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A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. (English) Zbl 0596.65004

The special spline smoothing model to be considered is given by ${y}_{i}=f\left({t}_{i}\right)+{ϵ}_{i}$, $i=1,···,n$, ${t}_{i}\in \left[0,1\right]$, where $ϵ=\left({ϵ}_{1},···,{ϵ}_{n}\right)\sim N\left(0,{\sigma }^{2}{I}_{n×n}\right)$, ${\sigma }^{2}$ is unknown, and f($·\right)$ is some function in the Sobolev space ${W}_{2}^{m}\left[0,1\right]=\left\{f:$ $f,{f}^{1},···,{f}^{\left(m-1\right)}$ absolutely continuous, ${f}^{\left(m\right)}\in {L}_{2}\left[0,1\right]\right\}$ and the smoothing spline estimate ${f}_{n,\lambda }$ of f is the minimizer in ${W}_{2}^{m}\left[0,1\right]$ of

$1/n\sum _{i=1}^{n}{\left(f\left({t}_{i}\right)-{y}_{i}\right)}^{2}+\lambda {\int }_{0}^{1}{\left({f}^{\left(m\right)}\left(t\right)\right)}^{2}dt·$

A generalization of the maximum likelihood (GML) estimate for the smoothing parameter $\lambda$ is obtained, and this estimate is compared with the generalized cross validation (GCV) estimate both analytically and by Monte Carlo methods. The theoretical results are shown to extend to the generalized spline smoothing model, which includes the estimate of functions given noisy values of various integrals of them.

Reviewer: Y.Sun

##### MSC:
 65D10 Smoothing, curve fitting 65R20 Integral equations (numerical methods)