We propose a new penalized spline method for bivariate smoothing. Tensor product B-splines with row and column penalties are used as in the bivariate P-spline of Marx and Eilers (2005). What is new here is the introduction of a third penalty term and a modification of the row and column penalties. We call the new estimator a Bivariate Penalized Spline or BPS. The modified penalty used by the BPS results in considerable simplifications that speed computations and facilitate asymptotic analysis. We derive a central limit theorem for the BPS, with simple expressions for the asymptotic bias and variance, by showing that the BPS is asymptotically equivalent to a bivariate kernel regression estimator with a product kernel. As far as we are aware, this is the first central limit theorem for a bivariate spline estimator of any type. We also derive a fast algorithm for the BPS. Our simulation study shows that the mean square error of the BPS is comparable to or smaller than that of other methods for bivariate spline smoothing. Examples are given to illustrate the BPS.
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