Shi Peide, Cheng Ping
Consider the nonparametric regression model
Y=
g0(
T)+
u, where
Y is real-valued,
u is a random error,
T ranges over a nondegenerate compact interval, say [0,1], and
g0(·) is an unknown regression function, which is
m(
m≥0) times continuously differentiable and its
mth derivative,
g0(m), satisfies a Hölder condition of order γ(
m+γ>1/2). A piecewise polynomial
L1-norm estimator of
g0 is proposed. Under some regularity conditions including that the random errors are independent but not necessarily have a common distribution, it is proved that the rates of convergence of the piecewise polynomial
L1-norm estimator are
almost surely and
in probability, which can arbitrarily approach the optimal rates of convergence for nonparametric regression, where σ is any number in (0, min((
m+γ-1/2)/3,γ)).