Journal of Nonparametric Statistics
ISSN: 1048-5252 (Print) 1029-0311 (Online) Journal homepage: http://www.tandfonline.com/loi/gnst20
Erratum of: ‘Non-parametric models for functional data, with application in regression, time-series prediction and curve discrimination’ F. Ferraty & P. Vieu To cite this article: F. Ferraty & P. Vieu (2008) Erratum of: ‘Non-parametric models for functional data, with application in regression, time-series prediction and curve discrimination’, Journal of Nonparametric Statistics, 20:2, 187-189, DOI: 10.1080/10485250801999453 To link to this article: http://dx.doi.org/10.1080/10485250801999453
Published online: 21 Apr 2008.
Submit your article to this journal
Article views: 109
View related articles
Citing articles: 8 View citing articles
Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=gnst20 Download by: [198.23.176.202]
Date: 10 January 2016, At: 10:46
Journal of Nonparametric Statistics Vol. 20, No. 2, February 2008, 187–189
Erratum of: ‘Non-parametric models for functional data, with application in regression, time-series prediction and curve discrimination’
Downloaded by [198.23.176.202] at 10:46 10 January 2016
F. Ferraty* and P. Vieu Institut de Mathématiques de Toulouse, Equipe LSP, Université Paul Sabatier, Toulouse, France (Received 13 November 2006; final version received 18 February 2008 ) A topological hypothesis was omitted during the statement of the results in [P. Ferraty and P. Vieu, Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination, Nonparamet. Statist. 16 (2004), pp. 111–125]. This erratum states this additional hypothesis. It also provides a wide class of examples of functional spaces of statistical interest for which this new hypothesis is satisfied. Keywords: functional data; nonparametric regression; time-series prediction; curves discrimination; fractal dimension; semi-metric space
1. Aims of the erratum The paper [1] studies asymptotic properties for the kernel regression estimate in the general situation when the explanatory variable X is valued in some functional semi-metric space (H, d). Precisely, Theorem 3.1 states that this functional non-parametric estimate is almost surely consistent (with rates). One important feature of this theorem is that the consistency property holds uniformly over any compact set C. The proof of this result is decomposed in two parts. First, the pointwise consistency is shown (see last formula on p. 121 in [1]). Then, by Lispchitz considerations, this result is extended uniformly over the compact set C. It turns out that some important hypothesis about the topological structure of the set C has been forgotten to make valid the second step of the proof. This is corrected in Remark 1 below. Proposition 3.1 shows how the additional hypothesis is satisfied for a wide class of semi-metric spaces of interest in statistical applications. These topological considerations will be helpful in the future in many functional non-parametric settings, and not only in regression. From now on, the same notations as in [1] are used.
*Corresponding author. Email:
[email protected]
ISSN 1048-5252 print/ISSN 1029-0311 online © 2008 Taylor & Francis DOI: 10.1080/10485250801999453 http://www.informaworld.com
188
F. Ferraty and P. Vieu
2. The new topological condition As indicated at the bottom of p. 121 in [1], the gap between pointwise and uniform results is passed by means of the Lipschitz condition on the kernel and by covering the compact set C with a finite number of balls. The same thing is done, with more explicit details, along the proof of Lemma 4.5 in [2]. Precisely, as indicated on p. 339 of [2], the compact set is written in such a way that, for any l > 0, τ C⊂ B(tk , l), (1) k=1
Downloaded by [198.23.176.202] at 10:46 10 January 2016
and the key point is to assume a geometrical link between the number τ of balls and the radius l of each of them. Precisely (similarly to what appears in [2, p. 339, line 8]), it is necessary that ∃α > 0,
∃C > 0,
τ l α = C.
(2)
While the condition (2) is trivially satisfied in standard non-parametric problems when H = Rp , and d is the Euclidian metric on Rp (since it suffices to choose α = p), this topological property is not true for any abstract semi-metric space (H, d). This is the reason why the following remark has to be stated. Remark 1 To make Theorem 3.1 in [3] (and hence Corollaries 4.1, 4.4 and 4.5) valid, it is necessary to assume that the compact set C can be written in the form (1) and in such a way that Equation (2) holds.
3.
Examples of projection-based semi-metric spaces
As discussed in the recent monography [3], when dealing with non-parametric functional data analysis, the choice of the semi-metric d turns out to be of first importance. As indicated in Section 13.3.3 of [3], the projection-based ideas allow the construction of a wide class of semi-metrics, which can be easily used in applied statistical problems. We will see in Proposition 3.1 that for the topological spaces associated with these semi-metrics, the conditions (1) and (2) are always satisfied. The projection-based semi-metrics are constructed in the following way. Assume that H is a separable Hilbert space, with inner product ·, · and with orthonormal basis {e1 , . . . , ej , . . .}, and let k be a fixed integer, k > 0. As shown in Lemma 13.6 of [3], a semi-metric dk on H can be defined as follows: k dk (x, x ) = x − x , ej 2 . (3) j =1
PROPOSITION 3.1 Any compact subset C of (H, dk ) can be written in the form of Equations (1) and (2) with α = k. Proof
Let φ be the operator defined from H into Rk by φ(x) = (x, e1 , . . . , x, ek ).
Let deucl be the Euclidian distance on Rk , and let Beucl (·, ·) be an open ball of Rk for deucl , and Bk (·, ·) be an open ball of H for dk . Because φ is a continuous application between the topological
Journal of Nonparametric Statistics
189
spaces (H, dk ) and (Rk , deucl ), we have that φ(C) is a compact subset of Rk . So, one has ∀l > 0, ∃τ, ∃z1 , . . . , zτ ∈ Rk ,
φ(C) ⊂
τ i=1
Beucl (zi , l),
with τ l k = C for some C > 0. (4)
For i = 1, . . . , τ , let xi be an element of H such that φ(xi ) = zi . The solution of the equation φ(x) = zi is not unique in general, but just take xi to be one of these solutions. The result follows from Equations (4) and (5), which is a consequence of Equation (3): φ −1 (Beucl (zi , l)) = Bk (xi , l).
(5)
Downloaded by [198.23.176.202] at 10:46 10 January 2016
References [1] P. Ferraty and P. Vieu, Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination, Nonparamet. Statist. 16 (2004), pp. 111–125. [2] P. Ferraty, A. Goia, and P. Vieu, Functional nonparametric model for time series: A fractal appoach for dimension reduction. Test 11 (2002), pp. 317–344. [3] P. Ferraty and P.Vieu, Nonparametric Functional Data Analysis: Theory and Applications, Springer Series in Statistics, New York, 2006.
