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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

Provided by:
Kent Academic Repository

Downloaded from
http://kar.kent.ac.uk/23952/1/ET1.pdf

- (1) n )2M (2) n ,λ n ={ r(n)M (1) n }−1/4C1, and η =
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- (1998). By the central limit theorem for strongly mixing processes (Bosq,
- Conditions on ϕ(.) as in Assumptions A1 and A2 are satisﬁed in almost all known robust and likelihood type regressions. For example, in the qth quantile regression, we have ϕ(t)
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- K2(u)μ(u) Hnα μ(u) Hn(α+β) μ(u) Hnβ C|t|dtf(x +hu)du=O hd(M (1) n )2M (2) n , (A.54) uniformly in x ∈
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- (2003). Note that the condition E{ϕ(εi)|Xi}=0 a.e. is necessary for model speciﬁcation. Moreover, if the conditional density f (y|x) of Y given X is also continuously differentiable with respect to y, then as shown in Hong
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- (1995). We claim that for elements Eβ∗ nr(x) of Eβ∗ n(x) with p −| r| even, the hp+1 term will vanish. This means for any given r with |r|≤p and r2 with |r2|=p+1, ∑ 0≤|r|≤p {S−1 p }N(r1),N(r) νr+r2 = 0. (A.8) To prove this, ﬁrst note that for any r1 with 0 ≤|
- (1995). We claim that for elements Eβ∗nr(x) of Eβ∗n(x) with p−|r| even, the hp+1 term will vanish. This means for any given r with |r| ≤ p and r2 with |r2| = p+ 1, ∑ 0≤|r|≤p {S−1p }N(r1),N(r) νr+r2 = 0. (A.8) To prove this, first note that for any
- We partition the set {1,...,n} into 2q ≡ 2qn consecutive blocks of size r ≡ rn with n =
- 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 ,
- μ ix(α+β) μ ixβ {ϕni(x;t)−ϕni(x;0)}dt and Rni(x;α,β)
- 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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