Penalised spline estimation for generalised partially linear single-index models
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Generalised linear models are frequently used in modeling the relationship of the response variable from the general exponential family with a set of predictor variables, where a linear combination of predictors is linked to the mean of the response variable. We propose a penalised spline (P-spline) estimation for generalised partially linear single-index models, which extend the generalised linear models to include nonlinear effect for some predictors. The proposed models can allow flexible dependence on some predictors while overcome the “curse of dimensionality”. We investigate the P-spline profile likelihood estimation using the readily available R package mgcv, leading to straightforward computation. Simulation studies are considered under various link functions. In addition, we examine different choices of smoothing parameters. Simulation results and real data applications show effectiveness of the proposed approach. Finally, some large sample properties are established.
KeywordsGeneralised linear model Generalised additive model Low rank approximation Penalised splines Profile likelihood
We thank the Editor for suggesting the profile likelihood approach and for many extremely valuable points. This paper would not be possible otherwise. We also thank an anonymous referee for the thoughtful comments. Finally, we thank Xia, Y. and Härdle W. for providing the French Bank dataset.
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