Auxiliary Lemmas

1.9 Supplementary Appendix

1.9.3 Auxiliary Results

1.9.3.2 Auxiliary Lemmas

Lemma 1.7 Suppose that the assumptions of Theorem 1.3 hold. The function classes,G1-G6,Gb,Gη, andGϕ as defined in (1.9.19), (1.9.20),(1.9.22), (1.9.23), (1.9.27), (1.9.28), (1.9.24), (1.9.25), and (1.9.29) are of VC type with bounded envelop.

Proof of Lemma 1.7. We first identify the sub-classes that constitute the above functional classes and show that the uniform entropy condition is satisfied for each of these sub-classes. Then, we illustrate on how we use results on the

12A triangular array of stochastic process{f_ni(t):i=1, ...,n,t∈T}is separable if, for alln≥1, there exists a countable setT_n∈Tsuch that

E^∗

"

1 (

sup

t∈Tsup

s∈T_n n i=1

∑

(f_ni(s)−f_ni(t))²>0 )#

=0.

sub-classes to show that the functional classes in the theorem is of VC type. Define

M1≡ {y7→1{y≤t}:t∈T˜},

M2≡ {x₁7→K((x₁γd−xγ_d)/h)1{|x₁γd−xγ_d| ≤h}:(x,h)∈X ×H}, M3≡ {x₁7→K⁽¹⁾((x₁γ_d−xγ_d)/h)1{|x₁γ_d−xγ_d| ≤h}:(x,h)∈X ×H}, M4,1≡ {y7→∂_xγ<sup>`</sup>G<sub>d</sub>(y,xγ<sub>d</sub>):`∈ {0,1,2, ...,s},(d,x)∈ {0,1} ×X}, M4,2≡ {y7→∂_xγ<sup>`</sup>G<sub>d,1</sub>(y,xγ<sub>d</sub>):`∈ {0,1,2, ...,s},(d,x)∈ {0,1} ×X}.

M4,3≡ {∂G_d(y,xγ_d)/∂y|_y=t:(d,t,x)∈ {0,1} ×T˜×X}.

M4,4≡ {∂G_d,1(y,xγ_d)/∂y|_y=t:(d,t,x)∈ {0,1} ×T˜×X}.

M5≡ {∂_xγ<sup>`</sup> f<sub>d</sub>(xγ<sub>d</sub>):`∈ {0,1,2, ...,s},(d,x)∈ {0,1} ×X}.

By Lemma 19.15 in Van der Vaart (1998),M1is of VC type with the constant envelop. Under Assumption 1.8.2, bothK(·)andK⁽¹⁾(·)are of bounded variation, Lemma 22(i) of Nolan and Pollard (1987) implies thatM2,1andM2,2

belong to the VC class with a constant envelop. Next, since∂_v<sup>`</sup>F<sub>Yd,Rd|D,Xγd</sub>(y,r|d,v),`=0, ...,s, is Lipschitz continuous with respect toxγ_d under Assumption 1.6.2(i), Lemma 2.13 of Pakes and Pollard (1989) impliesM4,1andM4,2are of VC type with bounded envelop functions. The proof forM4,3,M4,4, andM5follows the same arguments based on the Lipschitz continuity of∂_yF_Yd,Rd|D,Xγ_d(y,r|d,v)with respect toyandv, and of∂_vfd,γ_d(v)with respect tov, as implied by Assumption 1.6.2(iv), and 1.6.1(i), respectively.

Now we are ready to show why the functional classes in the lemma are of VC type. We illustrate onG1andGη. All others follow by same lines of reasoning.

We focus onG1first. Note that the class thatg₁₁belongs to is a product of a finite set{(r₁,d₁)7→r₁1{d₁=d},d∈ {0,1}},M1, andM2, and thus, it is of VC type by Corollary A.1 in Chernozhukov et al. (2014). Since all three sub- classes have finite envelops, their product also does. Regardingg₁₂, we first show thatMφ ≡ {y7→φ_θ⁰⁰(s_d(y,xγ_d)): (x,θ)∈X ×Θ}is also a VC class with bounded envelop. For anyx₁,x₂∈X andθ₁,θ₂∈Θ, we have

φ

00

θ₁(s_d(y,x1γ_d))−φ_θ⁰⁰

2(s_d(y,x2γ_d))

≤ φ

00

θ1(s_d(y,x₁γd))−φ_θ⁰⁰

2(s_d(y,x₁γd)) +

φ

00

θ2(s_d(y,x₁γd))−φ

00

θ2(s_d(y,x₂γd))

≤M₁|θ₁−θ₂|+ sup

(θ,u)∈Θ×[υ_o,1]

φ

000

θ(u) sup

(y,x)∈T˜×X

∂_xγG_d(y,xγ_d)

kx₁−x₂k kγ_dk

≤M₁|θ₁−θ2|+M2kx₁−x₂k ≤√

2 max{M₁,M2}

(θ₁,x⁰₁)⁰−(θ₂,x⁰₂)⁰ ,

whereM₁andM₂are positive constants. The second inequality is due to the Lipschitz continuity condition onφ_θ⁰⁰, and the third follows becauseφ_θ⁰⁰⁰and∂_xγG_dare uniformly bounded under Assumption 1.6.4, and 1.6.2. The last one is by H¨older’s inequality. Another application of Lemma 2.13 of Pakes and Pollard (1989) yields the desired result.

LetMx={x˜7→x˜`−x<sub>`</sub>:`=2, ...,k,x∈X}. SinceX is compact,Mxis a VC class becauseN (εsup_x∈X x[−1]

, Mx,L2(Q))≤C(diam(X)/ε),for a positive constantCindependent ofε. Applying Corollary A.1 in Chernozhukov et al. (2014) again on the product ofM1, M3,Mφ,Mx, and the finite set{(y₁,y₂,d₂)7→1{d₂=d,y₂≤y₁},d∈ {0,1}}yields that the first half of g₁₂ belongs to a VC class. Next, by Lemma 5 of Sherman (1994), we deduce that{y₁7→^R1{d₂=d,y₂≤y₁∧t}hK⁽¹⁾(xγ_d,x⁰₂γd)(x_2,l−x_l)dF(w₂):ω∈Ω}and{w₂7→^Rg₁₁(w₁,ω)g₁₂(w₂,y₁,ω) dF(w₁):ω∈Ω}are both of the VC type. Since f_d(xγ_d)is uniformly bounded away from 0,{1/f_d(xγ_d)²:(d,x)∈ {0,1} ×X}admits a finite envelop. Applying Corollary A.1 in Chernozhukov et al. (2014) yet again concludes the proof.

Turning toGη, we first show that for a fixedx, the setMθ≡ {1/φ_θ⁰(s_Td(t,xγ_d,θ)):(t,θ)×T˜×Θ}belongs to the VC class. Recall thats_Td(t,xγ_d,θ) =φ_θ⁻¹

Rt

0φ_θ⁰(s_d(y,xγ_d))s_d,1(dy,xγ_d) . Hence,

1/φ_θ⁰

1(s_Td(t₁,xγ_d,θ1))−1/φ_θ⁰

2(s_Td(t₂,xγ_d,θ₂))

≤n 1/φ_θ⁰

1(s_Td(t₁,xγ_d,θ1))−1/φ_θ⁰

2(s_Td(t₁,xγ_d,θ2))o +n

1/φ_θ⁰

2(s_Td(t₁,xγ_d,θ2))−1/φ_θ⁰

2(s_Td(t₂,xγ_d,θ2))o

≡∆₁+∆₂.

Decomposing the first term further into,

|∆₁| ≤

1/φ˙_θ⁻¹

1

_{Z t}

1

0

φ

0

θ₁(s_d(y,xγ_d))s_d,1(dy,xγ_d)

−1/φ˙_θ⁻¹

1

_{Z t}

1

0

φ

0

θ₂(s_d(y,xγ_d))s_d,1(dy,xγ_d)

+

1/φ˙_θ⁻¹

1

_{Z t1}

0

φ

0

θ₂(s_d(y,xγ_d))s_d,1(dy,xγ_d)

−1/φ˙_θ⁻¹

2

_{Z t1}

0

φ

0

θ₂(s_d(y,xγ_d))s_d,1(dy,xγ_d)

≡∆₁₁+∆₁₂.

For the first term, we have

|∆₁₁| ≤(1−υo) sup

(z,θ)∈[0,y^∗_o]×Θ

φ

00

θ(φ⁻¹

θ (z)) φ˙_θ⁻¹(z)3

sup

(u,θ)∈[υ_o,1]×Θ

φ

0

θ(u)

|θ₁−θ2|=M₃|θ₁−θ2|.

Under Assumption 1.10.(ii),∆₁₂≤M₄|θ₁−θ₂|.

|∆₂| ≤ sup

(z,θ)∈[0,y^∗_o]×Θ

φ¨_θ⁻¹(z) φ˙_θ⁻¹(z)3

·

Z _t1 0

φ

0

θ₂(s_d(y,xγ_d))s_d,1(dy,xγ_d)− Z _t2

0

φ

0

θ₂(s_d(y,xγ_d))s_d,1(dy,xγ_d)

≤ sup

(z,θ)∈[0,y^∗_o]×Θ

φ¨_θ⁻¹(z) φ˙_θ⁻¹(z)3

· sup

(u,θ)∈[υ_o,1]×Θ

φ

0

θ(u) sup

(y,x)∈T˜×X

∂_xγG_d,1(y,xγ_d)

|t₁−t₂|

=M₅|t₁−t₂|.

where inequalities hold by the mean value theorem and under Assumptions 1.6.2, and 1.6.4. Combining the bounds and applying H¨older’s inequality, we conclude by Lemma 2.13 of Pakes and Pollard (1989) thatMθ is a VC class.

Next, following similar analysis as in the previous part, we deduce from Corollary A.1 of Chernozhukov et al.

(2014) that{w₁7→1{d₁=d}(1{y₁≤y} −G_d(y,x₁γ_d))∂_yG_d,1(y,xγ_d):(d,y,θ)∈ {0,1} ×T˜×Θ}for a givenx∈X is a VC class with a finite envelop. Applying Lemma 5 of Sherman (1994), we get{w₁7→^R1{y₁≤t}1{d₁=d} · (1{y₁≤y} −G_d(y,x₁γd))∂_yG_d,1(y,xγ_d)dy:(d,t,θ)∈ {0,1} ×T˜×Θ}also belongs to the VC class with an envelop F_η,1=G_d,1(y_o∧y_c,xγ_d). This is due to

Z 1{y₁≤t}1{d₁=d}(1{y₁≤y} −G_d(y,x₁γ_d))∂_yG_d,1(y,xγ_d)dy

≤2 Z t

0 ∂yG_d,1(y,xγ_d)dy≤G_d,1(y_o∧y_c,xγ_d).

Analogous results can be established for the other two parts ofΨ_d.

Combining these results with the fact that {x17→K((x₁γ_d−xγ_d)/h):(t,h)∈T˜×H} is VC with an envelop C1{|x₁γ_d−xγ_d| ≤h}, we deduce thatGη is of the VC type, with the envelop given by∑d=0,1C_d1{|x₁γ_d−xγ_d| ≤h}

whereC0andC1are positive constants. SettingHη,d(x₁γ_d) =C_d1{|x₁γ_d−xγ_d| ≤h}concludes the proof.

Lemma 1.8 Under the assumptions of Theorem 1.3, for anyδn=O_p n^−1/2 ,

sup

k˜γd−γ_dk≤δ_n

sup

(t,x)∈T˜×X

Z t 0

φ¨_d,γ^θ

d(y,x)

∂γGˆ_d(y,xγ˜d)−∂γGˆ_d(y,xγ_d) s_d,1(dy,xγ_d)

=O_p

(logn)^1/2n^−1/2h^−5/2δ_n

+O(δ_n).

Proof of Lemma 1.8.Split the term inside the norm operator into

∆₁(t,x,θ)≡ Z t

0

φ¨_d,γ^θ

d(y,x)

∂γκˆd,y(xγ˜d)

fˆ_d(xγ˜d) −∂γκˆd,y(xγ_d) fˆ_d(xγ_d)

s_d,1(dy,xγ_d),

∆2(t,x,θ)≡ − Z t

0

φ¨_d,γ^θ

d(y,x)

(κˆ_d,y(xγ˜_d)∂_γfˆ_d(xγ˜_d)

fˆ_d²(xγ˜_d) −κˆ_d,y(xγ_d)∂_γfˆ_d(xγ_d) fˆ_d²(xγ_d)

)

s_d,1(dy,xγ_d).

Decomposing∆1and ignoring smaller order terms gives

∆₁(t,x,θ) =f_d(xγ_d)⁻¹ Z t

0

φ¨_d,γ^θ

d(y,x)

∂_γκˆ_d,y(xγ˜_d)−∂_γκˆ_d,y(xγ_d) s_d,1(dy,xγ_d) + fˆ_d(xγ˜d)−fˆ_d(xγ_d)

f_d(xγ˜_d)f_d(xγ_d) Z t

0

φ¨_d,γ^θ

d(y,x)∂_γκˆd,y(xγ_d)s_d,1(dy,xγ_d) + (s.o.).

We investigate the uniform rate of the first term only. The second term exhibits the same rate and is simpler. Define

∆₁₁(t,x,θ,γ˜_d)≡ Z t

0

φ¨_d,γ^θ

d(y,x)E[∂_γκˆ_d,y(xγ˜_d)−∂_γκˆ_d,y(xγ_d)]s_d,1(dy,xγ_d),

∆12(t,x,θ,γ˜_d)≡ Z _t

0

φ¨_d,γ^θ

d(y,x){∂_γκˆd,y(xγ˜_d)−∂γκˆd,y(xγ_d)

−E[∂_γκˆd,y(xγ˜_d)−∂_γκˆd,y(xγ_d)]}s_d,1(dy,xγ_d).

By Fubini’s theorem and standard change of variables,

∆11(t,x,θ,γ˜d) =h⁻² Z t

0

φ¨_d,γ^θ _d(y,x)·

E h

ρ_1,1^γ˜d(y,Xγ˜_d)K⁽¹⁾((Xγ˜_d−xγ˜_d)/h)−ρ_1,1^γd(y,Xγ_d)K⁽¹⁾((Xγ_d−xγ_d)/h)i

s_d,1(dy,xγ_d)

=h⁻¹ Z t

0

φ¨_d,γ^θ

d(y,x)· Z

R

K⁽¹⁾(u)ρ_1,1^γ˜d(y,xγ˜d+uh)f_d(xγ˜d+uh)du

− Z

R

K⁽¹⁾(u)ρ_1,1^γd(y,xγ_d+uh)f_d(xγ_d+uh)du

s_d,1(dy,xγ_d)

= Z

R

uK⁽¹⁾(u)du· Z t

0

φ¨_d,γ^θ

d(y,x)·n

∂zρ_1,1^γ˜d(y,z)|_z=x˜γdf_d(xγ˜d) +ρ_1,1^γ˜d(y,xγ˜d)∂_zf_d(z)|_z=x˜γd

−

∂zρ_1,1^γd(y,z)|_z=xγdf_d(xγ_d) +ρ_1,1^γd(y,xγ_d)∂_zf_d(z)|_z=xγdo

s_d,1(dy,xγ_d),

whereρ_1,1^γ is defined in (1.4.8). The second equality follows by Taylor expansion and the fact that^R_[−1,1]K⁽¹⁾(u)du=0.

By the Lipschitz continuity ofρ_1,1^γ (y,xγ),∂_xγρ_1,1^γ (y,xγ),f_d(xγ), and∂_xγf_d(xγ), with respect toγas implied by Assump- tion 1.7.1, and by the fact thatkγ˜_d−γ_dk ≤δn, we conclude that sup_k˜γ

d−γ_dk≤δ_nsup_{(t,x,θ)∈T˜×X×Θ}k∆₁₁(t,x,θ,γ˜_d)k= O(δ_n).

The centered term∆₁₂can be bounded using following empirical process

Gδ,n≡ {w7→g_δ(w,ω,γ˜_d):ω∈Ω,kγ˜_d−γ_dk ≤δ_n},

whereg_δ(W,ω,γ˜_d)≡g_δ,1(W,ω,γ˜_d)−^Rg_δ,1(W,ω,γ˜_d)dF_W(W),and

g_δ,1(W,ω,γ˜d)≡ Z _t

0

φ¨_d,γ^θ

d(y,x)1{D=d,Y ≤y}s_d,1(dy,xγ_d)

·n

K⁽¹⁾((Xγ˜d−xγ˜d)/h)−K⁽¹⁾((Xγd−xγ_d)/h)o .

Applying similar lines of arguments as in Lemma 1.7, it is straightforward to show thatGδ,nis a VC type class with bounded envelop, for eachδ_n. From the continuous differentiability ofK⁽¹⁾(·), we deduce by similar arguments to Lemma 8.4 in Maistre and Patilea (2019) that

K⁽¹⁾((Xγ˜_d−xγ˜_d)/h)−K⁽¹⁾((Xγ_d−xγ_d)/h)

≤δ_nh⁻¹||K⁽²⁾((Xγ_d− xγ_d)/h)||+Cδ_n²h⁻², for some positive constantC. Combine this fact with the uniform boundedness of ¨φ_d,γ^θ

d, andsd,1, and we find that sup_g

δ∈G_δE[g²

δ]is bounded from above at the rate ofO δ_n²h⁻¹

. We then conclude from applying the maximal inequality in Lemma 1.5 that sup_k˜γ

d−γ_dksup_{(t,x,θ)∈T˜×X×Θ}k∆₁₂(t,x,θ,γ˜_d)k=O_p

(logn)^1/2n^−1/2h^−5/2δn

. For∆2, we have

∆2(t,x,θ,γ˜d) =−f_d⁻²(xγ˜d) Z t

0

φ¨_d,γ^θ

d(y,x)

κˆd,y(xγ˜d)−κˆd,y(xγ_d) ∂γfˆ_d(xγ˜d)s_d,1(dy,xγ_d)

−f_d⁻²(xγ_d) Z t

0

φ¨_d,γ^θ

d(y,x)κ_d,y(xγ_d)

∂_γfˆ_d(xγ˜_d)−∂_γfˆ_d(xγ_d) s_d,1(dy,xγ_d) +(f_d(xγ˜d) +f_d(xγ_d))(fˆ_d(xγ˜d)−fˆ_d(xγ_d))

f_d⁻²(xγ˜d)f_d⁻²(xγ_d)

· Z t

0

φ¨_d,γ^θ

d(y,x)κ_d,y(xγ_d)∂_γfˆ_d(xγ_d)s_d,1(dy,xγ_d).

Arguing as in the case of∆₁, one finds that the second term in the above display dominates the other two with a uniform rate ofO_p

(logn)^1/2n^−1/2h^−5/2δ_n

+O(δ_n). Gathering results on∆1and∆2completes our proof.

Lemma 1.9 Suppose the conditions of Theorem 1.5 hold. Then

sup

(t,θ)∈T˜×Θ

n^−1/2h^1/2

n i=1

∑

g_d,γd,`(X_i,x)Ψˆ_d Eˆd,γˆ_d,i,Eˆd,1,γˆ_d,i

(t,x,θ)−Ψ_d Ed,γ_d,i,Ed,1,γ_d,i

(t,x,θ)

=O_p

(logn)^1/2n^−1/2h^−(2`+1)/2 ,

whereg<sub>d,γ,`</sub>(X,x) =h<sup>−(`+1)</sup>K<sup>(`)</sup>(xγ<sub>d</sub>,Xγ<sub>d</sub>)/f(xγ<sub>d</sub>,d), for`=0,1.

Proof of Lemma 1.9. Defineη_3,1(t,xγ)≡φ_θ⁰ s_Td(t,xγ,θ)

,η_3,2(t,xγ)≡φ_θ⁰⁰(s_d(t,xγ)),η_3,3(W,t,γ)≡1{D=d} · (1{Y ≤t} −G_d(t,Xγ)), η3,4(t,xγ)≡s_d,1(t,xγ), η_3,5(W,t,γ)≡1{D=d} R1{Y ≤t} −G_d,1(t,Xγ)

,and for `= 1, ...,5, let the estimator ofη<sub>3,`</sub>be denoted by ˆη_3,`. Their definitions should be apparent. Index onθis suppressed.

From Theorem 1.3 and Lemma 1.4, we have

sup

(t,θ)∈T˜×Θ

ηˆ3,`(t,xγ<sub>d</sub>)−η3,`(t,xγ_d) =O_p

(logn)^1/2n^−1/2h^−1/2 ,

for`=1,2,4.

Given these notations, we divideΨ_dinto

Ψd,1(W,t,xγ) = 1 η3,1(t,xγ)

Z _t 0

η3,2(y,xγ)η_3,3(W,y,xγ)η_3,4(dy,xγ), Ψd,2(W,t,xγ) = −1

η3,1(t,xγ)η3,4(t,xγ)η_3,5(W,t,xγ), Ψd,3(W,t,xγ) = 1

η3,1(t,xγ) Z _t

0

η3,2(y,xγ)η_3,5(W,y,xγ)η_3,4(dy,xγ),

and thusΨ_d(Ed,γ,Ed,1,γ) =∑³<sub>`=1</sub>Ψd,`. We illustrate onΨd,1, since the other two terms share a similar structure. From tedious manipulation, it can be shown that ˆΨd,1(W,t,xγˆd)−Ψd,1(W,t,xγ_d) =∑¹⁰_{`=1</sub>A<sub>3,`}(W,t,x), where

A_3,1(W,t,x) =−ηˆ3,1(t,xγˆ_d)−η3,1(t,xγ_d) η3,1(t,xγ_d)ηˆ3,1(t,xγˆd)

Z t

0 η3,2(y,xγ_d)η_3,3(W,y,γd)η_3,4(dy,xγ_d), A3,2(W,t,x) = 1

η_3,1(t,xγ_d) Z _t

0

(ηˆ3,2(y,xγˆ_d)−η3,2(y,xγ_d))η_3,3(W,y,γ_d)η_3,4(dy,xγ_d), A_3,3(W,t,x) = 1

η3,1(t,xγ_d) Z t

0

η3,2(y,xγ_d)(ηˆ3,3(W,y,γˆd)−η3,3(W,y,γd))η_3,4(dy,xγ_d), A3,4(W,t,x) = 1

η3,1(t,xγ_d) Z t

0

η3,2(y,xγ_d)η_3,3(W,y,γ_d)(ηˆ3,4(dy,xγˆ_d)−η3,4(dy,xγ_d)), A_3,5(W,t,x) =−ηˆ3,1(t,xγˆ_d)−η3,1(t,xγ_d)

η3,1(t,xγ_d)ηˆ3,1(t,xγˆd)

· Z _t

0

(ηˆ3,2(y,xγˆ_d)−η_3,2(y,xγ_d))η_3,3(W,y,γ_d)η_3,4(dy,xγ_d), A_3,6(W,t,x) =−ηˆ3,1(t,xγˆ_d)−η3,1(t,xγ_d)

η3,1(t,xγ_d)ηˆ3,1(t,xγˆd)

· Z _t

0

η3,2(y,xγ_d)(ηˆ3,3(W,y,γˆd)−η3,3(W,y,γd))η_3,4(dy,xγ_d), A3,7(W,t,x) =−ηˆ3,1(t,xγˆ_d)−η3,1(t,xγ_d)

η_3,1(t,xγ_d)ηˆ_3,1(t,xγˆ_d)

· Z t

0 η3,2(y,xγ_d)η_3,3(W,y,γd)(ηˆ3,4(dy,xγˆd)−η3,4(dy,xγ_d)),

A_3,8(W,t,x) = 1 η3,1(t,xγ_d)

Z t 0

(ηˆ_3,2(y,xγˆ_d)−η_3,2(y,xγ_d))

·(ηˆ3,3(W,y,γˆ_d)−η3,3(W,y,γ_d))η_3,4(dy,xγ_d), A3,9(W,t,x) = 1

η3,1(t,xγ_d) Z t

0

η3,2(y,xγ_d)(ηˆ3,3(W,y,γˆ_d)−η3,3(W,y,γ_d))

·(ηˆ3,4(dy,xγˆd)−η3,4(dy,xγ_d)), A3,10(W,t,x) = 1

η3,1(t,xγ_d) Z t

0

(ηˆ3,2(y,xγˆ_d)−η3,2(y,xγ_d))

·η3,3(y,xγ_d)(ηˆ3,4(dy,xγˆd)−η3,4(dy,xγ_d)), A3,11(W,t,x) =−ηˆ3,1(t,xγˆ_d)−η3,1(t,xγ_d)

η_3,1(t,xγ_d)ηˆ_3,1(t,xγˆ_d)

· Z t

0

(ηˆ3,2(y,xγˆd)−η_3,2(y,xγ_d))(ηˆ3,3(W,y,γˆd)−η_3,3(W,y,γd))η_3,4(dy,xγ_d), A_3,12(W,t,x) = 1

η3,1(t,xγ_d) Z t

0

(ηˆ3,2(y,xγˆd)−η3,2(y,xγ_d))

·(ηˆ3,3(W,y,γˆ_d)−η3,3(W,y,γ_d))(ηˆ3,4(dy,xγˆ_d)−η_3,4(dy,xγ_d)), A_3,13(W,t,x) =−ηˆ3,1(t,xγˆ_d)−η3,1(t,xγ_d)

η3,1(t,xγ_d)ηˆ3,1(t,xγˆd)

· Z _t

0

(ηˆ3,2(y,xγˆ_d)−η_3,2(y,xγ_d))η_3,3(W,y,γ_d)(ηˆ3,4(dy,xγˆ_d)−η_3,4(dy,xγ_d)), A_3,14(W,t,x) =−ηˆ3,1(t,xγˆ_d)−η3,1(t,xγ_d)

η3,1(t,xγ_d)ηˆ3,1(t,xγˆd)

· Z _t

0

η3,2(y,xγ_d)(ηˆ3,3(W,y,γˆ_d)−η3,3(W,y,γ_d))(ηˆ3,4(dy,xγˆ_d)−η_3,4(dy,xγ_d)), A_3,15(W,t,x) =−ηˆ3,1(t,xγˆ_d)−η3,1(t,xγ_d)

η3,1(t,xγ_d)ηˆ3,1(t,xγˆd) · Z t

0

(ηˆ3,2(y,xγˆ_d)−η3,2(y,xγ_d))

·(ηˆ3,3(W,y,γˆd)−η3,3(W,y,γd))(ηˆ3,4(dy,xγˆd)−η_3,4(dy,xγ_d)).

Following the same type of analysis we have used so far, namely performing Taylor expansion, integration by parts, and applying the maximal inequality from Lemma 1.5 whenever appropriate, we get

sup

(t,θ)∈T˜×Θ

n^−1/2h^1/2

n

∑

i=1

g_{d,γd,`</sub>(X<sub>i</sub>,x)A<sub>3,`1}(W_i,t,x)

=O_p

(logn)^1/2n^−1/2h^−(2`+1)/2 ,

sup

(t,θ)∈T˜×Θ

n^−1/2h^1/2

n i=1

∑

g_{d,γd,`</sub>(X<sub>i</sub>,x)A<sub>3,`2}(W_i,t,x)

=O_p

logn·n⁻¹h^−(`+1) ,

sup

(t,θ)∈T˜×Θ

n^−1/2h^1/2

n i=1

∑

gd,γ_d,`(X<sub>i</sub>,x)A<sub>3,`3</sub>(W_i,t,x)

=Op

(logn)^3/2n^−3/2h^−(2`+3)/2

,

sup

(t,θ)∈T˜×Θ

n^−1/2h^1/2

n i=1

∑

g_d,γd,`(X_i,x)A_3,15(W_i,t,x)

=O_p

(logn)²n⁻²h^−(`+2) .

for`=0,1,`₁=1,2,3,4,`<sub>2</sub>=5,6,7,8,9,10, and`₃=11,12,13,14. As a result,

sup

(t,θ)∈T˜×Θ

n^−1/2h^1/2

n i=1

∑

g_d,γd,`(X_i,x) Ψˆ_d,1(W_i,t,xγˆ_d)−Ψ_d,1(W_i,t,xγ_d)

=O_p

(logn)^1/2n^−1/2h^−(2`+1)/2 .

Analogous results hold forΨd,2andΨd,3, concluding the proof.