  A basis expansion with regularization methods is much appealing to the
flexible or robust nonlinear regression models for data with complex
structures. When the underlying function has inhomogeneous smoothness, it is
well known that conventional reguralization methods do not perform well. In
this case, an adaptive procedure such as a free-knot spline or a local
likelihood method is often introduced as an effective method. However, both
methods need intensive computational loads. In this study, we consider a new
efficient basis expansion by proposing a smoothly varying regularization method
which is constructed by some special penalties. We call them adaptive-type
penalties. In our modeling, adaptive-type penalties play key rolls and it has
been successful in giving good estimation for inhomogeneous smoothness
functions. A crucial issue in the modeling process is the choice of a suitable
model among candidates. To select the suitable model, we derive an approximated
generalized information criterion (GIC). The proposed method is investigated
through Monte Carlo simulations and real data analysis. Numerical results
suggest that our method performs well in various situations.
