Wavelet‐Based Single‐Index Additive Models With Irregular Link and Additive Functions

Because of the complexity of data sets in practice, there has been much interest in developing statistical analysis tools for problems involving high-dimensional covariates. Examples of these models include partial linear additive models (PLAMs) and single-index models (SIMs). A common feature of these models is that they achieve dimension reduction to circumvent the “curse of dimensionality” while retaining the flexibility of the nonparametric regression. In the statistical and machine learning literature, fitting the additive parts in PLAM models and the link function in SIM models by nonparametric methods usually requires smooth additive components and regular link functions, and it is usually achieved using kernel methods or spline smoothing. In this work, we present a novel intrinsically interpretable combination of these two models with competitive predictive performance. We relax the smoothness assumptions and develop a nonparametric estimation procedure of the additive components and the link function that uses wavelet bases expansions adapted to non-equispaced designs. Simulation studies and real data analyses are employed to demonstrate the usefulness of the approach. Computer codes are provided as Supporting Information.