Estimation of a Functional Single Index Model

Ferraty, Frédéric; Park, Juhyun; Vieu, Philippe

doi:10.1007/978-3-7908-2736-1_17

Cited by 17 publications

(8 citation statements)

References 2 publications

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“…Once the directional parameter is estimated, it suffices to plug this estimate

\overset{θ}{true}

(

\overset{θ}{true}

being either

{\overset{false^}{θ}}_{1}

{\overset{false˜}{θ}}_{1}

) into the definition ) to construct a fully automatic estimate

\overset{m}{true} = {\overset{false^}{g}}_{\overset{false^}{θ}}

of the regression operator in the model ). Consistency properties of these estimate have been provided in Ferraty et al (2011) when the estimated direction

\overset{θ_{1}}{true}

is used and in Ferraty, Goia, Salinelli, and Vieu (2013) when the estimated direction

{\overset{false˜}{θ}}_{1}

is used. In both cases it is shown that the rates of convergence are the same as if Δ was a standard one‐dimensional real random variable, showing that the functional single index model reaches its main goal of being insensitive to the dimensionality of the problem.…”

Section: Single Functional Index Regressionmentioning

confidence: 99%

“…See also the early work by Amato, Antoniadis, and De Feis (2006) for a slightly similar presentation of the model. A nice way allowing both for checking identifiability and for interpreting the outputs of the model has been provided in Ferraty, Park, and Vieu (2011), is based on average derivatives ideas such as developed in multivariate analysis in Härdle and Stoker (1989) and can be summarized as follows. Assume that for x ∈ E the functional operator m (.)…”

Section: Single Functional Index Regressionmentioning

confidence: 99%

“…This approach was developed in Ferraty et al (2011) and the directional estimate defined by Equations ) and () is consistent.…”

Section: Single Functional Index Regressionmentioning

confidence: 99%

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On semiparametric regression in functional data analysis

Ling

Vieu

2020

WIREs Computational Stats

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The aim of this paper is to provide a selected advanced review on semiparametric regression which is an emergent promising field of researches in functional data analysis. As a deliberate strategy, we decided to focus our discussion on the single functional index regression (SFIR) model in order to fix the ideas about the stakes linked with infinite dimensional problems and about the methodological challenges that one has to solve when building statistical procedure: one of the most challenging issue being the question of dimensionality effects reduction. This will be the first (and the main) part of this discussion and a complete survey of the literature on SFIR model will be presented. In a second attempt, other semiparametric models (and more generally, other dimension reduction models) will be shortly discussed with the double goal of presenting the state of art and of defining challenging tracks for the future. At the end, we will discuss how additive modeling is an appealing idea for more complicated models involving multifunctional predictors and some tracks for the future will be pointed in this setting. This article is categorized under: Statistical Models > Semiparametric Models Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods

show abstract

“…Once the directional parameter is estimated, it suffices to plug this estimate

\overset{θ}{true}

(

\overset{θ}{true}

being either

{\overset{false^}{θ}}_{1}

{\overset{false˜}{θ}}_{1}

) into the definition ) to construct a fully automatic estimate

\overset{m}{true} = {\overset{false^}{g}}_{\overset{false^}{θ}}

of the regression operator in the model ). Consistency properties of these estimate have been provided in Ferraty et al (2011) when the estimated direction

\overset{θ_{1}}{true}

is used and in Ferraty, Goia, Salinelli, and Vieu (2013) when the estimated direction

{\overset{false˜}{θ}}_{1}

Section: Single Functional Index Regressionmentioning

confidence: 99%

Section: Single Functional Index Regressionmentioning

confidence: 99%

See 1 more Smart Citation

On semiparametric regression in functional data analysis

Ling

Vieu

2020

WIREs Computational Stats

Self Cite

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show abstract

“…Alternatively, a kernel estimator can be used for h (·) (e.g. Ait‐Saïdi et al , ; Ferraty et al , ).…”

Section: Non‐linear Scalar‐on‐function Regressionmentioning

confidence: 99%

Methods for Scalar‐on‐Function Regression

Reiss

Goldsmith

Shang

et al. 2016

Int Statistical Rev

139

109

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Recent years have seen an explosion of activity in the field of functional data analysis (FDA), in which curves, spectra, images, etc. are considered as basic functional data units. A central problem in FDA is how to fit regression models with scalar responses and functional data points as predictors. We review some of the main approaches to this problem, categorizing the basic model types as linear, nonlinear and nonparametric. We discuss publicly available software packages, and illustrate some of the procedures by application to a functional magnetic resonance imaging dataset.

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“…Thus, it couples the advantages of both parametric and nonparametric regression models. Because of these advantages, it has received an increasing amount of attention in the functional regression literature (e.g., Ferraty et al, 2003;James and Silverman, 2005;Ait-Saïdi et al, 2008;Ferraty et al, 2011;Chen et al, 2011;Jiang and Wang, 2011;Goia and Vieu, 2015;Fan et al, 2015).…”

Section: Introductionmentioning

confidence: 99%