Auxiliary Lemmas and Proofs

2.8 Supplementary Appendix

2.8.2 Auxiliary Lemmas and Proofs

the influence functions directly. Let

ψp,x,1(a;θ₀,G)≡E

(1−R)`(Z)Λ_x(W;β0) (1−Q₀)f_X|R(X|1) ·

(1−r)`(z)1(x≤X)

1−Q₀ −F_X|R(X|1)

, (2.8.25)

ψp,x,3(a;θ₀,G)≡ −r−Q₀ Q0

·E

(1−R)`(Z)Λ_x(W;β0) 1−Q0

· G(X) fX|R(X|1)

. (2.8.26)

Combining (2.8.25), (2.8.23), and (2.8.26), we deduce that

∆p,3=1 n

n

∑

i=1

ψp3,i=−1 n

n

∑

i=1

1 f_Y|R(q_τ|1)

ψg,p,i+(1−R_i)`(Z_i)Λ_x(W_i;β₀)g_p(X_i) 1−Q₀

−d_p(θ₀,G)) +o_p(n^−1/2). (2.8.27)

whereψg,pis defined in (2.7.10). Collecting the results in (2.8.20), (2.8.21), and (2.8.27), it follows that the asymptotic linear representation forU QE[_pis given as in (2.7.3). This completes our proof.

verify two conditions: (i) the classF is Donsker, and (ii) the variance of∆m₄(A;θ,qb_τ,q_τ)tends to zero.

To show (i), we let

F1≡ {y∈Y 7→1(y≤q)−1(y≤qτ): q∈Y},

F2≡ {z∈Z 7→L(k(z)⁰γ)/L(k(z)⁰γ+t(z)⁰λ_s): (γ,λ_s)∈Θ_γ,λs}, F3≡ {(r,z)∈ {0,1} ×Z 7→re_j(z): j=1, ...,d_β}.

Hence,F⊆F1·F2·F3.

Here,F1is a collection of indicator functions overR. Under Assumptions 2.7(a)(ii) and (e)(i) ,F2is pointwise compact; for definition, see Example 19.8 of Van der Vaart (1998). It is well-known that the pointwise compact class of functions and collections of indicators of cells in Euclidean space are pointwise measurable; see the discussion in Section 2.3 of Van Der Vaart and Wellner (1996) for definition. The pointwise measurability ofF3follows by definition. Then, by Lemma 8.10 in Kosorok (2008),F is pointwise measurable. F1is a VC-subgraph class with VC-index equal to 2. In addition,F1is uniformly bounded byF1≡2. Theorem 2.6.7 in Van Der Vaart and Wellner (1996) implies that

sup

Q

N(εkF₁k_Q,2,F1,L₂(Q))≤K₁ε⁻². (2.8.29)

Due to Assumption 2.7(e)(ii),

L(k(z)⁰γ1)

L(k(z)⁰γ1+t(z)⁰λs1)−_L(k(z)^L(k(z)0 ^0γ2)

γ2+t(z)⁰λs2)

≤K2· kk(z)k kγ₁−γ2k+K₃· kt(z)k kλ_s1−λs2k.

Combining this fact and Assumption 2.7(f),F2admits an integrable envelop function,F2≡K2γ¯kk(z)k+K₃λ¯skt(z)k+

K₄, where ¯γ ≡sup_γ∈Θ

γkγk and ¯λs≡sup_λ

s∈Θ_λskλ_sk are finite under Assumption 2.7(a)(ii). It then follows from Theorem 2.10.20 of Van Der Vaart and Wellner (1996) that for allδ >0,

Z δ

0

sup

Q

q

logN(εkF₂k_Q,2,F2,L₂(Q))dε≤

∑

j∈{γ,λ_s} Z δ

0

q

logN(εj,¯Θj,k·k)dε

≤

∑

j∈{γ,λ_s}

pd_j Z δ

0

s log

1+4diam(Θ_j) εj¯

dε<∞, (2.8.30)

wherediam(Θ)is the diameter ofΘ, andQis any finite discrete measure. By Assumption 2.7(f),E[F₂²]<∞, and therefore,F2is Donsker. Another application of Theorem 2.10.20 of Van Der Vaart and Wellner (1996) toF yields that

Z δ

0

sup

Q

q

logN(4εkFk_Q,2,F,L₂(Q))dε

≤

3

∑

j=1 Z δ

0

sup

Q

q

logN(ε F_j

_Q,2,Fj,L₂(Q))dε<∞, (2.8.31)

whereF3(z)≡ ke(z)k, is an envelop function forF3. The last inequality follows from (2.8.29), (2.8.30), and the fact thatF3is a finite set, and therefore, BUEI.

To show thatFis square integrable, note

E[|F₁F₂F₃|²]≤4E[ke(Z)k²(K₂γ¯kk(Z)k+K₃λ¯skt(Z)k+K₄)²]<∞,

where the second inequality follows by the Cauchy–Schwartz inequality and Assumption 2.7(f). By Theorem 2.10.20 and Theorem 2.10.1 in Van Der Vaart and Wellner (1996),F is Donsker.

In the next step, we show (ii). That is,^Rk∆m₄(a;θ,qb_τ,q_τ)k²dF_A(a)converges in probability to 0. Towards this end,

Z

||∆m₄(a;θ,bqτ,q_τ)||²dF(a)≤ c²₅

2c²₄· sup

y∈Y,z∈Z|fY|ZR(y|z,1)|

!2

E[ke(Z)k²](qbτ−qτ)²

≤K₅(qb_τ−q_τ)²=O_p(n⁻¹_s ), (2.8.32)

where the second inequality is by MVT and Assumption 2.7(b). The last equality follows by (2.8.7).

Combining (i) and (ii), Theorem 2.1 in Wellner and van der Vaart (2007) yields (2.8.28). To complete the proof, it suffices to show thatE[m₄(A;θ,bqτ)−m4(A;θ,q_τ)|Y_n] =O_p(n^−1/2). By a first-order expansion,

E[km₄(A;θ,qbτ)−m4(A;θ,qτ)k |Y_n] =E[

R f_Y|ZR(eqτ|Z,1)e(Z)

|Y_n]· |qbτ−q_τ|

≤K₆ sup

y∈Y,z∈Z|f_Y|ZR(y|Z,1)|E[ke(Z)k]·O_p(n^−1/2) =O_p(1)·O_p(n^−1/2) =O_p(n^−1/2),

where the second line follows by MVT, Assumptions 2.7(b), and (f).

Lemma 2.5 Under the assumptions of Lemma 2.2,

H1≡ (

(r,z)∈ {0,1} ×Z 7→(r−L(k(z)⁰γ))L⁰(k(z)⁰γ)ek(z)

L(k(z)⁰γ)(1−L(k(z)⁰γ)) :ek(z)∈ {k₁(z), ...,k_dγ(z)},γ∈Θγ

o , H2≡

(r,z)∈ {0,1} ×Z 7→

r

L(k(z)⁰γ+t(z)⁰λ_s)−1

L(k(z)⁰γ)et(z): et(z)∈ {t₁(z), ...,t_d

λ(z)},(γ,λ_s)∈Θ_γ,λs ,

H3≡

(r,z)∈ {0,1} ×Z 7→

1−r

1−L(k(z)⁰γ+t(z)⁰λ_a)−1

L(k(z)⁰γ)et(z): et(z)∈ {t₁(z), ...,t_d

λ(z)},(γ,λ_a)∈Θ_γ,λa , H4≡

(r,z,x,y)∈ {0,1} ×Z X Y 7→

r·1(y≤q)

L(k(z)⁰γ+t(z)⁰λs)− (1−r)·Λ(w,β) 1−L(k(z)⁰γ+t(z)⁰λa)

·L(k(z)⁰γ)ee(z):

ee(z)∈ {e₁(z), ...,e_d

β(z)},q∈Y0,θ∈Θ o

,

are Glivenko-Cantelli (GC).

Proof of Lemma 2.5: Under Assumptions 2.7(e)(iii) and (f),H1,H2, andH3, admit integrable envelop functions, H₁,H₂, andH₃, respectively, whereH₁(r,z)≡K₁kk(z)k,H₂(r,z)≡K₂kt(z)k, andH₃(r,z)≡K₃kt(z)k. Hence, by Example 19.8 in Van der Vaart (1998),Hj,j=1,2,3,are GC. To show thatH4is also GC, we define

H5≡ {y∈ {0,1} ×Y 7→1(y≤q),q∈Y0}, H6≡

(r,z,x)∈ {0,1} ×Z X Y 7→(1−r)Λ(w,β)L(k(z)⁰γ)e(z)e 1−L(k(z)⁰γ+t(z)⁰λa) , ee(z)∈ {e₁(z), ...,e_d

β(z)},q∈Y0,θ∈Θ o

.

Notice thatH4=F2F3H5−H6. Under Assumptions 2.7(c)(iii), (e)(iii), and (f),H6admits an integrable envelop, H₆≡K₄ke(z)k. By Example 19.8 in Van der Vaart (1998),H4is GC. It is straightforward to checkF2F3H5admits an integrable envelop function. From the proof of Lemma 2.4, we know thatF2andF3are Donsker. Then, by the Donskerness ofH5andH6, and Corollary 9.26 in Kosorok (2008),H4is also GC.

Lemma 2.6 Under the assumptions of Theorem 2.3,

dbq,n(θb,G)−dq(θ₀,G) =op(1).

Proof of Lemma 2.6: Letη(θ) = (η₁(·,θ),η₂)⁰, f(a;θ,η(θ))≡ f₁(a;θ,η(θ))·G⁻¹(η₁(x,θ)/η₂) +f₂(a;θ,η(θ)), where

f₁(a;θ,η(θ))≡ 1−r Q₀η2

· L(k(z)⁰γ+t(z)⁰λ_s)

1−L(k(z)⁰γ+t(z)⁰λa)·Λx(w;β), and f₂(a;η(θ))≡ −x·f₁(a;θ,η(θ)). With these quantities, we can write

dbq,n(bθ,G)−d_q(θ₀,G) =En[f(A;θ₀,ηb_n(θ))]−E[f(A;η₀(θ₀))],

where

ηb_n(θ)≡ En

"

(1−R) Q₀

L(k(z)⁰γb+t(z)⁰bλs) 1−L(k(z)⁰γb+t(z)⁰bλa)

1(X≤ ·)

# ,En[R]

Q₀

!0

,

η₀(θ)≡

E

(1−R) Q₀

L(k(z)⁰γ+t(z)⁰λ_s)

1−L(k(z)⁰γ+t(z)⁰λa)1(X≤x)

,1 0

.

LetNθ0 denote a neighborhood ofθ0. We proceed by showing, (i) whenηbn(θ)is sufficiently close toη0(θ), we have sup_θ∈N

θ0

kf(a;θ,η(θ))−f(a;θ,η0(θ))k ≤c(a)·sup_x∈X,θ∈N

θ0

kη(θ)−η₀(θ)k, for somec(·)satisfying, (ii)

E[c(A)] sup

x∈X,θ∈N_θ0kηbn(θ)−η0(θ)k→^p 0, and (iii)E[sup_θ∈N

θ0,kη−η₀k≤εkf(A;θ,η(θ))k]<∞.

By Assumptions 2.7(c) and (e),L(·)andΛx(·)are uniformly bounded. Under Assumption 2.8(d),G⁻¹is Lipschitz with a bounded Lipschitz constant. Then, (i) follows immediately by

kf(a;θ,η(θ))−f(a;θ,η₀(θ))k ≤K₁sup

x∈Xkη₁(x,θ)−η1,0(x,θ)k+K₂kη₂−η2,0k, and lettingc(·) =max{K1,K2}.

Next, we shall verify (ii). To this end, it suffices to show thatηb1,nconverges toη1,0uniformly. This follows from Gηbeing GC, whereGη≡Gη,1·Gη,2, and

Gη,1≡

(r,z)∈ {0,1} ×Z 7→ (1−r)L(k(z)⁰γ+t(z)⁰λs)

Q0(1−L(k(z)⁰γ+t(z)⁰λa)):θ∈Θ

, Gη,2≡ {x∈X 7→1(x≤q):q∈X}.

SinceGη,1is uniformly bounded, it follows from Example 19.2 in Van der Vaart (1998) that it is GC.Gη,2is VC subgraph with VC index equal to 2, and therefore, it is GC. Then, by Corollary 9.27 in Kosorok (2008),Gη is also uniformly bounded GC.

Lastly, (iii) holds under Assumption 2.1(d) and Assumption 2.7(e).

By (i), (ii), and the Markov inequality, sup_θ∈N

θ0kf(A;θ,ηb(θ))−f(A;θ,η₀(θ))k→^p 0. Then the desired result is

obtained by (iii) and Lemma 4.3 in Newey and McFadden (1994).

Lemma 2.7 Under the assumptions of Theorem 2.3, we have thatEn[∆φ_q(A;θ0,G)] =n⁻¹∑ⁿ_i=1ψg,q(A_i;θ0,G) +o_p (n^−1/2),where∆φgandψg,qare defined in (2.8.17) and (2.7.9), respectively.

Proof of Lemma 2.7:Letf₀(a;η)≡_1−Q^1−r

0`(z)Λ_x(w;β₀)G⁻¹(η₁/η₂),and hence,∆φ_q(a;θ₀,G) =f₀(a;ηb_n)−f₀(a;η₀), whereηb_n(·)≡

En

h_(1−R)`(Z)

1−Q₀ 1(X≤ ·)i

,En[R]/Q₀0

,andη₀(·)≡(F_X|R(·|1),1)⁰. The first step of our proof is to show that

√n(En−E)(f0(a;ηbn)−f0(a;η0)) =op(1), (2.8.33)

by invoking Theorem 2.1 in Wellner and van der Vaart (2007).

Towards this end, we need to (i) define the functional space Hη such that P(ηbn∈Hη)→1, (ii) verify that Fη≡ {f₀(·;η):η∈Hη}is Donsker, and (iii) show that

Z

(f₀(a;η)b −f₀(a;η0))²dF_A(a)→^p 0. (2.8.34)

For (i), letHη≡(Hη,1,Hη,2), where

Hη,1≡ {x∈X 7→ f(x):f non-decreasing, bounded between 0 and 1},

andHη,2≡[1−δ_η,1+δη], for someδη∈(0,1/2). GivenH, Condition (i) is implied byηb→^p η0uniformly, which is shown in Lemma 2.6.

Next, we establish the Donsker property ofFη. Note thatFη=_1−Q^1−r

0`(z)Λ_x(w;β0)·G⁻¹(Hη₁/Hη₂).By Lemma 9.11 in Kosorok (2008),Hη,1is BUEI, relative to the envelopH_η,1≡1.H2is a bounded convex set in the Euclidean space, and hence, trivially BUEI. Pointwise measurability is immediate from the definitions of the two sets. By Assumption 2.8(d), G⁻¹ is a Lipschitz continuous function with a bounded Lipschitz constant. Given that H2 is bounded away from 0, we conclude from Lemma 9.14 and Theorem 9.15 in Kosorok (2008) thatG⁻¹(Hη1/Hη2)is also BUEI and pointwise measurable relative to the envelopHη,3≡sup{x∈X}. By Theorem 2.10.1 in Van Der Vaart and Wellner (1996),G⁻¹(Hη₁/Hη₂)is uniformly bounded Donsker. Finally, under Assumptions 2.7(c)(iii) and (e), Corollary 9.32 then implies thatFηis uniformly bounded Donsker.

Now, to show (2.8.34), note that Z

(f₀(a;η)b −f₀(a;η0))²dF_A(a)≤K₁sup

x∈X|ηb1,n(x)−η1,0(x)|²+K₂|ηb2,n−η2,0|²=o_p(1),

where the first inequality follows by carefully bounding the coefficients associated with each term and by the fact that fourth moments ofkk(z)kandkt(z)kexist under Assumption 2.7(f). The last one follows becauseηb_nconverges uniformly toη₀.

Next, we prove that

√n











 ηb1,n(·)

ηb_2,n





−





 η1,0(·)

η_2,0











 Gη₁,η₂(·),

whereGη1,η2(·)is a tight, two-dimensional mean zero Gaussian process with covariance functionΣ(x₁,x₂) =E[(ψ_η,1 (x₁),ψ_η,2)(ψ_η,1(x₂),ψ_η,2)⁰],ψ_η,1(·) =ψ_η,1(a,γ₀;·)≡h_η(r,z;γ₀)1(x≤ ·)−F_X|R(x|1),ψ_η,2≡(r−Q₀)/Q₀,andh_η(r,z;

γ₀)≡^(1−r)`(z)_1−Q

0 .

The weak convergence follows fromG0,η being Donsker, where

G0,η≡ {(r,x,z)∈ {0,1} ×X ×Z 7→hη(r,z;γ0)·1(x≤q):q∈X}.

Under Assumption 2.7(e),h_ηis uniformly bounded. Since the class of indicator functions is uniformly bounded Donsker, by Corollary 9.32 in Kosorok (2008), we conclude thatGηis also uniformly bounded Donsker.

In view of (2.8.33), we deduce that

En[∆φ_q(A;θ₀,G)] =E[f₀(A;ηb_n)−f₀(A;η₀)] +o_p(n^−1/2). (2.8.35)

In the next step, we derive the asymptotic linear representation of the first term on the right hand side of (2.8.35) . The proof continues by showing that the map, φ_G(η)≡^R f(a;η)dF_A(a),is Hadamard differentiable in η atη0. Observe that we can decomposeφGas follows

(η₁,η₂)7→η1

η2

7→G⁻¹◦(η₁/η₂)7→h_G◦(G⁻¹◦(η₁/η₂)),

whereh_G(g)≡^R _1−Q^1−r

0`(z)Λ_x(w;β0)gdF_A(a).The first map is continuous and uniformly bounded onHη,2, and thus, Hadamard differentiable. The second and third maps are both composition maps. SinceG⁻¹(·)is continuously differ- entiable with bounded first-order derivative under Assumption 2.8(d), by Lemma 3.9.25 in Van Der Vaart and Wellner (1996), it is Hadamard differentiable. Since integration is a linear functional,h_Gis also Hadamard differentiable. Now we invoke the chain rule, e.g. Theorem 20.9 in Van der Vaart (1998), and conclude thatφ_Gis Hadamard differentiable, with the functional derivative ofφ_Gatη0in the direction of(ψ_η,1,ψη,2)given as follows

(ψ_η,1,ψη,2)7→h_g◦ 1

G⁰◦G⁻¹◦ η1,0

η_2,0

· ψη,1

η_2,0−η1ψη,2

η_2,0²

!!

.

Given the Hadamard differentiability, we then apply the delta method (as in Theorem 3.9.4 in Van Der Vaart and

Wellner (1996)) to get the linear expansion,ψ_g,q,

ψg,q(a;θ0) =

Z 1−r

1−Q₀· `(z)Λ_x(w;β₀)

G⁰(G⁻¹(η_1,0/η_2,0))· ψη,1

η2,0

−η1,0ψη,2

η_2,0²

! dF_A(a)

=E

"

1−R

1−Q₀· `(Z)Λ_x(W;β0) G⁰ G⁻¹(F_X|R(X|1))·

(1−r)`(z)1(x≤X)

1−Q₀ −rF_X|R(X|1) Q₀

# .

This concludes our proof.