Universal nonparametric kernel-type estimators for the mean and covariance functions of a stochastic process

Yu. Yu. Linke   ; I. S. Borisov

Teoriâ veroâtnostej i ee primeneniâ, Tome 69 (2024) no. 1, pp. 46-75

Résumé

Let $f_1(t), \dots, f_n(t)$ be independent copies of some a.s. continuous stochastic process $f(t)$, $t\in[0,1]$, which are observed with noise. We consider the problem of nonparametric estimation of the mean function $\mu(t) = \mathbf{E}f(t)$ and of the covariance function $\psi(t,s)=\operatorname{Cov}\{f(t),f(s)\}$ if the noise values of each of the copies $f_i(t)$, $i=1,\dots,n$, are observed in some collection of generally random (in general) time points (regressors). Under wide assumptions on the time points, we construct uniformly consistent kernel estimators for the mean and covariance functions both in the case of sparse data (where the number of observations for each copy of the stochastic process is uniformly bounded) and in the case of dense data (where the number of observations at each of $n$ series is increasing as $n\to\infty$). In contrast to the previous studies, our kernel estimators are universal with respect to the structure of time points, which can be either fixed rather than necessarily regular, or random rather than necessarily formed of independent or weakly dependent random variables.

Détail

Keywords: nonparametric regression, estimator of the mean function, estimator of the covariance function, kernel estimator, uniform consistency.

Bibliographie

[1] A. Cuevas, “A partial overview of the theory of statistics with functional data”, J. Statist. Plann. Inference, 147 (2014), 1–23 | DOI | MR | Zbl

[2] P. Kokoszka, M. Reimherr, Introduction to functional data analysis, Texts Statist. Sci. Ser., CRC Press, Boca Raton, FL, 2017, xvi+290 pp. | DOI | MR | Zbl

[3] H.-G. Müller, “Functional modelling and classification of longitudinal data”, Scand. J. Statist., 32:2 (2005), 223–240 | DOI | MR | Zbl

[4] Tailen Hsing, R. Eubank, Theoretical foundations of functional data analysis, with an introduction to linear operators, Wiley Ser. Probab. Stat., John Wiley Sons, Ltd., Chichester, 2015, xiv+334 pp. | DOI | MR | Zbl

[5] F. Bunea, A. E. Ivanescu, M. H. Wegkamp, “Adaptive inference for the mean of a Gaussian process in functional data”, J. R. Stat. Soc. Ser. B Stat. Methodol., 73:4 (2011), 531–558 | DOI | MR | Zbl

[6] T. Tony Cai, Ming Yuan, “Optimal estimation of the mean function based on discretely sampled functional data: phase transition”, Ann. Statist., 39:5 (2011), 2330–2355 | DOI | MR | Zbl

[7] Guanqun Cao, Li Wang, Yehua Li, Lijian Yang, “Oracle-efficient confidence envelopes for covariance functions in dense functional data”, Statist. Sinica, 26:1 (2016), 359–383 | DOI | MR | Zbl

[8] Guanqun Cao, Lijian Yang, D. Todem, “Simultaneous inference for the mean function of dense functional data”, J. Nonparametr. Stat., 24:2 (2012), 359–377 | DOI | MR | Zbl

[9] P. Hall, H.-G. Müller, Jane-Ling Wang, “Properties of principal component methods for functional and longitudinal data analysis”, Ann. Statist., 34:3 (2006), 1493–1517 | DOI | MR | Zbl

[10] Seonjin Kim, Zhibiao Zhao, “Unified inference for sparse and dense longitudinal models”, Biometrika, 100:1 (2013), 203–212 | DOI | MR | Zbl

[11] Yehua Li, Tailen Hsing, “Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data”, Ann. Statist., 38:6 (2010), 3321–3351 | DOI | MR | Zbl

[12] Zhenhua Lin, Jane-Ling Wang, “Mean and covariance estimation for functional snippets”, J. Amer. Statist. Assoc., 117:537 (2022), 348–360 | DOI | MR | Zbl

[13] Shujie Ma, Lijian Yang, R. J. Carroll, “A simultaneous confidence band for sparse longitudinal regression”, Statist. Sinica, 22:1 (2012), 95–122 | DOI | MR | Zbl

[14] Q. Song, R. Liu, Q. Shao, L. Yang, “A simultaneous confidence band for dense longitudinal regression”, Comm. Statist. Theory Methods, 43:24 (2014), 5195–5210 | DOI | MR | Zbl

[15] Jiangyan Wang, Guanqun Cao, Li Wang, Lijian Yang, “Simultaneous confidence band for stationary covariance function of dense functional data”, J. Multivariate Anal., 176 (2020), 104584, 21 pp. | DOI | MR | Zbl

[16] Hulin Wu, Jin-Ting Zhang, Nonparametric regression methods for longitudinal data analysis. Mixed-effects modeling approaches, Wiley Ser. Probab. Stat., Wiley-Interscience [John Wiley Sons], Hoboken, NJ, 2006, xxii+369 pp. | DOI | MR | Zbl

[17] Fang Yao, “Asymptotic distributions of nonparametric regression estimators for longitudinal or functional data”, J. Multivariate Anal., 98:1 (2007), 40–56 | DOI | MR | Zbl

[18] D. Degras, “Asymptotics for the nonparametric estimation of the mean function of a random process”, Statist. Probab. Lett., 78:17 (2008), 2976–2980 | DOI | MR | Zbl

[19] Fang Yao, H.-G. Müller, Jane-Ling Wang, “Functional data analysis for sparse longitudinal data”, J. Amer. Statist. Assoc., 100:470 (2005), 577–590 | DOI | MR | Zbl

[20] Jin-Ting Zhang, Jianwei Chen, “Statistical inferences for functional data”, Ann. Statist., 35:3 (2007), 1052–1079 | DOI | MR | Zbl

[21] Xiaoke Zhang, Jane-Ling Wang, “Optimal weighting schemes for longitudinal and functional data”, Statist. Probab. Lett., 138 (2018), 165–170 | DOI | MR | Zbl

[22] Xiaoke Zhang, Jane-Ling Wang, “From sparse to dense functional data and beyond”, Ann. Statist., 44:5 (2016), 2281–2321 | DOI | MR | Zbl

[23] Shuzhuan Zheng, Lijian Yang, W. K. Härdle, “A smooth simultaneous confidence corridor for the mean of sparse functional data”, J. Amer. Statist. Assoc., 109:506 (2014), 661–673 | DOI | MR | Zbl

[24] Ling Zhou, Huazhen Lin, Hua Liang, “Efficient estimation of the nonparametric mean and covariance functions for longitudinal and sparse functional data”, J. Amer. Statist. Assoc., 113:524 (2018), 1550–1564 | DOI | MR | Zbl

[25] G. M. James, T. J. Hastie, “Functional linear discriminant analysis for irregularly sampled curves”, J. R. Stat. Soc. Ser. B Stat. Methodol., 63:3 (2001), 533–550 | DOI | MR | Zbl

[26] Jane-Ling Wang, Jeng-Min Chiou, H.-G. Müller, “Functional data analysis”, Annu. Rev. Stat. Appl., 3 (2016), 257–295 | DOI

[27] D. Gervini, “Free-knot spline smoothing for functional data”, J. R. Stat. Soc. Ser. B Stat. Methodol., 68:4 (2006), 671–687 | DOI | MR | Zbl

[28] J. A. Rice, Colin O. Wu, “Nonparametric mixed effects models for unequally sampled noisy curves”, Biometrics, 57:1 (2001), 253–259 | DOI | MR | Zbl

[29] Fang Yao, T. C. M. Lee, “Penalized spline models for functional principal component analysis”, J. R. Stat. Soc. Ser. B Stat. Methodol., 68:1 (2006), 3–25 | DOI | MR | Zbl

[30] J. A. Rice, B. W. Silverman, “Estimating the mean and covariance structure nonparametrically when the data are curves”, J. Roy. Statist. Soc. Ser. B, 53:1 (1991), 233–243 | MR | Zbl

[31] I. S. Borisov, Yu. Yu. Linke, P. S. Ruzankin, “Universal weighted kernel-type estimators for some class of regression models”, Metrika, 84:2 (2021), 141–166 | DOI | MR | Zbl

[32] Y. Y. Linke, I. S. Borisov, P. S. Ruzankin, “Universal kernel-type estimation of random fields”, Statistics, 57:4 (2023), 785–810 | DOI | MR | Zbl

[33] Y. Linke, I. Borisov, P. Ruzankin, V. Kutsenko, E. Yarovaya, S. Shalnova, “Universal local linear kernel estimators in nonparametric regression”, Mathematics, 10:15 (2022), 2693, 28 pp. | DOI

[34] Yu. Yu. Linke, “Towards insensitivity of Nadaraya–Watson estimators to design correlation”, Theory Probab. Appl., 68:2 (2023), 198–210 | DOI | DOI | MR | Zbl

[35] Yu. Linke, I. Borisov, “Insensitivity of Nadaraya–Watson estimators to design correlation”, Comm. Statist. Theory Methods, 51:19 (2022), 6909–6918 | DOI | MR | Zbl

[36] Yu. Yu. Linke, “Asymptotic properties of one-step weighted $M$-estimators with applications to regression”, Theory Probab. Appl., 62:3 (2018), 373–398 | DOI | DOI | MR | Zbl

[37] Yu. Yu. Linke, I. S. Borisov, “Constructing explicit estimators in nonlinear regression problems”, Theory Probab. Appl., 63:1 (2018), 22–44 | DOI | DOI | MR | Zbl

[38] Yu. Linke, “Asymptotic properties of one-step $M$-estimators”, Comm. Statist. Theory Methods, 48:16 (2019), 4096–4118 | DOI | MR | Zbl

[39] Yu. Yu. Linke, I. S. Borisov, “Constructing initial estimators in one-step estimation procedures of nonlinear regression”, Statist. Probab. Lett., 120 (2017), 87–94 | DOI | MR | Zbl

[40] Yu. Yu. Linke, I. S. Borisov, “An approach to constructing explicit estimators in nonlinear regression”, Siberian Adv. Math., 33:4 (2023), 338–346 | DOI

[41] Yu. Yu. Linke, “Asymptotic normality of one-step $M$-estimators based on non-identically distributed observations”, Statist. Probab. Lett., 129 (2017), 216–221 | DOI | MR | Zbl

[42] N. N. Chentsov, “Weak convergence of stochastic processes whose trajectories have no discontinuities of the second kind and the “heuristic” approach to the Kolmogorov–Smirnov tests”, Theory Probab. Appl., 1:1 (1956), 140–144 | DOI | MR