1 TT
(15)
rτ ≡-------∑ ∑ Kεεs
Tm tsts
τ(τ- 1)h t=1 s≠t
τhen
Thm/2 Jτ → N(0, σ0 ) in distribution,
(16)
where σ02=2E{σε4(zt)fz(zt)}{∫K2(u)du}
and fz (∙) is the marginal density function of
zt .
Technical conditions required to derive the asymptotic distribution of Jτ are given in
Appendix, which are adopted from Li (1999) and Franke and Mwita (2003). In the
assumptions we use the definitions of Robinson (1988) for the class of kernel functions Yv
and the class of functions A v∞ , defined in Appendix.
Conditions (A1)(i)-(iv) and (A2)(i)-(ii) are adopted from condition (A1) and (A2) of Li
(1999), which are used to derive the asymptotic normal distribution of a second-order
degenerate U-statistic. Conditions (A1)(v)-(vii) and (A2)(iii) are conditions (A1), (A2), (B1),
(B2), (C1) and (C2) of Franke and Mwita (2003), which are required to get the uniform
convergence rate of nonparametric kernel estimator of conditional quantile with mixing data.
Finally Conditions (A2)(iv)-(v) are adopted from conditions of Lemma 2 of Yoshihara (1976),
which are required to get the asymptotic equivalence of nondegenerate U-statistic and its
projection under the β -mixing process.
We consider testing for local departures from the null that converge to the null at the rate
T12 h- m/4. More precisely we consider the sequence of local alternatives:
H,T : Fyiz {Qθ(X) + dτ>(z,)|z,} = θ, (17)
where dT = T-1/2h-m/4 and the function l(∙) and its first-derivatives are bounded.
Theorem 1. Assume the conditions (A1) and (A2). Then
(i) Under the null hypothesis (3), Thm/2Jτ → N(0,σ2) in distribution, where