Asymptotics of the “minimumL ₁-norm” estimates in nonparametric regression models

Abstract

Consider the nonparametric regression modelY=g ₀(T)+u, whereY is real-valued,u is a random error,T ranges over a nondegenerate compact interval, say [0,1], andg ₀(·) is an unknown regression function, which ism(m≥0) times continuously differentiable and itsmth derivative,g ^(m)₀ , satisfies a Hölder condition of order γ(m+γ>1/2). A piecewise polynomialL ₁-norm estimator ofg ₀ 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 polynomialL ₁-norm estimator are$o(n^{ - \frac{{m + \gamma - 1/2 - \delta }}{{2\left( {m + \gamma } \right) + 1}}} )$ almost surely and$o(n^{ - \frac{{m + \gamma - \delta }}{{2\left( {m + \gamma } \right) + 1}}} )$ 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)).