Wavelet penalized likelihood estimation in generalized partially linear models

The paper deals with a semiparametric 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 partial linear 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 $\ell^1$-penalty leads to an adaptive estimation. An algorithm is also proposed for implementation and simulations are used to illustrate the results.