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Wavelet penalized likelihood estimation in generalized functional models

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  • Irène Gannaz

Abstract

The 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 literature. We establish in particular that the ℓ 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. Copyright Sociedad de Estadística e Investigación Operativa 2013

Suggested Citation

  • Irène Gannaz, 2013. "Wavelet penalized likelihood estimation in generalized functional models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 122-158, March.
    • Handle: RePEc:spr:testjl:v:22:y:2013:i:1:p:122-158
    DOI: 10.1007/s11749-012-0310-6
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    References listed on IDEAS

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    1. Anestis Antoniadis & Irène Gijbels & Mila Nikolova, 2011. "Penalized likelihood regression for generalized linear models with non-quadratic penalties," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(3), pages 585-615, June.
    2. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    3. Chang, Xiao-Wen & Qu, Leming, 2004. "Wavelet estimation of partially linear models," Computational Statistics & Data Analysis, Elsevier, vol. 47(1), pages 31-48, August.
    4. Antoniadis A. & Fan J., 2001. "Regularization of Wavelet Approximations," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 939-967, September.
    5. Rice, John, 1986. "Convergence rates for partially splined models," Statistics & Probability Letters, Elsevier, vol. 4(4), pages 203-208, June.
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