Asymptotic optimality of a multivariate version of the generalized cross validation in adaptive smoothing splines

Abstract

We consider an adaptive smoothing spline with a piecewise-constant penalty function $\lambda(x)$, in which a univariate smoothing parameter $\lambda$ in the classic smoothing spline is converted into an adaptive multivariate parameter $\boldsymbol{\lambda}$. Choosing the optimal value of $\boldsymbol{\lambda}$ is critical for obtaining desirable estimates. We propose to choose $\boldsymbol{\lambda}$ by minimizing a multivariate version of the generalized cross validation function; the resulting estimator is shown to be consistent and asymptotically optimal under some general conditions—i.e., the counterparts of the nice asymptotic properties of the generalized cross validation in the ordinary smoothing spline are still provable. This provides theoretical justification of adopting the multivariate version of the generalized cross validation principle in adaptive smoothing splines.