an:03872479 Zbl 0547.62024 Silverman, B. W. Spline smoothing: The equivalent variable kernel method EN Ann. Stat. 12, 898-916 (1984). 00149160 1984
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62G05 65D10 62J02 65C99 46E35 variable kernel; roughness penalty; penalized maximum likelihood; curve estimation; density estimation; Sobolev spaces; cubic spline estimator; kernel approach; spline-smoothing methods; approximation of the weight functions; hat matrix The cubic spline estimator of the regression curve is related with the existence of a weight function which enables to perform a nonparametric estimation of the function. The relation between this method and the kernel approach is fixed under suitable conditions with the aims of giving an intuitive insight into spline-smoothing methods. The main result is that for a sequence of probability distribution functions $$F_ n$$, the weight function corresponding to a design point is similar to a specific kernel function k if n is sufficiently large, the smoothing parameter $$\lambda$$ is small and the design point t is not near the boundaries of the interval (a,b) on which the sequence is defined. k is centred in t with bandwidth $$(\lambda /F'(t))^{1/4}$$, where $$F=\lim_{n\to\infty }F_ n$$ is an absolutely continuous distribution function on (a,b). This result is given in theorem A under assumptions related with the existence of F, the boundness of its first and second derivatives and restrictions on how fast $$\lambda$$ should tend to zero. From three lemmas and two propositions given in Section 4 the theorem is obtained. As the approximation of the weight function is not good when t is close to the boundaries of (a,b) a solution is stated in theorem B of Section 5. Illustrations of the performance of the approximation of the weight functions are given. Some applications are derived in connection with the hat matrix and the estimation of a density function. C.N.Bouza