Wavelet penalized likelihood estimation in generalized functional models. (English) Zbl 1259.62017

Summary: This paper deals with generalized functional regression. The aim is to estimate the influence of covariates on observations, drawn from an exponential distribution. The link considered has a semiparametric expression: if we are interested in a functional influence of some covariates, we authorize others to be modeled linearly. We thus consider a generalized partially linear regression model with unknown regression coefficients and an unknown nonparametric function. We present a maximum penalized likelihood procedure to estimate the components of the model introducing penalty based wavelet estimators. Asymptotic rates of the estimates of both the parametric and the nonparametric part of the model are given and quasi-minimax optimality is obtained under usual conditions in the literature. We establish in particular that the \(\ell^1\)-penalty leads to an adaptive estimation with respect to the regularity of the estimated function. An algorithm based on backfitting and Fisher-scoring is also proposed for implementation. Simulations are used to illustrate the finite sample behavior, including a comparison with kernel- and spline-based methods.


62G08 Nonparametric regression and quantile regression
65T60 Numerical methods for wavelets
62J05 Linear regression; mixed models
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)
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