where the last equality holds since k(u) = —k(—u)), we can notice that Wn(θ*) = λ∕-Un.
Applying first Lemma 3.1 and then Theorem 3.1 in Powell, Stock, and Stoker (1989), or similarly Lemma
3.3b. in Zheng (1996) we can show that Wn(θ∙) is asymptotically normally distributed and that it is
Op (√1n) . Let define Hn in the following way:
Hn(xi,Xj ) = h K ( xj
x
-i)[s(θ∙,xi) — s(θ∙,xj∙)].
(42)
First, we need to verify that E ∣j∣Hn(xi,Xj)∣∣2j = o(n). Let define v2(θ∙,x) = E(s2(θ∙, x)/x) and v(θ*,x) =
E(s(θ∙, x)/x),
Hn(xi,xj)∣∣2∣ = E E ^∣Hn(xi,xj)∣2 /xi,xj∙)] =
(43)
1 ° xj
= h2 K( -j
— xi
h
°2
) [v2(θ*,xi) + v2(θ∙,xj) — 2v(θ∙,xi)v(θ*,xj)] g(xi)g(xj)dxidxj =
now using the change of variable from (xi, xj ) to (xi, u = —
—) we obtain
= — ʃ ∣∣K(u)∣∣2 £v2(θ*, xi) + v2(θ*, xi + hu) — 2v(θ∙, xi)v(θ*, xi + hu)] g(xiι)g(xi + hu)dxihdu =
= O(—) = O(n(nh) 1 ) = o(n) since nh → ∞.
h
(44)
This implies that ʌ/-(Un — Un) = op(1). Thus, we need just to study the behavior of Un which is given by
2
Un = E(rn(xi)) + - 2^rn(xi) — Ε(rn(xi)).
i
Let compute rn (xi) which is defined in the following way:
(45)
rn(xi) — E (Hn(xi,xj )/xi) — hK (
x
— ) [s(θ ,xi) — v(θ ,xj)] g(xj)dxj =
(46)
where
and
= — K K(u)[s(θ∙,xi) — v(θ∙,xi + hu)] g(xi + hu)hdu = r(xi) + tn(xi)
h
r(xi) = [s(θ∙, xi) — v(θ∙, xi)] g(xi)
tn(xi) = ,θ θ. u2K (u)du = h
(47)
(48)
(49)
This last expression has been obtained applying the mean value theorem to v(θ*, Xi + hu) and g(xi + hu),
whichyields v(θ*,Xi) + huv0(xi, θ*) and g(θ*,xi) + hug0(xi,θ*) where xit lies in [xi,xi + hu].
Further, we need to compute E(rn (xi)) = E(Hn(xi, xj))
24