g is (d - 1) -times partially differentiable for d- 1 ≤μ≤d; for some ρ> 0 ,
sup У ∈φzρ g ( У ) - g (z ) - Gg ( y, z )|/l y - z Γ ≤ Dg (z ) for all z , where φ ρ ={ У | |У - z∖ < p}
; Gg = 0 when d = 1 ; Gg is a (d - 1)th degree homogeneous polynomial in y- z with
coefficients the partial derivatives of g at z of orders 1 through d - 1 when d > 1 ; and
g(z), its partial derivatives of order d -1 and less, and Dg(z), has finite αth moments.
Proof of Theorem (i)
In the proof, we use several approximations to JT . We define them now and recall a few
already defined statistics for convenience of reference.
|
TT ^ I _____ _____ JU ≡--------∑ ∑ Ksεtεs t T(T -1)h ~ ⅛ ts t s |
(A.1) |
|
JT ≡ —1— ∑ ]∑ Kεε T(T- 1)h t=1 s≠t |
(A.2) |
|
1 TT TUu 'τ,f'τ' m∖ι∑ ^∑ ^∑ s^tsUtuUUsu T(T- 1)h t=1 s≠t |
(A.3) |
|
JTL ≡ —1— ∑ ∑ K,εLεL T(T- 1)h t=1 s≠t |
(A.4) |
where εt = I {yt ≤ Qθ (xt )}- θ,
εt=I{yt≤Qθ(xt)}-θ,
εtU =I{yt +CT ≤Qθ(xt )} -θ,
εtL =I{yt-CT ≤Qθ(xt)}-θ and
CT is an upper bound consistent with the uniform convergence rate of the nonparametric
estimator of conditional quantile given in equation (13). The proof of Theorem 1 (i) consists
of three steps.
Step 1. Asymptotic normality:
Thm/2JT → N(0,σ02), (A.5)
11