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Abstract:
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The asymptotic behavior of penalized spline estimators is studied using B-splines and a m-th order penalty. Li and Ruppert (2008) showed that univariate penalized splines behave like binned kernel estimators with "equivalent" kernels depending upon m. Explicit expressions were obtained for the asymptotic bias and variance. However, their work was limited to piecewise constant or linear splines and penalties on first or second differences. Li, Apanasovich, and Ruppert (2008) have extended these results to arbitrary degree splines and penalties. The asymptotic distribution does not depend on the degree of the spline, only the penalty order. P-splines are not design-adaptive; their asymptotic bias depends on the predictor's density. The asymptotic theory has been extended to additive models by Li and Ruppert (2008). The weighted, penalized estimators have the oracle property.
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