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Uniform Bahadur Representation for Local Polynomial Estimates\ud of M-Regression and Its Application to The Additive Model

By Efang Kong, Oliver Linton and Yingcun Xia

Abstract

We use local polynomial fitting to estimate the nonparametric M-regression function for strongly mixing\ud stationary processes {(Yi,Xi)}. We establish a strong uniform consistency rate for the Bahadur representation\ud of estimators of the regression function and its derivatives. These results are fundamental for\ud statistical inference and for applications that involve plugging in such estimators into other functionals\ud where some control over higher order terms are required. We apply our results to the estimation of an\ud additive M-regression model

Topics: QA276
Publisher: Econometric Theory
Year: 2010
OAI identifier: oai:kar.kent.ac.uk:23952

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  51. Zni ≡ Rni(x k;αjl,βkl)− Rni(x k;αjl+1,βjl+1). (A.13) Note that by Assumption A2 we have, uniformly in x,α , and β , that | ni(x;α,β)|≤CM (1) n . (A.14) Therefore, |Zni|≤CM (1) n . Using Lemma 6, we can apply Lemma 4 to each Vl with B1 = C1M (1) n ,
  52. μ ix(α+β) μ ixβ {ϕni(x;t)−ϕni(x;0)}dt and Rni(x;α,β)
  53. ϕni(x k;t +u Hn(β2−β1))−ϕni(x k;0)}dt ≡ 1+ 2. Therefore, E{Zni}2 = hd K2(u) f (x k +hu)E{( 1 + 2)2|Xi = x k +hu}du. The conclusion is thus obvious, observing that by Cauchy inequality and

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