 21 (4), 298–313 (2012).

    Article  MathSciNet  Google Scholar 

  96. D. M. Mason, ‘‘Proving consistency of non-standard kernel estimators,’’ Stat. Inference Stoch. Process. 15 (2), 151–176 (2012).

    Article  MathSciNet  Google Scholar 

  97. D. M. Mason and M. A. Newton, ‘‘A rank statistics approach to the consistency of a general bootstrap,’’ Ann. Statist. 20 (3), 1611–1624 (1992).

    Article  MathSciNet  Google Scholar 

  98. D. M. Mason and J. W. H. Swanepoel, ‘‘A general result on the uniform in bandwidth consistency of kernel-type function estimators,’’ TEST 20 (1), 72–94 (2011).

    Article  MathSciNet  Google Scholar 

  99. D. M. Mason and J. W. H. Swanepoel, ‘‘Uniform in bandwidth consistency of kernel estimators of the density of mixed data,’’ Electron. J. Stat. 9 (1), 1518–1539 (2015).

    Article  MathSciNet  Google Scholar 

  100. E. Masry, ‘‘Nonparametric regression estimation for dependent functional data: asymptotic normality,’’ Stochastic Process. Appl. 115 (1), 155–177 (2005).

    Article  MathSciNet  Google Scholar 

  101. E. Mayer-Wolf and O. Zeitouni, ‘‘The probability of small Gaussian ellipsoids and associated conditional moments,’’ Ann. Probab. 21 (1), 14–24 (1993).

    Article  MathSciNet  Google Scholar 

  102. M. Mohammedi, S. Bouzebda, and A. Laksaci, ‘‘On the nonparametric estimation of the functional expectile regression,’’ C. R. Math. Acad. Sci. Paris 358 (3), 267–272 (2020).

    Article  MathSciNet  Google Scholar 

  103. M. Mohammedi, S. Bouzebda, and A. Laksaci, ‘‘The consistency and asymptotic normality of the kernel type expectile regression estimator for functional data,’’ J. Multivariate Anal. 181 (104673), 24 (2021).

    Article  MathSciNet  Google Scholar 

  104. A. Mokkadem and M. Pelletier, ‘‘Moderate deviations principles for the kernel estimator of nonrandom regression functions,’’ Afr. Stat. 11 (2), 995–1021 (2016).

    MathSciNet  Google Scholar 

  105. A. Mokkadem, M. Pelletier, and B. Thiam, ‘‘Large and moderate deviations principles for kernel estimators of the multivariate regression,’’ Math. Methods Statist. 17 (2), 146–172 (2008).

    Article  MathSciNet  Google Scholar 

  106. H.-G. Müller, Nonparametric Regression Analysis of Longitudinal Data, Vol. 46: Lecture Notes in Statistics (Springer-Verlag, Berlin, 1988).

  107. E. A. Nadaraja, ‘‘On a regression estimate,’’ Teor. Verojatnost. i Primenen. 9, 157–159 (1964).

    MathSciNet  Google Scholar 

  108. E. A. Nadaraja, Nonparametric Estimation of Probability Densities and Regression Curves, Vol. 20: Mathematics and Its Applications (Soviet Series) (Kluwer Academic Publishers Group, Dordrecht, 1989).

  109. R. B. Nelsen, An introduction to copulas. Springer Series in Statistics (Springer, New York, 2nd ed., 2006).

  110. W. K. Newey and J. L. Powell, ‘‘Asymmetric least squares estimation and testing,’’ Econometrica 55 (4), 819–847 (1987).

    Article  MathSciNet  Google Scholar 

  111. H. Niederreiter, Random Number Generation and Quasi-Monte-Carlo Methods, Vol. 63: CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM) (Philadelphia, PA, 1992).

  112. Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests (Cambridge University Press, Cambridge, 1995).

    Book  Google Scholar 

  113. D. Nolan and D. Pollard, ‘‘\(U\)-processes: Rates of convergence,’’ Ann. Statist. 15 (2), 780–799 (1987).

    Article  MathSciNet  Google Scholar 

  114. S. M. Ould Maouloud, ‘‘Some uniform large deviation results in nonparametric function estimation,’’ J. Nonparametr. Stat. 20 (2), 129–152 (2008).

    Article  MathSciNet  Google Scholar 

  115. E. Parzen, ‘‘On estimation of a probability density function and mode,’’ Ann. Math. Statist. 33, 1065–1076 (1962).

    Article  MathSciNet  Google Scholar 

  116. D. Pollard, Convergence of Stochastic Processes, Springer Series in Statistics (Springer-Verlag, New York, 1984).

  117. A. Puhalskii and V. Spokoiny, ‘‘On large-deviation efficiency in statistical inference,’’ Bernoulli 4 (2), 203–272 (1998).

    Article  MathSciNet  Google Scholar 

  118. M. Rachdi and P. Vieu, ‘‘Nonparametric regression for functional data: Automatic smoothing parameter selection,’’ J. Statist. Plann. Inference 137 (9), 2784–2801 (2007).

    Article  MathSciNet  Google Scholar 

  119. M. E. Radavichyus, ‘‘Probabilities of large deviations for maximum likelihood estimators,’’ Dokl. Akad. Nauk SSSR 268 (3), 551–556 (1983).

    MathSciNet  Google Scholar 

  120. J. O. Ramsay and B. W. Silverman, Functional Data Analysis, Springer Series in Statistics (Springer, New York, 2nd ed., 2005).

  121. G. G. Roussas, ‘‘Nonparametric estimation of the transition distribution function of a Markov process,’’ Ann. Math. Statist. 40, 1386–1400 (1969).

    Article  MathSciNet  Google Scholar 

  122. M. Samanta, ‘‘Nonparametric estimation of conditional quantiles,’’ Statist. Probab. Lett. 7 (5), 407–412 (1989).

    Article  MathSciNet  Google Scholar 

  123. I. N. Sanov, ‘‘On the probability of large deviations of random magnitudes,’’ Mat. Sb. (N.S.) 42(84), 11–44 (1957).

  124. D. W. Scott, Multivariate Density Estimation, Wiley Series in Probability and Statistics (John Wiley and Sons, Inc., Hoboken, NJ, 2nd ed., 2015).

  125. A. Sieders and K. Dzhaparidze, ‘‘A large deviation result for parameter estimators and its application to nonlinear regression analysis,’’ Ann. Statist. 15 (3), 1031–1049 (1987).

    Article  MathSciNet  Google Scholar 

  126. B. W. Silverman, Density Estimation for Statistics and Data Analysis, Monographs on Statistics and Applied Probability (Chapman and Hall, London, 1986).

  127. I. Soukarieh and S. Bouzebda, ‘‘Renewal type bootstrap for increasing degree \(U\)-process of a Markov chain,’’ J. Multivariate Anal. 195 (105143), 25 (2023).

    Article  Google Scholar 

  128. I. Soukarieh and S. Bouzebda, ‘‘Weak convergence of the conditional \(U\)-statistics for locally stationary functional time series,’’ Stat. Inference Stoch. Process. 1–78 (2024). https://doi.org/10.1007/s11203-023-09305-y

  129. W. Stute, ‘‘Conditional empirical processes,’’ Ann. Statist. 14 (2), 638–647 (1986).

    Article  MathSciNet  Google Scholar 

  130. R. A. Tapia and J. R. Thompson, Nonparametric Probability Density Estimation, Vol. 1: Johns Hopkins Series in the Mathematical Sciences (Johns Hopkins University Press, Baltimore, Md., 1978).

  131. A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, Springer Series in Statistics (Springer-Verlag, New York, 1996).

  132. S. R. S. Varadhan, Large deviations and applications, in: École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–1987, Vol. 1362: Lecture Notes in Math. (Springer, Berlin, 1988), pp. 1–49.

  133. S. R. S. Varadhan, Large Deviations, Vol. 27: Courant Lecture Notes in Mathematics (Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2016).

  134. G. Vitali, ‘‘Sui gruppi di punti e sulle funzioni di variabili reali,’’ Torino Atti 43, 229–246 (1908).

    Google Scholar 

  135. A. G. Vituškin, O mnogomernykh variatsiyakh (Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1955).

    Google Scholar 

  136. M. P. Wand and M. C. Jones, Kernel Smoothing, Vol. 60: Monographs on Statistics and Applied Probability (Chapman and Hall, Ltd., London, 1995).

  137. G. S. Watson, ‘‘Smooth regression analysis,’’ Sankhyā Ser. A 26, 359–372 (1964).

    MathSciNet  Google Scholar 

  138. W. Wertz, Statistical Density Estimation: A Survey, Vol. 13: Angewandte Statistik und Ökonometrie [Applied Statistics and Econometrics] (Vandenhoeck and Ruprecht, Göttingen, With German and French Summaries, 1978).

  139. C.Wu, N. Ling, P. Vieu, and W. Liang, ‘‘Partially functional linear quantile regression model and variable selection with censoring indicators MAR,’’ J. Multivariate Anal. 197, Paper No. 105189 (2023).

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ACKNOWLEDGEMENTS

The authors express their gratitude to the Editor-in-Chief, an Associate Editor, and the referee for their invaluable comments. These remarks have significantly enhanced the original work, leading to a more focused and improved presentation.

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APPENDIX A

APPENDIX A

This appendix contains supplementary information that is essential to providing a more comprehensive understanding of the paper.

Theorem A.1 (Theorem 3.1 [6]). Let \(\left\{U_{n}(t):t\in T\right\}\) be a sequence of stochastic processes, where \(T\) is an index set. Let \(\left\{\varepsilon_{n}\right\}\) be a sequence of positive numbers that converge to zero. Let \(I:l_{\infty}(T)\rightarrow[0,\infty]\) and let \(I_{t_{1},\ldots,t_{m}}:\mathbb{R}^{m}\rightarrow[0,\infty]\) be a function for each \(t_{1},\ldots,t_{m}\in T\). Let \(d(\cdot,\cdot)\) be a pseudometric in \(T\). Consider the following conditions:

(a.1) \((T,d)\) is totally bounded;

(a.2) for each \(t_{1},\ldots,t_{m}\in T,\left(U_{n}\left(t_{1}\right),\ldots,U_{n}\left(t_{m}\right)\right)\) satisfies the LDP with the rate \(\varepsilon_{n}^{-1}\) and good rate function \(I_{t_{1},\ldots,t_{m}}\);

(a.3) for each \(\tau>0\),

$$\lim_{\eta\rightarrow 0}\limsup_{n\rightarrow\infty}\varepsilon_{n}\log\left(\mathbb{P}^{*}\left\{\sup_{d(s,t)\leq\eta}\left|U_{n}(t)-U_{n}(s)\right|\geq\tau\right\}\right)=-\infty;$$

(b.1) for each \(0\leq c<\infty,\bigg{\{}z\in l_{\infty}(T):I(z)\leq c\bigg{\}}\) is a compact set of \(l_{\infty}(T)\);

(b.2) for each \(A\subset l_{\infty}(T)\),

$$-\inf_{z\in A^{\circ}}I(z)\leq\liminf_{n\rightarrow\infty}\varepsilon_{n}\log\left(\mathbb{P}_{*}\left\{U_{n}\in A\right\}\right)$$
$${}\leq\limsup_{n\rightarrow\infty}\varepsilon_{n}\log\left(\mathbb{P}^{*}\left\{U_{n}\in A\right\}\right)\leq-\inf_{z\in\bar{A}}I(z).$$

If the set of conditions (a) is satisfied, then the set of conditions (b) holds with \(I(\cdot)\) given by

$$I(z)=\sup\bigg{\{}I_{t_{1},\ldots,t_{m}}\left(z\left(t_{1}\right),\ldots,z\left(t_{m}\right)\right):t_{1},\ldots,t_{m}\in T,m\geq 1\bigg{\}}.$$

If the set of conditions (b) is satisfied, then the set of conditions (a) holds with

$$I_{t_{1},\ldots,t_{m}}\left(u_{1},\ldots,u_{m}\right)=\inf\bigg{\{}I(z):z\in l_{\infty}(T),\left(z\left(t_{1}\right),\ldots,z\left(t_{m}\right)\right)=\left(u_{1},\ldots,u_{m}\right)\bigg{\}}$$

and the pseudometric \(d(\cdot,\cdot)\) is defined

$$d(s,t)=\sum_{k=1}^{\infty}k^{-2}\min\left(d_{k}(s,t),1\right),$$

where

$$d_{k}(s,t)=\sup\left\{\left|u_{2}-u_{1}\right|:I_{s,t}\left(u_{1},u_{2}\right)\leqq k\right\}.$$

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Berrahou, NE., Bouzebda, S. & Douge, L. Functional Uniform-in-Bandwidth Moderate Deviation Principle for the Local Empirical Processes Involving Functional Data. Math. Meth. Stat. 33, 26–69 (2024). https://doi.org/10.3103/S1066530724700030

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