compact neighborhood Θ0 of 0, the set Θ = {v = Qθ (x) + μ | x ∈ G, μ ∈ Θ0} is compact
and for some constant c0 > 0 , fy|x(v|x) ≥ c0 for all x∈ G, v∈Θ . (vii) There is an
increasing sequence sT of positive integers such that for some finite A ,
TT
— βsT/(3τ)(sτ) ≤ A , 1 ≤ sτ ≤ T for all T ≥ 1.
(A2) (i) we use product kernels for both L (∙) and K (∙) , let l and k be their
corresponding univariate kernel which is bounded and symmetric, then l(∙) is non-negative,
l(∙) ∈ Yv, k(∙) is non-negative and k(∙) ∈ Y2.
(ii) h = O(T α ) for some 0 < α ' < (7 / 8)m .
p1
(iii) a= o(1) and ST=Ta(sTlogT)→∞ for some sT →∞
(iv) there exists a positive number δ such that for r = 2 +δ and a generic number M0
∫∫ 1 K
hm
z1
z2
r
dFz(z1)dFz(z2)≤M0<∞ and
h
≤M0 <∞
(v) for some δ ' (0 < δ ' < δ), β(T) = O(T-(2+δ ')/δ ).
The following definitions are due to Robinson (1988).
Definition (D1) Yλ, λ≥ 1 is the class of even functions k: R→ R satisfying
∫ uik(u)du=δi0 (i =0,1,K,λ-1),
R
k(u) = O ( (1 + ∣uр+1+г )-1 ), for some ε > 0,
where δi is the Kronecker’s delta.
ij
Definition (D2) A αμ , α>0, μ>0 is the class of functions g: Rm → R satisfying that
10
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