M-type regression splines involving time series

Abstract

Consider a strictly stationary time series Z ≙ {(X i, Y i): i = 1,2,…} with X i being R d -valued and Y i real valued. The nonparametric M-type regression function g 0(·) is defined by E( Ψ( Y 1 − g 0( X 1))| X 1 = x) = 0, where Ψ(·) is a function chosen suitably. Tensor products of B-splines are adopted to approximate g 0 and a class of M-type regression spline estimators of this function are obtained based on a segment, ( X 1, Y 1),…, ( X n , Y n ), of Z. Suppose that g 0(·) is smooth up to order r (> d 2 ) . Under certain regularity conditions, the M-type regression spline estimators can achieve the optimal rates of convergence n −r (2r+d) in L 2-norms restricted to a compact domain when the spline knots are deterministically given. The M-estimators considered here include Huber's estimator, L 1-norm estimator, regression quantile estimator and L p -norm estimator as special cases.