Kernel estimators for the mean function of a stochastic process under sparse design conditions

Yu. Yu. Linke

Matematičeskie trudy, Tome 25 (2022) no. 2, pp. 149-161.

Résumé

The problem of nonparametric estimation of the mean function for a continuous random process is considered, when the noisy values of each of its independent trajectories are observed in some random time points — design elements. Under broad conditions on the dependence of design elements, uniformly consistent estimates are constructed for the mean function in the case of one of the versions of so-called sparse design, when the number of design elements for each of the trajectories is the same and independent of the growing number of series of observations. Unlike the papers of predecessors, we do not require that the set of design elements consist of independent identically distributed or weakly dependent random variables. Regarding the design, it is only assumed that the entire set of design points with a high probability forms a refining partition of the domain of the random process under consideration.

Bibliographie

[1] Linke Yu.Yu., “K voprosu o nechuvstvitelnosti otsenok Nadaraya- – Vatsona otnositelno korrelyatsii elementov dizaina // Teoriya veroyatn. i ee primen.” (to appear)

[2] Linke Yu.Yu., “Asimptoticheskie svoistva odnoshagovykh vzveshennykh M-otsenok s prilozheniyami k zadacham regressii”, Teoriya veroyatn. i ee primen., 62:3 (2017), 468–498 | DOI

[3] Linke Yu.Yu., Borisov I.S., “Postroenie yavnykh otsenok v zadachakh nelineinoi regressii”, Teoriya veroyatn. i ee primen., 63:1 (2018), 29–56 | DOI | MR | Zbl

[4] Borisov I.S., Linke Yu.Yu., Ruzankin P.S., “Universal weighted kernel-type estimators for some class of regression models”, Metrika, 61:2 (2021), 141–166 | DOI | MR

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

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

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

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

[9] Hsing T., Eubank R., Theoretical foundations of functional data analysis, with an introduction to linear operators, John Wiley and Sons, 2015 | MR | Zbl

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

[11] Kokoszka P., Reimherr M., Introduction to functional data analysis, Chapman and Hall/CRC, 2017 | MR | Zbl

[12] Li Y., Hsing T., “Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data”, Ann. Statist., 38 (2010), 3321–3351 | MR | Zbl

[13] Lin Z., Wang J.-L., “Mean and covariance estimation for functional snippets”, J. Amer. Statist. Assoc., 117 (2022), 348–360 | DOI | MR | Zbl

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

[15] Linke Yu.Yu., Borisov I.S., “Insensitivity of Nadaraya–Watson estimators to design correlation”, Communications in Statistics – Theory and Methods, 2021 | MR

[16] Linke Yu.Yu., “Asymptotic properties of one-step M-estimators”, Communications in Statistics – Theory and Methods, 48:16 (2019), 4096–4118 | DOI | MR | Zbl

[17] Linke Yu.Yu., Borisov I.S., “Constructing initial estimators in one-step estimation procedures of nonlinear regression”, Statist. Probab. Lett., 120:1 (2017), 87–94 | DOI | MR | Zbl

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

[19] Muller H.-G., “Functional modelling and classification of longitudinal data”, Scand. J. Statist., 32 (2005), 223–246 | DOI | MR

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

[21] Wang J.-L., Chiou J.-M., Muller H.-G., “Review of functional data analysis”, Annu. Rev. Stat. Appl., 3 (2016), 257–295 | DOI

[22] Wu H., Zhang J.-T., Nonparametric regression methods for longitudinal data analysis: mixed-effects modeling approaches, John Wiley and Sons, 2006 | MR | Zbl

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

[24] Yao F., Muller H.-G., Wang J.-L., “Functional data analysis for sparse longitudinal data”, J. Amer. Statist. Assoc., 100 (2005), 577–590 | DOI | MR | Zbl

[25] Zhang J.-T., Chen J., “Statistical inferences for functional data”, Ann. Statist., 35:1 (2007), 1052–1079 | MR | Zbl

[26] Zhang X., Wang J.-L., “Optimal weighting schemes for longitudinal and functional data”, Stat. Prob. Lett., 138 (2018), 165–170 | DOI | MR | Zbl

[27] Zhang X., Wang J.-L., “From sparse to dense functional data and beyond”, Ann. Statist., 44:5 (2016), 2281–2321 | MR | Zbl

[28] Zheng S., Yang L., Hardle W., “A smooth simultaneous confidence corridor for the mean of sparse functional data”, J. Amer. Statist. Assoc., 109 (2014), 661–673 | DOI | MR | Zbl

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