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Nonparametric modelling for functional data: selected survey and tracks for future. (English) Zbl 1411.62084
The paper under review is to first survey the state of the art in the nonparametric functional data analysis (NPFDA) and to present some open questions in order to promote the field. NPFDA deals with infinite dimensional data and nonparametric modelling with infinite-dimensional assumptions.
Section 2 starts with a basic functional regression problem with sample of independent variables in some infinite-dimensional space \({\mathcal F}\). The estimating of the nonlinear functional operator \(r\) follows a finite-dimensional method in averaging locally functional kernel and smoothing factor, \[ Y_i = r(\chi_i)+\varepsilon_i, \;\;\hat{r}(x) = \frac{\sum_{i=1}^nY_iK(\frac{d(x, \chi_i)}{h})}{\sum_{i=1}^nK(\frac{d(x, \chi_i)}{h})}, \] for \(h=h_n\) and \(\lim_{n\to \infty} h_n=0\). Ferraty and Vicu (2006) [Nonparametric functional data analysis. Theory and practice. New York: Springer-Verlag] (Well-Popularized Monography on NPFDA) shows that there is a convergence with almost complete point-wise rates under mild conditions on the kernel functional \(K\) and smoothing factor \(h\). F. Ferraty et al. [J. Stat. Plann. Inference 140, No. 2, 335–352 (2010; Zbl 1177.62044)] present a uniform converging rate by the entropy function. There are several improvements in the directions (1) \(L_p\) rates of convergence, (2) asymptotic distribution, (3) uniform in bandwidth results, (4) deviation principles and (5) asymptotic for random and/or data-driven parameters.
The authors address further extensions on kernel estimate from (i) robust kernel functional regression, (ii) kNN functional regression, (iii) local linear functional regression, (iv) recursive kernel functional regression, (v) delta-sequence estimate, (vi) other versions of kernel estimates. Estimate based on reproducing kernel Hilbert space and the Stein-type estimate are essential ideas in these extensions of NPFDA. Regression with dependent functional variables and the functional bootstrapping procedure are developed, regression with functional response and recursive estimate converging rate are analyzed, fixed design functional regression and \(L_2\) errors expansions and asymptotic normality are obtained.
The paper focuses on the basic regression model and basic estimate to specify asymptotic results in nonparametric infinite-dimensional space (i) by pending a very slow but optimal choice of h in the multivariate case, (ii) by using the Gaussian process and the small ball probability function, (iii) by feasibility of nonparametric ideas in FDA with semi-metrics to gain much trustable rates of convergence in the infinite-dimensional setting.
Section 3 discusses the estimate of nonlinear regression operators under i.i.d assumptions. The nonparametric conditional distribution function (c.d.f) is estimated (1) by kernel with unrestrictive nonparametric modeling assumptions, (2) by asymptotic for the basic kernel estimate involving random and/or data-driven bandwidths, (3) by asymptotic for modified version of kernel estimate, and (4) by direct impact for estimating real parameters with conditional quantiles of c.d.f and Weibull tail. The nonparametric conditional density is estimated (1) by kernel with basic kernel functional conditional density, (2) by estimating basic kernel asymptotic, (3) by modified version of kernel and MAR, and (4) by statistical procedures with impact on conditional mode estimation. The nonparametric conditional hazard function is provided (rates of complete convergence, asymptotic normality and \(L_2\) error expansion), the nonparametric functional discrimination (supervised classification) is developed, and density estimation and small ball probability estimation are summarized with lack of reference measure (as Lebesgue measure in finite-dimensional spaces) by assuming absolute continuity of the probability distribution of the functional variable, by-product of density estimation develops nonparametric estimates of the concentration function which is useful for analyzing the distribution of the process generating the functional data set.
Section 4 discusses for open questions in NPFDA. The most important point for practice of nonparametric functional estimates is the choice of the semi-metric. Building semi-metric on the functional data set is more complex. Nonparametric ideas allow the derivation of estimates with nice properties under very unrestrictive constraints on the data, can be a pilot tool in FDA. Semi-parametric FDA intents to propose intermediary models between linear and nonparametric FDAs, and is a part of the family of dimensionality reduction models. Using nonparametric for developing new parametric estimates and testing is another trend for nonparametric regressor. The challenge question remains on developing new models and new methods by nonparametric functional data analysis (NPFDA). The functional semi-parametric is just in bud, and the testing from the nonparametric procedure is rather underdeveloped.

MSC:
62G05 Nonparametric estimation
62G08 Nonparametric regression and quantile regression
Software:
fda (R)
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References:
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