On The Asymptotics Of Penalised Splines

The asymptotic behavior of penalised spline estimators is studied in the univariate case. B-splines are used and a penalty is placed on mth order differences of the coefficients. The number of knots is assumed to converge to ∞ as the sample size increases. We show that penalised splines behave similarly to Nadaraya-Watson kernel estimators with an “equivalent” kernels depending upon m. The equivalent kernels we obtain for penalised splines are the same as those found by Silverman for smoothing splines. The asymptotic distribution of the penalised spline estimate is Gaussian and we give simple expression for the asymptotic mean and variance. Providing that it is fast enough, the rate at which the number of knots converges to ∞ does not affect the asymptotic distribution. The optimal rate of convergence of the penalty parameter is given. Penalised splines are not design-adaptive.