Generalized functional linear models

By H. Muller, U. Stadtmuller. 2005

658 citations recorded by the source corpus.

We propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function. A linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function, and the expected value of the response is related to this linear predictor via a link function. If, in addition, a variance function is specified, this leads to a functional estimating equation which corresponds to maximizing a functional quasi-likelihood. This general approach includes the special cases of the functional linear model, as well as functional Poisson regression and functional binomial regression. The latter leads to procedures for classification and discrimination of stochastic processes and functional data. We also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data, using a semiparametric quasi-likelihood procedure. An essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated Karhunen-Loeve expansion. We develop asymptotic inference for the proposed class of generalized regression models. In the proposed asymptotic approach, the truncation parameter increases with sample size, and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics. We establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable L^2 metric and is defined through a generalized covariance operator.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c8bb48ff79c5b41803e5ffa1e5fd7e05d3db436f

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Citations in the corpus

Single and multiple index functional regression models with nonparametric link (2011, methodology): “Advances in this direction include those made in [1, 9, 10, 14] and [17], where both parametric and nonparametric link functions were introduced in order to connect the response to a linear functional model in the explanatory variables.”
Shrinkage estimation and selection for multiple functional regression (2011, background): “How to extend the shrinkage estimation results to generalized functional linear model (James, 2002; Muller and Stadtmuller, 2005; Cardot and Sarda, 2005) is another interesting topic for further study.”
Functional linear regression that's interpretable (2009, background): “See Ramsay and Silverman (2002) and Muller and Stadtmuller (2005) for numerous specific applications.”
Nonparametric inference in generalized functional linear models (2014, methodology): “We assume that G has bounded second- and thirdorder derivatives, and G̈≥ δ for some constant δ > 0; see similar conditions on page 738 of [20].”
Determining the order of the functional autoregressive model (2013, positive): “The last assumption is made in most contributions studying the functional linear model, see Yao et al. (2005a, 2005b), Müller and Stadtmüller (2005), Cai and Hall (2006), Li and Hsing (2007), Horvath et al. (2009), Mckeague and Sen (2010) and Li et al. (2010), to name only a handful of recent…”
Functional Additive Models (2008, background): “These include generalized functional linear models (James 2002; Escabias, Aguilera, and Valderrama 2004; Cardot and Sarda 2005; Müller and Stadtmüller 2005); modifications of functional regression for longitudinal, (i.”
CLT in functional linear regression models (2005, neutral): “We copy verbatim the arguments used to bound ( 29 ) : rst of all we reintroduce the operator Gn (z) below and”
Estimation in functional linear quantile regression (2012, methodology): “Other than functional linear mean regression, [33] developed estimation methods for generalized functional linear models using series expansions of covariates and slope functions.”
Locally Sparse Estimator for Functional Linear Regression Models (2017, methodology): “Extensions of classic FLR include generalized functional linear models studied by Müller and Stadtmüller (2005), PACE proposed by Yao, Müller, and Wang (2005) for sparse longitudinal © American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North…”
Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains (2015, methodology): “Moreover, they can be used as a building block for further statistical analyses such as functional clustering methods or functional principal component regression with multiple covariates (see Müller and Stadtmüller 2005, for the univariate case).”