On the rates of convergence of “minimum l1-norm” estimates in a partly linear model
Titel:
On the rates of convergence of “minimum l1-norm” estimates in a partly linear model
Auteur:
Shi, Peide Guoying, Li
Verschenen in:
Communications in statistics
Paginering:
Jaargang 23 (1994) nr. 1 pagina's 175-196
Jaar:
1994
Inhoud:
Consider the partly linear model Y = X'β0+go(T)+u, where Y is a real-valued response, X is a d-vector of explanatory variables, u is a random error β0 is a d-vector of parametersT is another explanatory variable ranging over a nondegenerate compact interval, say [0, 1], and g0(·) is an unknown function, which is m (≥ 0) times continuously differentiable and its mth derivative satisfies a Holder condition with exponent γ ∈ [0,1]. Based on i.i.d. observations (T1,X'1,Y1),…, (Tn,X'n,Yn) of (T,X',Y), this article studies the rates of convergence of the L1-norm estimators for β0 and g0 obtained from the minimization problem [image omitted] where Fn is a space of B-spline functions of order m + 1. Under some mild conditions, it is shown that the estimator of g0 achieves the convergence rate OP(n-(m+γ/[2(m+γ) + 1]), which is Stone's optimal global rate of convergence of estimators for nonparametric regression, and the estimator of β0 achieves the convergence rate n-1/2.
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