Abstract
The paper gives a unified study of the large sample asymptotic theory of penalized splines including the O-splines using B-splines and an integrated squared derivative penalty [22], the P-splines which use B-splines and a discrete difference penalty [13], and the T-splines which use truncated polynomials and a ridge penalty [24]. Extending existing results for O-splines [7], it is shown that, depending on the number of knots and appropriate smoothing parameters, the $L_{2}$ risk bounds of penalized spline estimators are rate-wise similar to either those of regression splines or to those of smoothing splines and could each attain the optimal minimax rate of convergence [32]. In addition, convergence rate of the $L_{\infty }$ risk bound, and local asymptotic bias and variance are derived for all three types of penalized splines.
Highlights
Penalized spline smoothing has become popular in the last two decades
Penalized splines exploit the mixed effect presentation of smoothing splines but are computationally much simpler because they use low rank bases; penalized splines inherit the computational simplicity of regression splines but relieve the overfitting of regression splines as they employ a smoothness penalty
Penalized spline methods have been well developed in functional data analysis, e.g., [44, 15, 43, 40]
Summary
Penalized spline smoothing has become popular in the last two decades. The approach uses a flexible choice of bases and penalty and is often viewed as a bridge between regression splines and smoothing splines. The L∞ convergence rates of all three types of penalized splines shall be established and the rates are optimal for the small number of knots scenario. The local variance of the three types of penalized splines can be studied via the bounds on the diagonals of (I + ηD)−1, where η > 0 is a scalar and D is the difference penalty matrix mentioned above. The rates of the local asymptotic variance of penalized splines are established for all three types of penalized splines; see Propositions 6.1, 6.2 and 6.3. These three results ensure that a unified theoretic study is attainable. We introduce three types of penalized splines that are commonly used and formulate a unified estimator that contains all of them
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