Comparative Analysis for Robust Penalized Spline Smoothing Methods

Algorithm 1

𝑀-type iterative algorithm for penalized regression splines.

Step  0.  Input an initial curve estimate 𝐦_𝑝⁽⁰⁾, termination tolerance 𝜀, and the maximum iteration

   number Iter_max. Meanwhile, set 𝑘 = 0 and conduct the following loop iterations.

Step  1.  Estimate 𝜎_𝜀 by using MADN in (17);

Step  2.  Produce the empirical pseudo data according to (19);

Step  3.  Calculate the penalized least-square estimator 𝐦_𝑝^(𝑘+1) defined in (20) for the pseudo data,

    where the penalty parameter 𝜆 is chosen according to the generalized cross-validation

    (GCV) criterion formulated by (11) in the foregoing subsection.

Step  4.  If ‖𝐦_𝑝^(𝑘+1)−𝐦_𝑝^(𝑘)‖<𝜀‖𝐦_𝑝^(𝑘)‖ or 𝑘 = Iter_max, terminate and output the robust

    𝑀-type penalized spline estimate 𝐦_𝑀 = 𝐦_𝑝^(𝑘+1); else set 𝑘 = 𝑘 + 1 and continue Step  1.