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Uniform consistency and uniform in bandwidth consistency for nonparametric regression estimates and conditional \(U\)-statistics involving functional data. (English) Zbl 07214281
Summary: W. Stute [(1991), Annals of Probability, 19, 812-825] introduced a class of so-called conditional \(U\)-statistics, which may be viewed as a generalisation of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to
\[ m(\mathbf{t} := \mathbb{E}[\varphi(Y_1, \dots, Y_m)|(X_q, \dots, X_m) = \mathbf{t}], \quad \text{for } \mathbf{t} \in \mathbb{R}^{dm}. \]
We apply the methods developed in Dony and Mason [(2008), Bernoulli, 14(4), 1108-1133] to establish uniformity in \(\mathbf{t}\) and in bandwidth consistency (i.e. \(h_n, h_n \ in [a_n, a_n]\) where \(0 < a_n < b_n \to 0\) at some specific rate) to \(m(\mathbf{t}\) of the estimator proposed by Stute when \(Y\) and covariates \(X\) are functional taking value in some abstract spaces. In addition, uniform consistency is also established over \(\varphi \in \mathscr{F}\) for a suitably restricted class \(\mathscr{F}\). The theoretical uniform consistency results, established in this paper, are (or will be) key tools for many further developments in functional data analysis. Applications include the Nadaraya-Watson kernel estimators and the conditional distribution function. Our theorems allow data-driven local bandwidths for these statistics.
MSC:
62G08 Nonparametric regression and quantile regression
62R10 Functional data analysis
Software:
fda (R)
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