On Generalized Cross-Validation for Multivariate Smoothing Spline Functions

Abstract

Let there be given data $y_i = f(t_i ) + \varepsilon _i ,i = 1, \cdots ,n$, where f is an element of $H^m (\Omega )$, $\Omega $ is a bounded subset of $R^d $ with smooth boundary and satisfying a uniform cone condition, and the errors $\varepsilon _i $ are i.i.d. random variables with zero mean. To estimate f we use the Thin Plate Smoothing Spline $S_{m,n,\lambda } $ defined as the unique solution to the minimization problem \[ \mathop {{\text{Minimize }}}\limits_{u \in D^{ - m} L^2 \left(R^d \right)} \lambda | u |_m^2 + \frac{1}{n}\sum_{i = 1}^n {\left[ {u\left( {t_i } \right) - y_i } \right]^2 } \]where the smoothing parameter $\lambda $ is obtained using the method of Generalized Cross-Validation. Under mild assumptions on the limiting behavior of the knots, we prove that this method is asymptotically optimal, and \[\mathop {\lim }\limits_{n \to \infty } E\left[ {\left| {f - S_{m,n,\lambda _n^ * } } \right|_{0,\Omega }^2 } \right] = 0\] where $\lambda _n^ * $ is the minimizes of the Generalized Cross-Validation function.