Proofs - Nonparametric Transformation Models

Before proving the main results of the Sections 2.3 and 2.4 an auxiliary lemma is given.

2.8.1 An Auxiliary Result

The following Lemma yields an asymptotic expansion for the difference of the conditional quantile function and its estimator ˆF_Y⁻¹_|X(τ|x). Recall model (2.1) under the local alterna-tivesH1,n in (2.15) andY0 =gβ0(X) +c0+ε.

Lemma 2.8.1 Assume (A1)–(A5). Then, F_Y⁻¹

0|X(τ|x)−Fˆ_Y⁻¹_|X(τ|x)

= 1

f_Y₀|X(F_Y⁻¹

0|X(τ|x)|x) 1

f_X(x)p(Fˆ _Y⁻¹

0|X(τ|x), x)−p0(F_Y⁻¹

0|X(τ|x), x) f_X(x)² fˆ_X(x)

+op

√n

=op n⁻¹⁴ , F_Y⁻¹

0|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x)

2.8. Proofs

= 1

f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x) 1

fX(x)pˆ₀(F_Y⁻¹

0|X(τ|x), x)−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)²

fˆ_X(x)

(2.45)

+o_p 1

√n

=op n⁻¹⁴ and

Fˆ_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x)

= cn

nf_Y₀|X(F_Y⁻¹

0|X(τ|x)|x)f_X(x)

i=1

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(Xi)−c0−εi) K_h_x(x−Xi)∆n(Xi) +op

√n

(2.46) uniformly in x∈supp(v) and τ ∈supp(µ).

Proof: Denote thej-th derivatives of K and fε by K^(j) and fε^(j), respectively. For appro-priatey_i^∗ ∈Rone has

p(y, x)−pˆ0(y, x)

= 1 n

i=1

K_h_y(y−gβ0(Xi)−c0−εi−cn∆n(Xi))− K_h_y(y−gβ0(Xi)−c0−εi) K_h_x(x−X_i)

= 1 n

i=1 r−1

j=0

K^(j)

y−gβ0(Xi)−c0−εi

h_y

K_h_x(x−Xi)(−1)^j+1c^j+1n ∆n(Xi)^j+1 h^j+1y (j+ 1)!

+ 1 n

i=1

K^(r)(y_i^∗)K_h_x(x−X_i)(−1)^r+1c^r+1_n ∆_n(X_i)^r+1 h^r+1y (r+ 1)! .

Thanks to (A2) and (2.41), one has (for an appropriate constantC >0)

1 n

i=1

K^(r)(y^∗_i)K_h_x(x−Xi)(−1)^r+1c^r+1_n ∆n(Xi)^r+1 h^r+1y (r+ 1)!

≤ Cc^r+1_n h^r+1y

1 n

i=1

|K_h_x(x−Xi)|

=o_p 1

√n

. Moreover, integration by parts yields

E 1

h^j+1y

K^(j)

y−gβ0(X1)−c0−εi

h_y

K_h_x(x−X₁)∆_n(X₁)^j+1

= Z Z

1 h^j+1y

K^(j)

y−g_β₀(w)−c₀−e h_y

Khx(x−w)∆n(w)^j+1fε(e)de fX(w)dw

= Z

− 1

h^j_yK^(j−1)

y−g_β₀(w)−c₀−e hy

f_ε(e)

∞

−∞

+ Z 1

h^jy

K^(j−1)

y−gβ0(w)−c0−e h_y

f_ε⁽¹⁾(e)de

K_h_x(x−w)∆_n(w)^j+1f_X(w)dw

= ...

...

= Z 1

h_yK

y−gβ0(w)−c0−e h_y

f_ε^(j)(e)deK_h_x(x−w)∆_n(w)^j+1f_X(w)dw

= Z

K(e)f_ε^(j)(y−g_β₀(w)−c0−hye)deK_h_x(x−w)∆n(w)^j+1fX(w)dw

=f_ε^(j)(y−g_β₀(w)−c₀)∆_n(x)^j+1f_X(x) +o(1)

uniformly iny∈R, x∈supp(v) for allj = 1, ..., r−1. (2.41) and (2.42) imply c^2(j+1)n log(n)

h^dx^Xh^2j+1y

→0 for allj = 1, ..., r−1, so that

1 n

i=1

K^(j)

y−g_β₀(X_i)−c₀−ε_i hy

Khx(x−Xi)(−1)^j+1c^j+1n ∆n(Xi)^j+1 h^j+1y (j+ 1)!

= (−1)^j+1c^j+1n

nh^j+1y (j+ 1)!

i=1

K^(j)

y−g_β₀(X_i)−c₀−ε_i hy

Khx(x−Xi)∆n(Xi)^j+1

−E

K^(j)

y−g_β₀(X1)−c0−ε1

h_y

K_h_x(x−X₁)∆_n(X₁)^j+1

+o_p 1

√n

= c^j+1n

h^jy

O_p

s log(n) nh^dx^Xhy

! +op

√n

=op

√n

for all j = 1, ..., r−1 and uniformly with respect to x ∈ supp(v) and with respect to y in some compact set, where the second to last equality follows from the results of Hansen (2008) (see section 1.1). Hence,

p(y, x)−pˆ₀(y, x) =−c_n n

i=1

K_h_y(y−g_β₀(X_i)−c₀−ε_i)K_h_x(x−X_i)∆_n(X_i) +o_p 1

√n

=O_p(cn)

=o_p n⁻¹⁴

. (2.47)

uniformly on compact sets with respect to y and uniformly in x ∈ supp(v). Since (2.43) and (2.44) imply

nlog(n)⁻^2j+1² h

dX(j+4)

x2j+1 h²_y → ∞

⇒ c^2jn log(n) n¹⁴h^d_x^Xh^2j+1_y

→0

2.8. Proofs for all j= 1, ..., r−1,a similar reasoning leads to

fˆ_Y_|X(y|x)−f_Y₀_|X(y|x) =op n⁻¹⁴

uniformly on compact sets. These asymptotic expressions will be used to obtain a similar expression for ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹_|X(τ|x). Again, Theorem 2 of Hansen (2008) or more precisely the adjustments discussed later in the proof of Lemma 4.2.12 combined with (A4) ensure that

p₀(y, x)−p₀(y, x) =o_p n⁻¹⁴

, fˆ_X(x)−f_X(x) =o_p n⁻¹⁴

and ∂

∂yfˆ_Y_|X(y|x) =O_p(1) uniformly on compact sets, so that Lemma 1.1.2 leads to

Fˆ_Y₀_|X(y|x)−F_Y₀_|X(y|x)

= pˆ₀(y, x)

fˆ_X(x) −p₀(y, x) fX(x)

= 1

f_X(x)(ˆp₀(y, x)−p₀(y, x))−p₀(y, x)

f_X(x)²( ˆf_X(x)−f_X(x))

−fˆX(x)−fX(x) fˆX(x)fX(x)

p0(y, x)−p0(y, x)−p0(y, x)( ˆfX(x)−fX(x)) fX(x)

= 1

f_X(x)(ˆp0(y, x)−p0(y, x))−p0(y, x)

f_X(x)²( ˆfX(x)−fX(x)) +op

√n

=o_p n⁻¹⁴ . and

Fˆ_Y_|X(y|x)−F_Y₀_|X(y|x) = 1

fX(x)(ˆp(y, x)−p₀(y, x))−p₀(y, x)

fX(x)²( ˆf_X(x)−f_X(x)) +op

√n

=o_p n⁻¹⁴ .

Since for an appropriate y^∗ between ˆF_Y⁻¹_|X(τ|x) andF_Y⁻¹

0|X(τ|x) 0 = ˆF_Y|X( ˆF_Y⁻¹_|X(τ|x)|x)−F_Y₀|X(F_Y⁻¹

0|X(τ|x)|x)

= ˆF_Y_|X(F_Y⁻¹

0|X(τ|x)|x) + ˆf_Y_|X(F_Y⁻¹

0|X(τ|x)|x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x) + ∂

∂yfˆ_Y_|X(y^∗|x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x)2

−F_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x)

= ˆF_Y_|X(F_Y⁻¹

0|X(τ|x)|x)−F_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x) +f_Y_|X(F_Y⁻¹

0|X(τ|x)|x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x) + fˆ_Y_|X(F_Y⁻¹

0|X(τ|x)|x)−f_Y_|X(F_Y⁻¹

0|X(τ|x)|x) Fˆ_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x)

+ ∂

∂yfˆ_Y_|X(y^∗|x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x)2

due to the continuity of ˆF_Y_|X and F_Y₀_|X, it holds that ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x) =op n⁻¹⁴ uniformly inx∈supp(v) and τ ∈supp(µ). Moreover, note that

f_Y_|X(y|x)−f_Y₀_|X(y|x) =O(c_n) uniformly on compact sets and thatc_nn⁻¹⁴ =o n⁻¹²

. Hence, 0 = ˆF_Y_|X( ˆF_Y⁻¹_|X(τ|x)|x)−F_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x)

= ˆF_Y_|X(F_Y⁻¹

0|X(τ|x)|x)−F_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x) +f_Y|X(F_Y⁻¹

0|X(τ|x)|x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x)

+O_p

Fˆ_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x)2 +O_p

fˆ_Y_|X(F_Y⁻¹

0|X(τ|x))−f_Y₀_|X(F_Y⁻¹

0|X(τ|x)) Fˆ_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x) +o_p

√n

= ˆF_Y|X(F_Y⁻¹

0|X(τ|x)|x)−F_Y₀|X(F_Y⁻¹

0|X(τ|x)|x) +f_Y₀|X(F_Y⁻¹

0|X(τ|x)|x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x) +op

√n

uniformly inx∈supp(v) andτ ∈supp(µ). Due to (A6), this in turn implies F_Y⁻¹

0|X(τ|x)−Fˆ_Y⁻¹_|X(τ|x)

Fˆ_Y_|X(F_Y⁻¹

0|X(τ|x)|x)−F_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x) f_Y|X(F_Y⁻¹

0|X(τ|x)|x) +op

√n

= 1

f_Y₀|X(F_Y⁻¹

0|X(τ|x)|x) 1

f_X(x)p(Fˆ _Y⁻¹

0|X(τ|x), x)−p0(F_Y⁻¹

0|X(τ|x), x) f_X(x)² fˆ_X(x)

+op

√n

. The same expression can be obtained forF_Y⁻¹

0|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x) when replacing ˆp by ˆp0, so that (see (2.47))

Fˆ_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x)

= 1

f_Y₀|X(F_Y⁻¹

0|X(τ|x)|x)f_X(x) pˆ0(F_Y⁻¹

0|X(τ|x), x)−p(Fˆ _Y⁻¹

0|X(τ|x), x) +op

√n

= c_n

f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x)f_X(x)n

i=1

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X_i)−c₀−ε_i) K_h_x(x−X_i)∆_n(X_i) +o_p

√n

2.8.2 Proof of Lemma 2.3.2

Thanks to Lemma 2.8.1 the difference ˆF_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x) can be written as Fˆ_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x)

2.8. Proofs

= 1

f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x) 1 n

i=1

fX(x)K_h_y(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)

−p₀(F_Y⁻¹

0|X(τ|x), x) f_X(x)²

Khx(x−Xi) +op

√n uniformly in x∈supp(v), so that

v(x) ˆF_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x)2

=nh

Z Z v(x)

1 f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x) 1 n

i=1

fX(x)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X_i)−c₀−ε_i)

−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)²

Khx(x−Xi) +op

√n 2

dx µ(dτ)

=nh

Z Z v(x)

1 f_Y₀|X(F_Y⁻¹

0|X(τ|x)|x) 1 n

i=1

f_X(x)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(Xi)−c0−εi)

−p0(F_Y⁻¹

0|X(τ|x), x) f_X(x)²

K_h_x(x−Xi) 2

dx µ(dτ) + Z Z

√ nh

v(x) 1

f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x) 1 n

i=1

f_X(x)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X_i)−c₀−ε_i)

−p0(F_Y⁻¹

0|X(τ|x), x) fX(x)²

K_h_x(x−X_i)

dx µ(dτ) +o_p(1).

Recall

κ(x, τ) = v(x) f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x)²f_X(x)². Because of H¨older’s inequality it suffices to show the assertion for

Z Z

v(x) f_Y₀|X(F_Y⁻¹

0|X(τ|x)|x)²fX(x)² 1 n

i=1

K_h_y(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)

− p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

Khx(x−Xi)

dx µ(dτ)

= h

i=1

Z Z

κ(x, τ) K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(Xi)−c0−εi)−p0(F_Y⁻¹

0|X(τ|x), x) f_X(x)

K_h_x(x−X_i)²dx µ(dτ) + h

i=1 n

j=1 j6=i

Z Z

κ(x, τ)

K_h_y(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

Khx(x−Xi)

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X_j)−c₀−ε_j)−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

K_h_x(x−X_j)dx µ(dτ)

=T₁+T₂.

Asymptotic Behaviour of T₁

First, note that using integration by parts and Lemma 1.1.1, one has Z

K_h_y(F_Y⁻¹

0|X(τ|x)−z)²f_Y₀_,X(z, x)dz

= Z

2K_h_y(F_Y⁻¹

0|X(τ|x)−z)K_h_y(F_Y⁻¹

0|X(τ|x)−z)p₀(z, x)dz

= Z

2K(z)K(z)p₀(F_Y⁻¹

0|X(τ|x)−hyz, x)dz

= Z

2K(z)K(z)dz p0(F_Y⁻¹

0|X(τ|x), x)−hy

2zK(z)K(z)dz fY0,X(F_Y⁻¹

0|X(τ|x), x) +O h²_y as well as

2K(z)K(z)dz = lim

u→∞

Z _u

−∞

2K(z)K(z)dz= lim

u→∞K(u)² = 1 and

K_h_y(F_Y⁻¹

0|X(τ|x)−z)f_Y₀_,X(z, x)dz=p₀(F_Y⁻¹

0|X(τ|x), x) +o 1

√n uniformly inx∈supp(v) and τ ∈supp(µ), so that

K_h_y(F_Y⁻¹

0|X(τ|x)−z)−p0(F_Y⁻¹

0|X(τ|x)) f_X(x)

f_Y₀_,X(z, x)dz

=p₀(F_Y⁻¹

0|X(τ|x), x)−h_y Z

2zK(z)K(z)dz f_Y₀_,X(F_Y⁻¹

0|X(τ|x), x)

−p₀(F_Y⁻¹

0|X(τ|x), x)² f_X(x) +o

√n

+O h²_y

=p0(F_Y⁻¹

0|X(τ|x), x)

1−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

−hy

2zK(z)K(z)dz fY0,X(F_Y⁻¹

0|X(τ|x), x) +o

√n

+O h²_y

uniformly inx∈supp(v) and τ ∈supp(µ). Similar calculations yield Z

K_h_y(F_Y⁻¹

0|X(τ|x)−z)−p₀(F_Y⁻¹

0|X(τ|x)) fX(x)

∂

∂xfY0,X(z, x)dz

= ∂

∂up0(F_Y⁻¹

0|X(τ|x), u)

u=x−2

p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

∂

∂up0(F_Y⁻¹

0|X(τ|x), u) u=x

+p0(F_Y⁻¹

0|X(τ|x), x)² f_X(x)²

∂

∂xf_X(x)−h_y Z

2zK(z)K(z)dz ∂

∂xf_Y₀_,X(y, x) y=F_Y⁻¹

0|X(τ|x)

+o 1

√n

+O h²_y

2.8. Proofs

= ∂

∂u

p₀(F_Y⁻¹

0|X(τ|x), u)

1−p₀(F_Y⁻¹

0|X(τ|x), u) fX(u)

u=x

−hy

2zK(z)K(z)dz ∂

∂xfY0,X(y, x) y=F_Y⁻¹

0|X(τ|x)+o 1

√n

+O h²_y and

K_h_y(F_Y⁻¹

0|X(τ|x)−z)−p₀(F_Y⁻¹

0|X(τ|x)) f_X(x)

∂²

∂x²fY0,X(z, x)dz

= ∂²

∂u²p0(F_Y⁻¹

0|X(τ|x), u)

u=x−2

p₀(F_Y⁻¹

0|X(τ|x), x) f_X(x)

∂²

∂u²p0(F_Y⁻¹

0|X(τ|x), u) u=x

+p₀(F_Y⁻¹

0|X(τ|x), x)² fX(x)²

∂²

∂x²f_X(x)−h_y Z

2zK(z)K(z)dz ∂²

∂u²f_Y₀_,X(F_Y⁻¹

0|X(τ|x), u) u=x

+o 1

√n

+O h²_y

uniformly inx∈supp(v) andτ ∈supp(µ). Here, _∂x^∂ fX(x) and _∂x^∂²2fX(x) are the derivative and the Hessian off_X, that is a vector and a matrix. The expectation ofT₁ can be written as

E[T₁]

Z Z

κ(x, τ)E

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(Xi)−c0−εi)−p0(F_Y⁻¹

0|X(τ|x), x) f_X(x)

Khx(x−Xi)²

dx µ(dτ)

Z Z

κ(x, τ) Z

K_h_y(F_Y⁻¹

0|X(τ|x)−z)−p0(F_Y⁻¹

0|X(τ|x), x) f_X(x)

K_h_x(x−w)²f_Y₀_,X(z, w)dw dz dx µ(dτ)

=h⁻

x 2

Z Z

κ(x, τ) Z

K_h_y(F_Y⁻¹

0|X(τ|x)−z)− p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

K(w)²fY0,X(z, x−hxw)dw dz dx µ(dτ)

=h⁻

x 2

Z Z

κ(x, τ) Z

K_h_y(F_Y⁻¹

0|X(τ|x)−z)− p0(F_Y⁻¹

0|X(τ|x), x) f_X(x)

K(w)²dw fY0,X(z, x) +hx

∂

∂ufY0,X(z, u) u=x

K(w)²w dw

+h²_x Z

K(w)²w^t ∂²

∂x²f_Y₀_,X(z, x)w dw+O h³_x

dz dx µ(dτ)

=h⁻

x 2

K(w)²dw Z Z

κ(x, τ)p₀(F_Y⁻¹

0|X(τ|x), x)

1−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

dx µ(dτ)

−hyh⁻

x 2

K(w)²dw Z

2zK(z)K(z)dz Z Z

κ(x, τ)fY0,X(F_Y⁻¹

0|X(τ|x), x)dx µ(dτ) +h¹⁻

x 2

Z Z

κ(x, τ)

∂

∂u

p₀(F_Y⁻¹

0|X(τ|x), u)

1−p0(F_Y⁻¹

0|X(τ|x), u) f_X(u)

u=x

dx µ(dτ) Z

K(w)²w dw

+h²⁻

x 2

K(w)²w^t Z Z

κ(x, τ)

∂²

∂u²p0(F_Y⁻¹

0|X(τ|x), u) u=x

−2p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

∂²

∂u²p₀(F_Y⁻¹

0|X(τ|x), u) u=x

+p0(F_Y⁻¹

0|X(τ|x), x)² f_X(x)²

∂²

∂x²f_X(x)

dx µ(dτ)w dw+O

h_yh¹⁻

x 2

K(w)²w dw

+O hyh²⁻

x 2

+O h³⁻

x 2

h²_yh⁻

x 2

+op(1)

=b+O h³⁻

x 2 +h²_yh⁻

x 2 +h_yh²⁻

x 2

h_yh¹⁻

x 2

K(w)²w dw

=b+o_p(1)

by the bandwidth assumptions (2.19) and (2.20). LetC >0 be a sufficiently large constant.

Then, the variance ofT1 can be bounded by

Var h

i=1

Z Z

κ(x, τ)

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X_i)−c₀−ε_i)−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

K_h_x(x−Xi)²dx µ(dτ)

≤ h^d_x^X n E

Z Z

κ(x, τ)

K_h_y(F_Y⁻¹

0|X(τ|x)−gβ0(X1)−c0−ε1)− p₀(F_Y⁻¹

0|X(τ|x), x) f_X(x)

K_h_x(x−X₁)²dx µ(dτ) 2

≤ Ch^d_x^X n E

K_h_x(x−X₁)²dx 2

= C

nh^dx^X

K(x)²dx 2

=o(1),

so thatT1=b+op(1).

2.8. Proofs

Asymptotic Behaviour of T₂ Similar to Lemma 1.1.1, one has

K_h_x(x−X)

K_h_y((F_Y₀_|X)⁻¹(τ|x)−Y₀)−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

= Z Z

K_h_x(x−w)

K_h_y((F_Y₀_|X)⁻¹(τ|x)−z)− p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

f_Y₀_,X(z, w)dz dw

= Z

K(w) Z

K_h_y(F_Y⁻¹

0|X(τ|x)−z)f_Y₀_,X(z, x−h_xw)dz

−p0(F_Y⁻¹

0|X(τ|x), x)

f_X(x) f_X(x−h_xw)

= Z

K(w) Z Z

F−1 Y0|X(τ|x)−z

−∞

K(u)fY0,X(z, x−hxw)du dz

−p0(F_Y⁻¹

0|X(τ|x), x)

f_X(x) f_X(x−h_xw)

= Z

K(w) Z

K(u) Z (F_Y

0|X)⁻¹(τ|x)−hyu

−∞

fY0,X(z, x−hxw)dz du

−p0(F_Y⁻¹

0|X(τ|x), x)

f_X(x) f_X(x−h_xw)

= Z

K(w) Z

K(u)p0(F_Y⁻¹

0|X(τ|x)−hyu, x−hxw)dz du

−p0(F_Y⁻¹

0|X(τ|x), x)fX(x−hxw) fX(x)

=O h^q_x

=o 1

√n

(2.48) uniformly in x∈supp(v). Therefore, the expectation of T₂ can be written as

E[T2] = (n−1)h

Z Z

κ(x, τ)E

K_h_y(F_Y⁻¹

0|X(τ|x)−gβ0(X1)−c0−ε1)

−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

Khx(x−X1) 2

dx µ(dτ)

=o(1).

Define Z_i(x) =

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X_i)−c₀−ε_i)−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

K_h_x(x−X_i),

so that

T2 = h

i=1 n

j=1 j6=i

Z Z

κ(x, τ)Zi(x)Zj(x)dx µ(dτ).

Later, Theorem 2.1 of De Jong (1987) will be used to show asymptotic normality ofT₂. By the same reasoning as before, it can be proven that

Z₁(x)Z₂(x)Z₃(u)Z₄(u)

=o 1

n² and

Z1(x)Z2(x)Z2(u)Z3(u)

=o 1

nh^dx^X

uniformly inx, u∈supp(v), which results in

E[T₂²]

= 2(n−1)h^d_x^X

n E

Z Z

κ(x, τ)Z1(x)Z2(x)dx µ(dτ) 2

+o(1)

= 2(n−1)h^d_x^X

n E

Z Z

κ(x, τ)

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X1)−c0−ε1)

−p₀(F_Y⁻¹

0|X(τ|x), x) f_X(x)

K_h_x(x−X1)

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X2)−c0−ε2)

−p₀(F_Y⁻¹

0|X(τ|x), x) f_X(x)

Khx(x−X2)dx µ(dτ) 2

+o(1)

= 2h^−d_x ^XE

Z Z

κ(X₁−h_xx, τ)

K_h_y(F_Y⁻¹

0|X(τ|X₁−h_xx)−g_β₀(X₁)−c₀−ε₁)

−p₀(F_Y⁻¹

0|X(τ|X₁−h_xx), X₁−h_xx) fX(X1−hxx)

K(x)

K_h_y(F_Y⁻¹

0|X(τ|X₁−h_xx)−g_β₀(X₂)−c₀−ε₂)−p₀(F_Y⁻¹

0|X(τ|X₁−h_xx), X₁−h_xx) fX(X1−hxx)

x+X₁−X₂ h_x

dx µ(dτ) 2

+o(1)

= 2h^−d_x ^X

Z Z Z Z Z Z

κ(w1−hxx, τ)

K_h_y(F_Y⁻¹

0|X(τ|w₁−hxx)−z1)

−p₀(F_Y⁻¹

0|X(τ|w₁−h_xx), w₁−h_xx) f_X(w1−hxx)

K(x)

K_h_y(F_Y⁻¹

0|X(τ|w₁−hxx)−z2)

−p₀(F_Y⁻¹

0|X(τ|w₁−h_xx), w₁−h_xx) f_X(w1−hxx)

x+w1−w2

dx µ(dτ) 2

fY0,X(z1, w1) fY0,X(z2, w2)dw1dw2dz1dz2+o(1)

= 2

Z Z Z Z Z Z

κ(w₁−h_xx, τ)

K_h_y(F_Y⁻¹

0|X(τ|w₁−h_xx)−z₁)

2.8. Proofs

−p₀(F_Y⁻¹

0|X(τ|w₁−h_xx), w₁−h_xx) fX(w1−hxx)

K(x)

K_h_y(F_Y⁻¹

0|X(τ|w₁−h_xx)−z₂)

−p0(F_Y⁻¹

0|X(τ|w₁−hxx), w1−hxx) f_X(w₁−h_xx)

K(x+w₂)dx µ(dτ) 2

f_Y₀_,X(z₁, w₁) f_Y₀_,X(z₂, w₁−h_xw₂)dw₁dw₂dz₁dz₂+o(1).

Note that K_h_y(F_Y⁻¹

0|X(τ|w₁−h_xx)−z₁) = I_{z

1≤F⁻¹

Y0|X(τ|w₁)}+o(1) for Lebesgue all w₁, x∈ R^d^X, z∈Rand µ-allτ ∈(0,1), so that the dominated convergence theorem yields

E[T₂²] = 2

Z Z Z Z Z Z

κ(w₁, τ)

I_{z

1≤F⁻¹

Y0|X(τ|w₁)}−p0(F_Y⁻¹

0|X(τ|w₁), w1) f_X(w₁)

K(x)

I_{z

2≤F⁻¹

Y0|X(τ|w1)}−p0(F_Y⁻¹

0|X(τ|w₁), w1) f_X(w₁)

K(x+w₂)dx µ(dτ) 2

f_Y₀_,X(z₁, w₁)f_Y₀_,X(z₂, w₁)dw₁dw₂dz₁dz₂+o(1)

= 2

Z Z Z Z

κ(w1, τ)

I_{z

1≤F_Y⁻¹

0|X(τ|w1)}− p0(F_Y⁻¹

0|X(τ|w₁), w1) f_X(w₁)

I_{z

2≤F_Y⁻¹

0|X(τ|w1)}−p0(F_Y⁻¹

0|X(τ|w₁), w1) f_X(w₁)

µ(dτ)

fY0,X(z1, w1)fY0,X(z2, w1)dw1dz1dz2

Z Z

K(x)K(x+w2)dx 2

dw2+o(1).

(2.49) Later, it will be shown, that the asymptotically non negligible term is equal toV. Define

W_i,j = 2h

n Z Z

κ(x, τ) Z_i(x)−E[Z₁(x)]

Z_j(x)−E[Z₁(x)]

dx µ(dτ).

Then,

W(n) :=X

i<j

W_i,j =T₂

is what De Jong (1987) called clean, that is E[W_i,j|(Y_i, X_i)] = 0 for all i6= j ∈ {1, ..., n}.

In (2.48), it was proven that E[Z₁(x)] = o ^√¹_n

uniformly in supp(v). Moreover, one can showW(n) =T2+op(1) as well as E[W(n)²] =E[T₂²] +o(1) similarly to before. Therefore,

maxi<j E[W_i,j² ] E[W(n)²] =

4h^d_x^XE hR R

κ(x, τ)Z1(x)Z2(x)dx µ(dτ) 2i

nE[W(n)²] =O

1 n

=o(1), so that in order to prove normality of W(n) and thus normality of T2 it remains to show

E[W(n)⁴] E[W(n)²]² →3 (see Theorem 2.1 of De Jong (1987)). It holds that

E[W(n)⁴] =X

i<j

k<l

r<s

t<u

E[W_i,jW_k,lW_r,sW_t,u]

= n(n−1)

2 E[W_1,2⁴ ] +3n(n−1)(n−2)(n−3)

4 E[W_1,2² ]² + 3n(n−1)(n−2)E[W_1,2² W_2,3² ]

4 + 6n(n−1)(n−2)E[W_1,2W_2,3W_3,1² ] + 6n(n−1)(n−2)(n−3)E[W1,2W2,3W3,4W4,1], (2.50) where the prefactors are explained later.

In the following, consider

W˜_i,j = 2h

n Z Z

κ(x, τ)Z_i(x)Z_j(x)dx µ(dτ)

instead ofW_i,j as this makes calculations (a little bit) clearer and more convenient and the proof of the asymptotic negligibility of these replacements follows in a similar manner (for exampleE[ ˜W_1,2⁴ ] =E[W_1,2⁴ ] +o(1)).

First, one has for an appropriate constantC >0 n²E[ ˜W_1,2⁴ ] = 16h^2d_x ^X

n² E

Z Z

κ(x, τ)Z₁(x)Z₂(x)dx µ(dτ) 4

≤ Ch^2d_x ^X n² E

Z Z

κ(x, τ)|K_h_x(x−X₁)K_h_x(x−X₂)|dx µ(dτ) 4

= C

n²h^2dx ^X

Z Z

κ(X1−hxx, τ)

K(x)K

x+X1−X2

h_x

dx µ(dτ) 4

=o(1).

In equation (2.49) was shown that n⁴

16E[ ˜W_1,2² ]² =V²+o(1).

For a sufficiently large constant C >0,E[ ˜W_1,2² W˜_2,3² ] can be bounded by n³E[ ˜W_1,2² W˜_2,3² ]

≤ Ch^2d_x^X

n E

Z Z

κ(x₁, τ₁)

K_h_x(x₁−X₁)K_h_x(x₁−X₂)

dx₁µ(dτ₁) 2

Z Z

κ(x2, τ2)

K_h_x(x2−X3)K_h_x(x2−X2)

dx2µ(dτ2) 2

= C

nh^2dx ^X

Z Z

κ(X1−hxx1, τ1)

K(x1)K

x1+X1−X2

dx1µ(dτ1) 2

Z Z

κ(X₃−h_xx₂, τ₂)

K(x₂)K

x₂+X₃−X₂ h_x

dx₂µ(dτ₂) 2

= C

nh^2dx ^X

Z Z Z Z Z

κ(w1−hxx1, τ1)

K(x1)K

x1+w1−w2

h_x

dx1µ(dτ1) 2

Z Z

κ(w₃−h_xx₂, τ₂)

K(x₂)K

x₂+w₃−w₂ hx

dx₂µ(dτ₂) 2

2.8. Proofs

f_X(w₁)f_X(w₂)f_X(w₃)dw₁dw₂dw₃

= C n

Z Z Z Z Z

κ(w2+hxw1−hxx1, τ1)

K(x1)K(x1+w1)

dx1µ(dτ1) 2

Z Z

κ(w2+hxw3−hxx2, τ2)

K(x2)K(x2+w3)

dx2µ(dτ2) 2

fX(w2+hxw1)fX(w2)fX(w2+hxw3)dw1dw2dw3

=o(1).

E[ ˜W_1,2W˜_2,3W˜_3,1² ] can be treated similar since n³E[ ˜W_1,2W˜_2,3W˜_3,1² ]

≤ Ch^2d_x ^X n E

Z Z

κ(x₁, τ₁)

K_h_x(x₁−X₁)K_h_x(x₁−X₂)

dx₁µ(dτ₁) Z Z

κ(x2, τ2)

Khx(x2−X2)Khx(x2−X3)

dx2µ(dτ2) Z Z

κ(x₃, τ₃)

K_h_x(x₃−X₃)K_h_x(x₃−X₁)

dx₃µ(dτ₃) 2

= C

nh^2dx^X

E Z Z

κ(X1−hxx1, τ1)

K(x1)K

x1+ X1−X2

dx1µ(dτ1) Z Z

κ(X₂−h_xx₂, τ₂)

K(x₂)K

x₂+X₂−X₃ h_x

dx₂µ(dτ₂) Z Z

κ(X3−hxx3, τ3)K(x3)

x3+X3−X1

h_x

dx3µ(dτ3) 2

≤ C² nh^2dx^X

Z Z Z Z

K(x1)K

x1+w1−w2

dx1

K(x₂)K

x₂+w₂−w₃ h_x

dx₂

K(x₃)K

x₃+w₃−w₁ h_x

dx₃ 2

f_X(w₁)f_X(w₂)f_X(w₃)dw₁dw₂dw₃

≤ C³ n

Z Z Z Z

K(x1)K(x1+w1) dx1

K(x2)K(x2+w3)

dx2

fX(w2+hxw1)fX(w2)fX(w2−hxw3)dw1dw2dw3

=o(1)

for an appropriate constant C > 0. It remains to consider E[ ˜W_1,2W˜_2,3W˜_3,4W˜_4,1]. This expectation can be treated by

n⁴E[ ˜W1,2W˜2,3W˜3,4W˜4,1]

≤Ch^2d_x ^XE Z Z

κ(x₁, τ₁)

K_h_x(x₁−X₁)K_h_x(x₁−X₂)

dx₁µ(dτ₁)

Z Z

κ(x2, τ2)

Khx(x2−X2)Khx(x2−X3)

dx2µ(dτ2) Z Z

κ(x3, τ3)

Khx(x3−X3)Khx(x3−X4)

dx3µ(dτ3) Z Z

κ(x4, τ4)

K_h_x(x4−X4)K_h_x(x4−X1)

dx4µ(dτ4)

= C

h^2dx ^X

E Z Z

κ(X1−hxx1, τ1)

K(x1)K

x1+X1−X2

dx1µ(dτ1) Z Z

κ(X₂−h_xx₂, τ₂)

K(x₂)K

x₂+ X₂−X₃ h_x

dx₂µ(dτ₂) Z Z

κ(X3−hxx3, τ3)K(x3)

x3+ X3−X4

h_x

dx3µ(dτ3) Z Z

κ(X4−hxx4, τ4)K(x4)

x4+ X4−X1

dx4µ(dτ4)

≤ C² h^2d_x ^X

Z Z Z Z Z

K(x₁)K

x₁+w₁−w₂ hx

dx₁ Z

K(x2)K

x2+w2−w3

h_x

dx2

K(x3)K

x3+w3−w4

h_x

dx3

K(x3)K

x4+w₄−w₁ hx

dx4

fX(w1)fX(w2)fX(w3)fX(w4)dw1dw2dw3dw4

≤C³

Z Z Z Z Z

K(x₁)K(x₁+w₁) dx₁

K(x₂)K

x₂+w₂−w₃ hx

dx₂ Z

K(x₃)K(x₃+w₄) dx₃

f_X(w₂+h_xw₁)f_X(w₂)f_X(w₃)f_X(w₃−h_xw₄)dw₁dw₂dw₃dw₄

=C³h^d_x^X

Z Z Z Z Z

K(x1)K(x1+w1) dx1

K(x2)K(x2+w3) dx2

K(x₃)K(x₃+w₄) dx₃

fX(w2+hxw1)fX(w2)fX(w2−hxw3)fX(w2−hxw3−hxw4)dw1dw2dw3dw4

=o(1).

Finally, this leads to E[W(n)⁴] = 3n⁴

4 E[W_1,2² ]²+o(1) = 3V²+o(1) = 3E[T₂²]²+o(1) = 3E[W(n)²]²+o(1) and thusT₂ → N^D (0, V).

Note that the prefactor of 3n(n−1)(n−2)(n−3)

4 = ⁿ₄

·3·6 in (2.50) results from the fact that

• ⁿ₄

is the number of possibilities to choose a set of four indices out of {1, ..., n}

(without ordering them),

2.8. Proofs

• 3 is the number of possibilities to assign these indices to the corresponding four tuples (i, j),(k, l),(r, s),(t, u) to obtain E[W_1,2² ]² and

• 6 is the number of possible permutations of these tuples.

The other prefactors in (2.50) can be derived similarly, but do not matter for the asymptotic behaviour of _E[T^E[T2²⁴^]

2]².

Rewriting b and V

The expressions forV and b given in (2.22) and (2.25), respectively, follow from (compare (2.7))

p₀(F_Y⁻¹

0|X(τ|x), x) =F_ε(F_Y⁻¹

0|X(τ|x)−g(x))f_X(x) =τ f_X(x) and

f_Y,X(F_Y⁻¹

0|X(τ|x), x) =f_ε(F_Y⁻¹

0|X(τ|x)−g(x))f_X(x) =f_ε(F_ε⁻¹(τ))f_X(x).

To specify this, use the definition ofκ(x, τ) in (2.18) and write under the assumptions (2.23) and (2.24)

b=h⁻

x 2

K(w)²dw Z Z

κ(x, τ)p0(F_Y⁻¹

0|X(τ|x), x)

1− p0(F_Y⁻¹

0|X(τ|x), x) f_X(x)

dx µ(dτ) +o(1)

=h⁻

x 2

K(w)²dw

Z Z v(x)

f_ε(Fε⁻¹(τ))²f_X(x)τ(1−τ)dx µ(dτ) +o(1)

=h⁻

x 2

K(w)²dw

Z v(x) f_X(x)dx

Z τ(1−τ)

fε(Fε⁻¹(τ))² µ(dτ) +o(1) and (see (2.49))

V = 2 Z Z

K(x)K(x+s)dx 2

Z Z Z Z

κ(w, τ)

I_{z

1≤F_Y⁻¹

0|X(τ|w)}− p0(F_Y⁻¹

0|X(τ|w), w) f_X(w)

I_{z

2≤F_Y⁻¹

0|X(τ|w)}−p0(F_Y⁻¹

0|X(τ|w), w) f_X(w)

µ(dτ) 2

fY0,X(z1, w)fY0,X(z2, w)dw dz1dz2

= 2 Z Z

K(x)K(x+s)dx 2

Z Z Z Z

κ(w, τ) I_{z

1−g(w)≤F_ε⁻¹(τ)}−τ I_{z

2−g(w)≤F_ε⁻¹(τ)}−τ µ(dτ)

f_ε(z₁−g(w))f_ε(z₂−g(w))dz₁dz₂f_X(w)²dw

= 2 Z Z

K(x)K(x+s)dx 2

Z v(w)² f_X(w)²

Z Z Z I_{F_ε_(z₁_{−g(w))≤τ}}−τ fε(Fε⁻¹(τ)) I_{F_ε_(z₂_{−g(w))≤τ}}−τ

fε(Fε⁻¹(τ)) µ(dτ) 2

fε(z1−g(w))fε(z2−g(w))dz1dz2dw

= 2 Z Z

K(x)K(x+s)dx 2

Z v(w)² f_X(w)² dw Z 1

Z 1 0

Z I_{u₁_≤τ}−τ

I_{u₂_≤τ}−τ fε(Fε⁻¹(τ))² µ(dτ)

du1du2.

2.8.3 Proof of Theorem 2.3.4

Later, it will be shown, that the test statistic Tn defined in (2.13) is asymptotically equi-valent to ˜Tn+δ2,n+δ3,n, where

T˜n=nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x)2

dx µ(dτ),

δ2,n=− Z

v(x)D_βg_β₀(x)∆n(x)dx

Ω⁻¹ Z

v(x)D_βg_β₀(x)∆n(x)dx t

and

δ_3,n=−µ([0,1])

R v(x)∆_n(x)dx2

R v(w)dw . T˜n in turn can be split into

T˜_n=nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x) + ˆF_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x)2

dx µ(dτ)

=nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x)2

dx µ(dτ) +nh

Z Z

v(x) ˆF_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x)2

dx µ(dτ) + 2nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x) Fˆ_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x)

dx µ(dτ)

=T1+T2+T3.

While Lemma 2.3.2 can be applied forT₂ to obtain nh

Z Z

v(x) ˆF_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x)2

dx µ(dτ)−b→^D Z

with Z ∼ N(0, V) and b as well as V from Lemma 2.3.2, T1 can be treated as follows.

Remember

κ(x, τ) = v(x) f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x)²f_X(x)² as well as (2.46) and write

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x)2

dx µ(dτ)

=nh

Z Z

κ(x, τ) cn

i=1

Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)Khx(x−Xi)∆n(Xi)

2.8. Proofs

+o_p 1

√n 2

dx µ(dτ)

≤ c²_nh

i=1 n

j=1

Z Z

κ(x, τ)Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)Khx(x−Xi) Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xj)−c0−εj)Khx(x−Xj)∆n(Xi)∆n(Xj)dx µ(dτ) +op

√ncnh

i=1

Z Z

κ(x, τ)

Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi) K_h_x(x−X_i)∆_n(X_i)

dx µ(dτ) +o_p(1)

= 1 n²

i=1 n

j=1

Z Z

κ(x, τ)Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)Khx(x−Xi)Khx(x−Xj) Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xj)−c0−εj)∆n(Xi)∆n(Xj)dx µ(dτ) +o_p(1)

by the definition ofcn in (A2). Then, (2.42) leads to nh^d_x^Xhy → ∞and thus 1

nE Z Z

κ(x, τ)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X1)−c0−ε1)²K_h_x(x−X1)²∆n(X1)²dx µ(dτ)

= 1 n

Z Z Z Z

κ(x, τ)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(w)−c₀−e)²K_h_x(x−w)²∆_n(w)² fX(w)fε(e)dw de dx µ(dτ)

= 1

nh^dx^X

Z Z Z Z

κ(x, τ)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(x−hxw)−c0−e)²K(w)²∆n(x−hxw)² fX(x−hxw)fε(e)dw de dx µ(dτ)

= 1

nh^dx^Xhy

Z Z Z Z

κ(x, τ)K(e)²K(w)²∆n(x−hxw)² f_X(x−h_xw)f_ε(F_Y⁻¹

0|X(τ|x)−g_β₀(x−h_xw)−c₀−h_ye)dw de dx µ(dτ)

=o(1), that is

T₁ = 1 n²

i=1 n

j=1 j6=i

Z Z

κ(x, τ)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X_i)−c₀−ε_i)K_h_x(x−X_i) K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(Xj)−c0−εj)K_h_x(x−Xj)∆n(Xi)∆n(Xj)dx µ(dτ) +op(1).

Due to E

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X1)−c0−ε1)K_h_x(x−X1)∆n(X1)

= Z Z

Khy(F_Y⁻¹

0|X(τ|x)−gβ0(w)−c0−e)Khx(x−w)∆n(w)fε(e)fX(w)de dw

= Z Z

K(e)K(w)∆n(x−hxw)fε(F_Y⁻¹

0|X(τ|x)−gβ0(x−hxw)−c0−hye) f_X(x−h_xw)de dw

= ∆n(x)fε(F_Y⁻¹

0|X(τ|x)−gβ0(x)−c0)fX(x) +o(1)

uniformly inx∈supp(v) and τ ∈supp(µ), the expectation ofT1 can be written as E[T1] =

Z Z

κ(x, τ)E

Khy(F_Y⁻¹

0|X(τ|x)−gβ0(X1)−c0−ε1) K_h_x(x−X₁)∆_n(X₁)2

dx µ(dτ) +o(1)

=δ1,n+o(1) with

δ_1,n = Z Z

κ(x, τ)∆_n(x)²f_ε F_Y⁻¹

0|X(τ|x)−g_β₀(x)−c₀2

f_X²(x)dx µ(dτ)

= Z Z

v(x)∆_n(x)²dx µ(dτ)

=µ([0,1]) Z

v(x)∆_n(x)²dx.

Here, the definition ofκ(x, τ) and the fact were used that (compare (2.7)) f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x) =f_ε(F_Y⁻¹

0|X(τ|x)−g_β₀(x)−c₀) =f_ε(F_ε⁻¹(τ)).

In the following, it is shown that the variance of the asymptotically nonnegligible terms converges to zero. For reasons of clarity and comprehensibility, define

Zi,j = Z Z

κ(x, τ)K_h_x(x−Xi)K_h_x(x−Xj)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(Xi)−c0−εi) Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xj)−c0−εj)∆n(Xi)∆n(Xj)dx µ(dτ), so that T1 = _n²2

Pn i=1

j=i+1Zi,j +op(1). To show that the variance of T1 converges to zero, write

Var 2

n²

i=1 n

j=i+1

Z_i,j

= 4 n⁴

i=1 n

j=i+1 n

k=1 n

l=k+1

Cov(Z_i,j, Z_k,l)

= 4 n⁴

i=1 n

j=i+1

Var(Zi,j) + 4 n⁴

i=1 n

j=i+1 n

l=i+1 l6=j

Cov(Zi,j, Z_i,l)

+ 4 n⁴

k=1 n

i=k+1 n

j=i+1

Cov(Zi,j, Z_k,i) + 4 n⁴

i=1 n

j=i+1 n

l=j+1

Cov(Zi,j, Z_j,l)

2.8. Proofs

+ 4 n⁴

j=1 n

i=j−1 n

k=j−1 k6=i

Cov(Zi,j, Zk,j),

so that it suffices to prove that

E[Z_1,2² ], E[|Z_1,2Z_1,3|], E[|Z_1,2Z_2,3|] and E[|Z_1,3Z_2,3|] (2.51) converge to zero. For an appropriate constantC >0 it holds that

E[Z_1,2² ] n²

≤ C n²E

Z Z

κ(x, τ)

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X₁)−c₀−ε₁) K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X₂)−c₀−ε₂)K_h_x(x−X₁)K_h_x(x−X₂)

dx µ(dτ) 2

= C

n²h^2dx ^X

Z Z

κ(X1−hxx, τ)

K_h_y(F_Y⁻¹

0|X(τ|X₁−hxx)−g_β₀(X1)−c0−ε1) K_h_y(F_Y⁻¹

0|X(τ|X₁−h_xx)−g_β₀(X₂)−c₀−ε₂)K(x)K

x+X₁−X₂ hx

dx µ(dτ) 2

= C

n²h^2dx ^X

Z Z Z Z Z Z

κ(w₁−h_xx, τ)

K_h_y(F_Y⁻¹

0|X(τ|w₁−h_xx)−g_β₀(w₁)−c₀−e₁) Khy(F_Y⁻¹

0|X(τ|w₁−hxx)−gβ0(w2)−c0−e2)K(x)K

x+w₁−w₂ hx

dx µ(dτ) 2

fX(w1)fX(w2)fε(e1)fε(e2)dw1dw2de1de2

= C

n²h^d_x^X

Z Z Z Z Z Z

κ(w₁−h_xx, τ)

K_h_y(F_Y⁻¹

0|X(τ|w₁−h_xx)−g_β₀(w₁)−c₀−e₁) Khy(F_Y⁻¹

0|X(τ|w₁−hxx)−gβ0(w1−hxw2)−c0−e2)K(x)K(x+w2)

dx µ(dτ) 2

fX(w1)fX(w1−hxw2)fε(e1)fε(e2)dw1dw2de1de2

= C

n²h^dx^X

Z Z Z Z Z Z Z Z

κ(w₁−h_xx₁, τ₁)

K_h_y(F_Y⁻¹

0|X(τ₁|w₁−h_xx)−g_β₀(w₁)−c₀−e₁) K_h_y(F_Y⁻¹

0|X(τ₁|w₁−h_xx₁)−g_β₀(w₁−h_xw₂)−c₀−e₂)K(x₁)K(x₁+w₂)

κ(w1−hxx2, τ2)

K_h_y(F_Y⁻¹

0|X(τ2|w₁−hxx2)−g_β₀(w1)−c0−e1) Khy(F_Y⁻¹

0|X(τ2|w₁−hxx2)−gβ0(w1−hxw2)−c0−e2)K(x2)K(x2+w2)

fX(w1)fX(w1−hxw2)fε(e1)fε(e2)dw1dw2de1de2dx1dx2µ(dτ1)µ(dτ2)

= C² n²h^dx^Xh²_y

Z Z Z Z Z Z

κ(w₁−h_xx₁, τ₁)

K(e₁)K(e₂)K(x₁)K(x₁+w₂)

f_X(w₁)f_X(w₁−h_xw₂)f_ε(F_Y⁻¹

0|X(τ₁|w₁−h_xx)−g_β₀(w₁)−h_ye₁) fε(F_Y⁻¹

0|X(τ1|w₁−hxx1)−gβ0(w1−hxw2)−hye2)dw1dw2de1de2dx1µ(dτ1)

≤ C² n²h^dx^Xh²_y

Z Z Z Z Z Z

K(e1)K(e2)K(x1)K(x1+w2)

fX(w1)dx1dw1dw2de1de2

=o(1).

Again for an appropriate constant C >0, the second expectation in (2.51) can be written as

E[|Z_1,2Z_1,3|]

≤ C nh^2dx ^X

Z Z

κ(x, τ)

K_h_y(F_Y⁻¹

0|X(τ|X₁−hxx)−g_β₀(X1)−c0−ε1) K_h_y(F_Y⁻¹

0|X(τ|X₁−h_xx)−g_β₀(X₂)−c₀−ε₂)K(x)K

x+ X₁−X₂ hx

dx µ(dτ) Z Z

κ(x, τ)

K_h_y(F_Y⁻¹

0|X(τ|X₁−h_xx)−g_β₀(X₁)−c₀−ε₁) K_h_y(F_Y⁻¹

0|X(τ|X₁−hxx)−g_β₀(X1)−c0−ε3)K(x)K

x+ X1−X3

h_x

dx µ(dτ)

= C

nh^2dx ^X

Z Z Z Z Z Z Z Z Z Z

κ(x₁, τ₁)κ(x₂, τ₂)

Khy(F_Y⁻¹

0|X(τ1|w₁−hxx1)−gβ0(w1)−c0−e1) K_h_y(F_Y⁻¹

0|X(τ₁|w₁−h_xx₁)−g_β₀(w₂)−c₀−e₂)K(x₁)K

x₁+w₁−w₂ hx

K_h_y(F_Y⁻¹

0|X(τ₂|w₁−h_xx₂)−g_β₀(w₁)−c₀−e₁) Khy(F_Y⁻¹

0|X(τ2|w₁−hxx2)−gβ0(w3)−c0−e3)K(x2)K

x2+w1−w3

fX(w1) fX(w2)fX(w3)fε(e1)fε(e2)fε(e3)dw1dw2dw3de1de2de3dx1dx2µ(dτ1)µ(dτ2)

≤ C² nh_yh^2d_x ^X

Z Z Z Z Z Z Z Z

K(e₁)K(e₂)K(e₃)K(x₁)K

x₁+w₁−w₂ h_x

K(x₂)K

x₂+w₁−w₃ hx

f_X(w₁)f_X(w₂)f_X(w₃)dw₁dw₂dw₃de₁de₂de₃dx₁dx₂

≤ C³ nh_y

Z Z Z Z

|K(x₁)K(x₁+w₂)|dx₁ Z

|K(x₂)K(x₂+w₃)|dx₂

fX(w1)fX(w1−hxw2)fX(w1−hxw3)dw1dw2dw3

2.8. Proofs

=o(1).

Similarly, E[|Z_1,2Z_2,3|]

≤ C nh^2dx ^X

Z Z

κ(x, τ)

K_h_y(F_Y⁻¹

0|X(τ|X₁−hxx)−g_β₀(X1)−c0−ε1) K_h_y(F_Y⁻¹

0|X(τ|X₁−h_xx)−g_β₀(X₂)−c₀−ε₂)K(x)K

x+X₁−X₂ hx

dx µ(dτ) Z Z

κ(x, τ)

K_h_y(F_Y⁻¹

0|X(τ|X₂−h_xx)−g_β₀(X₂)−c₀−ε₂) K_h_y(F_Y⁻¹

0|X(τ|X₂−hxx)−g_β₀(X3)−c0−ε3)K(x)K

x+X2−X3

h_x

dx µ(dτ)

= C

nh^2dx ^X

Z Z Z Z Z Z Z Z

κ(x, τ)

Khy(F_Y⁻¹

0|X(τ|w₁−hxx)−gβ0(w1)−c0−e1) K_h_y(F_Y⁻¹

0|X(τ|w₁−h_xx)−g_β₀(w₂)−c₀−e₂)K(x)K

x+w₁−w₂ h_x

dx µ(dτ) Z Z

κ(x, τ)

K_h_y(F_Y⁻¹

0|X(τ|w₂−hxx)−g_β₀(w2)−c0−e2) Khy(F_Y⁻¹

0|X(τ|w₂−hxx)−gβ0(w3)−c0−e3)K(x)K

x+w2−w3

dx µ(dτ)

fX(w1)fX(w2)fX(w3)fε(e1)fε(e2)fε(e3)dw1dw2dw3de1de2de3

≤ C² nh_y

Z Z Z Z

|K(x)K(x+w₁)|dx Z

|K(x)K(x+w₃)|dx

fX(w2+hxw1)fX(w2)fX(w2−hxw3)dw1dw2dw3

=o(1) and

E[|Z_1,3Z_2,3|]

n ≤ C³

nh_y

Z Z Z Z

|K(x)K(x+w₂)|dx Z

|K(x)K(x+w₃)|dx

fX(w3+hxw1)fX(w2+hxw3)fX(w3)dw1dw2dw3

=o(1).

In total,

T₁=δ_1,n+o_p(1)

has been proven, so that only T₃ is left to be examined. Inserting equations (2.45) and (2.46) yields

T3= 2nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x) Fˆ_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x)

dx µ(dτ)

= 2nh

Z Z

κ(x, τ) cn

i=1

Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)Khx(x−Xi)∆n(Xi) +op

√n 1

i=1

K_h_y(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)

−p0(F_Y⁻¹

0|X(τ|x), x) fX(x)

K_h_x(x−X_i) +o_p 1

√n

dx µ(dτ)

= 2nh

Z Z

κ(x, τ) c_n

i=1

Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi) Khx(x−Xi)∆n(Xi)

1 n

i=1

K_h_y(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)

−p0(F_Y⁻¹

0|X(τ|x), x) f_X(x)

K_h_x(x−X_i)

dx µ(dτ) +o_p(1)

= 2cnh

i=1 n

j=1

Z Z

κ(x, τ)Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi) Khx(x−Xi)∆n(Xi)

K_h_y(F_Y⁻¹

0|X(τ|x)−gβ0(Xj)−c0−εj)−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

Khx(x−Xj)dx µ(dτ) +op(1),

where the second to last equality follows similarly to the proof of Lemma 2.3.2 and the treatment ofT1. For a sufficiently large constant C >0 one has (see (2.44))

cnh

x2 E

Z Z

κ(x, τ)Khy(F_Y⁻¹

0|X(τ|x)−gβ0(X1)−c0−ε1)Khx(x−X1)²∆n(X1)

K_h_y(F_Y⁻¹

0|X(τ|x)−gβ0(X1)−c0−ε1)−p₀(F_Y⁻¹

0|X(τ|x), x) f_X(x)

dx µ(dτ)

≤Cc_nh

x2 E Z Z

κ(x, τ)|K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X₁)−c₀−ε_i)|K_h_x(x−X₁)²dx µ(dτ)

=Ccnh

Z Z Z Z

κ(x, τ)|K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(w)−c0−e)|K_h_x(x−w)² f_X(w)f_ε(e)dw de dx µ(dτ)

= C

n¹²h

3dX

Z Z Z Z

κ(x, τ)|K(e)|K(w)² fX(x−hxw)fε(F_Y⁻¹

0|X(τ|x)−g_β₀(w)−c0−hye)dw de dx µ(dτ)

=o(1),

2.8. Proofs so that

T3 = 2cnh

i=1 n

j=1 j6=i

Z˜i,j+op(1)

with Z˜i,j =

Z Z

κ(x, τ)Khx(x−Xi)Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)∆n(Xi)Khx(x−Xj)

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X_j)−c₀−ε_j)−p0(F_Y⁻¹

0|X(τ|x), x) f_X(x)

dx µ(dτ).

To prove asymptotic negligibility of T₃, it suffices to show ^c²ⁿ_n^h2^dX^x E Pn i=1

Pn j=1 j6=i

Z˜_i,j2

= o(1). This leads to the proof of

c²_nh^d_x^XE[ ˜Z_1,2² ] =o(1), (2.52) c²_nh^d_x^XE[ ˜Z_1,2Z˜_2,1] =o(1),

c²_nnh^d_x^XE[ ˜Z1,2Z˜1,3] =o(1), (2.53) c²_nnh^d_x^XE[ ˜Z1,2Z˜3,1] =o(1),

c²_nnh^d_x^XE[ ˜Z_1,2Z˜_2,3] =o(1), c²_nnh^d_x^XE[ ˜Z1,2Z˜3,2] =o(1),

c²_nn²h^d_x^XE[ ˜Z_1,2Z˜_3,4] =o(1). (2.54) For the sake of brevity, only equations (2.52),(2.53) and (2.54) are proven. The other assertions follow similarly. Let C > 0 be a sufficiently large constant. Equation (2.52) results from

c²_nh^d_x^XE[ ˜Z_1,2² ]

≤Cc²_nh^d_x^XE

Z Z

κ(x, τ)|K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X₁)−c₀−ε₁) K_h_x(x−X₁)K_h_x(x−X₂)|dx µ(dτ)

≤ Cc²_n h^dx^X

Z Z

κ(X1−hxx, τ)

K_h_y(F_Y⁻¹

0|X(τ|X₁−hxx)−g_β₀(X1)−c0−ε1) K(x)K

x+X₁−X₂ hx

dx µ(dτ) 2

= Cc²_n h^dx^X

Z Z Z Z Z

κ(w₁−h_xx, τ)

K_h_y(F_Y⁻¹

0|X(τ|w₁−h_xx)−g_β₀(w₁)−c₀−e) K(x)K

x+w₁−w₂ hx

dx µ(dτ) 2

fX(w1)fX(w2)fε(e)dw1dw2de

=Cc²_n

Z Z Z Z Z

κ(w₁−h_xx, τ)

K_h_y(F_Y⁻¹

0|X(τ|w₁−h_xx)−g_β₀(w₁)−c₀−e)

K(x)K(x+w₂)

dx µ(dτ) 2

f_X(w₁)f_X(w₁−h_xw₂)f_ε(e)dw₁dw₂de

=Cc²_n

Z Z Z Z Z Z Z

κ(w1−hxx1, τ1)κ(w2−hxx2, τ2)

K(x1+w2)K(x2)K(x2+w2) K(x1)K_h_y(F_Y⁻¹

0|X(τ1|w₁−hxx1)−g_β₀(w1)−c0−e) K_h_y(F_Y⁻¹

0|X(τ₂|w₁−h_xx₂)−g_β₀(w₁)−c₀−e)

fX(w1)fX(w1−hxw2)fε(e)dw1dw2de dx1dx2µ(dτ1)µ(dτ2)

= C²c²_n hy

Z Z Z Z Z

κ(w₁−h_xx₁, τ₁)

K(e)K(x₁)K(x₁+w₂) fX(w1)fX(w1−hxw2)fε(F_Y⁻¹

0|X(τ1|w₁−hxx1)−g_β₀(w1)−hye)dw1dw2de dx1µ(dτ1)

=O n⁻¹h⁻

x 2 h⁻¹_y

=o(1).

In (2.48), it was shown that E

K_h_x(x−X₁)

K_h_y((F_Y₀_|X)⁻¹(τ|x)−g_β₀(X₁)−c₀−ε₁)−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

=o 1

√n

uniformly inx∈supp(v) and τ ∈supp(µ), so that c²_nnh^d_x^XE[ ˜Z1,2Z˜1,3]

=c²_nnh^d_x^X

Z Z Z Z

κ(x1, τ1)κ(x2, τ2)E

∆n(X1)²K_h_x(x1−X1)K_h_x(x2−X1) Khy(F_Y⁻¹

0|X(τ1|x₁)−gβ0(X1)−c0−ε1)Khy(F_Y⁻¹

0|X(τ2|x₂)−gβ0(X1)−c0−ε1) E

K_h_x(x1−X2)K_h_y(F_Y⁻¹

0|X(τ1|x₁)−g_β₀(X2)−c0−ε2)−p0(F_Y⁻¹

0|X(τ1|x₁), x1) f_X(x₁)

K_h_x(x2−X3)K_h_y(F_Y⁻¹

0|X(τ2|x₂)−g_β₀(X3)−c0−ε3)−p0(F_Y⁻¹

0|X(τ2|x₂), x2) f_X(x₂)

dx1dx2µ(dτ1)µ(dτ2)

=o h

!Z Z Z Z

κ(x₁, τ₁)κ(x₂, τ₂)E

|K_h_x(x₁−X₁)K_h_x(x₂−X₁) K_h_y(F_Y⁻¹

0|X(τ₁|x₁)−g_β₀(X₁)−c₀−ε₁)K_h_y(F_Y⁻¹

0|X(τ₂|x₂)−g_β₀(X₁)−c₀−ε₁)|

dx1dx2µ(dτ1)µ(dτ2)

=o 1

! E

Z Z Z Z

κ(X₁−h_xx₁, τ₁)κ(x₂, τ₂)|K(x₁)

2.8. Proofs K_h_y(F_Y⁻¹

0|X(τ₁|X₁−h_xx₁)−g_β₀(X₁)−c₀−ε₁)K_h_y(F_Y⁻¹

0|X(τ₂|x₂)−g_β₀(X₁)−c₀−ε₁)|

dx1dx2µ(dτ1)µ(dτ2)

=o 1

!Z Z Z Z Z Z

κ(x₂, τ₂)|K(x₁)K_h_y(F_Y⁻¹

0|X(τ₁|w−h_xx₁)−g_β₀(w)−c₀−e) Khy(F_Y⁻¹

0|X(τ2|x₂)−gβ0(w)−c0−e)|f_X(w)fε(e)dw de dx1dx2µ(dτ1)µ(dτ2)

=o 1

x2 h_y

!Z Z Z Z Z

|K(x₁)K(e)|

fX(w)fε(F_Y⁻¹

0|X(τ1|w−hxx1)−gβ0(w)−c0−hye)dw de dx1µ(dτ1)

=o(1).

Moreover, equation (2.54) follows from (2.48) by c²_nn²h^d_x^XE[ ˜Z1,2Z˜3,4]

=c²_nn²h^d_x^X

Z Z Z Z E

∆n(X1)K_h_x(x1−X1)K_h_y(F_Y⁻¹

0|X(τ1|x₁)−g_β₀(X1)−c0−ε1) E

∆n(X3)Khx(x2−X3)Khy(F_Y⁻¹

0|X(τ2|x₂)−gβ0(X3)−c0−ε3) E

K_h_x(x1−X2)

K_h_y(F_Y⁻¹

0|X(τ1|x₁)−g_β₀(X2)−c0−ε2)−p0(F_Y⁻¹

0|X(τ1|x₁), x1) f_X(x₁)

K_h_x(x2−X4)

K_h_y(F_Y⁻¹

0|X(τ2|x₂)−g_β₀(X4)−c0−ε4)−p₀(F_Y⁻¹

0|X(τ₂|x₂), x₂) f_X(x₂)

κ(x1, τ1)κ(x2, τ2)dx1dx2µ(dτ1)µ(dτ2)

=o h

Z Z Z Z

κ(x₁, τ₁)κ(x₂, τ₂)dx₁dx₂µ(dτ₁)µ(dτ₂)

=o(1).

All in all, it was proven that T3 =op(1) and thus T˜n−b=nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x)2

dx µ(dτ)−b

→D Z+δ_1,n

withZ ∼ N(0, V) and δ_1,n=µ([0,1])R

v(x)∆_n(x)²dx.

Asymptotic Equivalence of Tn and ˜Tn+δ2,n+δ3,n

Recall Remark 2.3.1 and the definition of ˆcβ,τ in (2.14). Due to δn=δ1,n+δ2,n+δ3,n

(see Remark 2.3.5) it remains to show asymptotic equivalence of ˜Tn+δ2,n+δ3,n and T_n=nh

x2 min

β∈B

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β(x)−ˆc_β,τ2

dx µ(dτ).

For that purpose define G(β)

=−2nh_x^dX² Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−gβ0(x)−cβ0,τ

dx(ˆcβ0,τ −cβ0,τ)µ(dτ)

−2nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−gβ0(x)−cβ0,τ

Dβ(gβ0(x) +cβ0,τ)dx µ(dτ)(β−β0) +nh

v(x)dx Z

(ˆcβ0,τ −cβ0,τ)²µ(dτ) +nh

x2 µ([0,1])(β−β0)^tΩ(β−β0) as well as ¯β = arg min

β∈B

G(β) and

βˆ=nh

x2 arg min

β∈B

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β(x)−ˆc_β,τ2

dx µ(dτ).

First, it will be shown that

||βˆ−β₀||=O_p n⁻¹²h⁻

x 4

, (2.55)

||β¯−β0||=O_p n⁻¹²h⁻

x 4

. (2.56)

Due to

Fˆ_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ = ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x) =o_p(1)

uniformly inx∈supp(v), τ ∈µ, assumption (A7) implies ˆβ−β₀ =o_p(1) and ¯β−β₀ =o_p(1).

Further, for all sequences βn ∈ B, n ∈ N with ||β_n−β0|| → 0, Lemma 2.8.1, assumption (A7) and equation (2.14) yield

c_β_n_,τ −c_β₀_,τ

Rv(x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x) +g_β₀(x)−g_β_n(x) dx Rv(x)dx

Rv(x) ˆF_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x) + ˆF_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x) dx

R v(x)dx +O_p(||β_n−β₀||)

= c_n

v(x)dx

i=1

Z v(x) f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x)f_X(x)Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi) K_h_x(x−X_i)∆_n(X_i)dx+O_p

√n+||β_n−β₀||

= c_n

Rv(x)dxE

Z v(x) f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x)f_X(x)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X_i)−c₀−ε_i)

2.8. Proofs

K_h_x(x−X_i)∆_n(X_i)dx

+o_p(c_n) +O_p 1

√n+||β_n−β₀||

=cn

Rv(x)∆_n(x)dx

R v(x)dx +op(cn) +O_p 1

√n+||β_n−β0||

(2.57) uniformly inτ ∈supp(µ), where the second to last equation can be shown analogously to the reasoning in the proof of Lemma 4.2.12 later. Moreover, note that

D_βcˆ_β,τ =−

R v(x)Dβgβ(x)dx

R v(x)dx =D_βc_β,τ (2.58)

and Z Z

v(x)(c_β₀_,τ −ˆc_β₀_,τ)D_β(g_β₀(x) +c_β₀_,τ)dx µ(dτ)

(2.57)

= cn

R v(x)∆_n(x)dx R v(x)dx

Z Z

v(x)Dβ(gβ0(x) +cβ0,τ)dx µ(dτ) +op(cn)

=cn

R v(x)∆n(x)dx R v(x)dx

Z Z v(x)

D_βg_β₀(x)−

R v(w)Dβgβ0(w)dw Rv(w)dw

dx µ(dτ) +op(cn)

=op(cn).

Therefore, a Taylor expansion ofβ7→ Fˆ_Y⁻¹_|X(τ|x)−g_β(x)−c_β,τ2

and the binomial formula yield for some β^∗ between ˆβ and β₀

0≤nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ2

dx µ(dτ)−T_n

=nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ2

dx µ(dτ)

−nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−ˆc_β₀_,τ2

dx µ(dτ) + 2nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−ˆc_β₀_,τ

D_β(g_β₀(x) + ˆc_β₀_,τ)dx µ(dτ)( ˆβ−β₀) +nh

x2 ( ˆβ−β0)^t

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β^∗(x)−ˆc_β^∗_,τ

Hess(g_β^∗(x) + ˆc_β^∗_,τ)

− D_β(g_β^∗(x) + ˆc_β^∗_,τ)t

D_β(g_β^∗(x) + ˆc_β^∗_,τ)

dx µ(dτ)( ˆβ−β₀)

= 2nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ

dx(ˆc_β₀_,τ −c_β₀_,τ)µ(dτ)

−nh

v(x)dx Z

(ˆc_β₀_,τ −c_β₀_,τ)²µ(dτ) + 2nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−gβ0(x)−ˆcβ0,τ

Dβ(gβ0(x) +cβ0,τ)dx µ(dτ)( ˆβ−β0)

−nh

x2 µ([0,1])( ˆβ−β0)^tΩ( ˆβ−β0) +op

x2 ||βˆ−β0||²

(2.59)

=−G( ˆβ) +o_p√ nh

x4 ||βˆ−β₀||

+o_p nh

x2 ||βˆ−β₀||² .

Here, it was used that

Fˆ_Y⁻¹_|X(τ|x)−g_β^∗(x)−ˆc_β^∗_,τ =o_p(1) uniformly inx∈supp(v), τ ∈supp(µ) and

Z Z

v(x) Dβ(gβ^∗(x) + ˆcβ^∗,τ)t

Dβ(gβ^∗(x) + ˆcβ^∗,τ)dx µ(dτ) = Ω +op(1) componentwise due toβ^∗−β0 =op(1).

Later, it will be shown that (again componentwise)

√nh

Z Z

Fˆ_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ

v(x)D_β(g_β₀(x) +c_β₀_,τ)dx

µ(dτ)

≤√ nh

Fˆ_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x)

v(x)Dβ(gβ0(x) +cβ0,τ)dx

µ(dτ) +√

Z Z

Fˆ_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x)

v(x)D_β(g_β₀(x) +c_β₀_,τ)dx

µ(dτ)

=O_p(1) (2.60)

as well as

√nh

Z Z

Fˆ_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ

v(x)dx

µ(dτ) =O_p(1).

Since (2.57) implies

c_β₀_,τ −c_β₀_,τ =O_p

c_n+ 1

√n

, equation (2.59) then leads to

0≤ −nh_x^dX² µ([0,1])( ˆβ−β0)^tΩ( ˆβ−β0) +O_p√

x4 ||βˆ−β0||

+op

x2 ||βˆ−β0||²

+O_p(1), that is ˆβ−β0=O_p n⁻¹²h⁻

x 4

To prove the equations from above, define

κ(x, τ) = v(x)Dβ(gβ0(x) +cβ0,τ) f_Y₀|X(F_Y⁻¹

0|X(τ|x)|x)f_X(x) (2.61) and write with Lemma 2.8.1

√nh

Z Z

Fˆ_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x)

v(x)Dβ(gβ0(x) +cβ0,τ)dx

µ(dτ)

= 1 n

i=1

Z Z

κ(x, τ)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(Xi)−c0−εi)K_h_x(x−Xi)∆n(Xi)dx

µ(dτ)

+op(1)

=E Z

κ(x, τ)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X₁)−c₀−ε₁)K_h_x(x−X₁)∆_n(X₁)dx

µ(dτ)

2.8. Proofs +o_p(1)

=O_p(1)

as well as (with Lemma 2.8.1)

√nh

Z Z

Fˆ_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x)

v(x)D_β(g_β₀(x) +c_β₀_,τ)dx

µ(dτ)

= Z

√n

i=1

˜ κ(x, τ)

K_h_y(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)−p₀(F_Y⁻¹

0|X(τ|x), x) fX(x)

K_h_x(x−X_i)dx

µ(dτ) +o_p(1)

= Z

√n

i=1

κ(X_i−h_xx, τ)

K_h_y(F_Y⁻¹

0|X(τ|X_i−h_xx)−g_β₀(X_i)−c₀−ε_i)

−p0(F_Y⁻¹

0|X(τ|X_i−hxx), Xi−hxx) f_X(X_i−h_xx)

K(x)dx

µ(dτ) +o_p(1).

Let C > 0 be a sufficiently large constant. Then, for each of the components ˜κ_k, k = 1, ..., dB,one has

Z h

√n

i=1

κk(Xi−hxx, τ)

K_h_y(F_Y⁻¹

0|X(τ|X_i−hxx)−gβ0(Xi)−c0−εi)

−p0(F_Y⁻¹

0|X(τ|X_i−hxx), Xi−hxx) f_X(X_i−h_xx)

K(x)dx

µ(dτ)

!2#

≤µ([0,1])E

Z h

√n

i=1

κ_k(X_i−h_xx, τ)

K_h_y(F_Y⁻¹

0|X(τ|X_i−hxx)−g_β₀(Xi)−c0−εi)−p0(F_Y⁻¹

0|X(τ|X_i−hxx), Xi−hxx) f_X(X_i−h_xx)

K(x)dx 2

µ(dτ)

≤µ([0,1])h

dX 2

X E

Z Z

κ_k(X₁−h_xx, τ)

K_h_y(F_Y⁻¹

0|X(τ|X₁−h_xx)−g_β₀(X₁)−c₀−ε₁)

−p₀(F_Y⁻¹

0|X(τ|X₁−h_xx), X₁−h_xx) f_X(X1−hxx)

K(x)dx

µ(dτ)

+µ([0,1])nh

dX 2

Z Z E

κ_k(X₁−h_xx, τ)

K_h_y(F_Y⁻¹

0|X(τ|X₁−h_xx)−g_β₀(X₁)−c₀−ε₁)

−p₀(F_Y⁻¹

0|X(τ|X₁−h_xx), X₁−h_xx) fX(X1−hxx)

K(x)dx µ(dτ)

≤Ch

dX 2

|K(x)|dx 2

+o h

dX 2

=o(1),

where the last inequality can be shown similarly to (2.48), so that

√nh

Z Z

Fˆ_Y⁻¹

0|X(τ|x)−F_Y⁻¹

0|X(τ|x)

v(x)Dβ(gβ0(x) +cβ0,τ)dx

µ(dτ) =op(1) (2.62) and

√nh

Z Z

Fˆ_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ

v(x)D_β(g_β₀(x) +c_β₀_,τ)dx

µ(dτ) =O_p(1).

Completely analogously, it can be shown that

√nh

Z Z

Fˆ_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ

v(x)dx

µ(dτ) =O_p(1).

Therefore, it holds that ˆβ−β0 =O_p n⁻¹²h⁻

x 4

. Especially, (2.59) implies T_n=nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ2

dx µ(dτ) +O_p(1).

β¯is defined as the due to (A8) unique minimizer ofG. Hence, 0 =D_βG( ¯β)

=−2nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ

D_β(g_β₀(x) +c_β₀_,τ)dx µ(dτ) + 2nh

x2 µ([0,1])(β−β₀)^tΩ, that is, (2.60) leads to

β¯=β₀+ 1

µ([0,1])Ω⁻¹ Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ D_β(g_β₀(x) +c_β₀_,τ)t

dx µ(dτ)

=β₀+O_p n⁻¹²h⁻

x 4

. (2.63)

Note that for allβ ∈B with||β−β0||=O_p n⁻¹²h⁻

x 4

, one has (see (2.59)) Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β(x)−ˆc_β,τ2

dx µ(dτ)

= Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ2

dx µ(dτ) +G(β) +op(1), so that

Tn=nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−gβˆ(x)−ˆcβ,τˆ

dx µ(dτ)

2.8. Proofs

=nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ2

dx µ(dτ) +G( ˆβ) +op(1)

≥nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−gβ0(x)−cβ0,τ

dx µ(dτ) +G( ¯β) +op(1)

=nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−gβ¯(x)−cˆβ,τ¯

dx µ(dτ) +op(1)

≥T_n+o_p(1).

Consequently, to obtain the asymptotic distribution of Tn it suffices to calculate that of nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−gβ0(x)−cβ0,τ

dx µ(dτ) +G( ¯β) = ˜Tn+G( ¯β).

Inserting ¯β from (2.63) into G( ¯β) yields T˜n+G( ¯β)

= ˜T_n−2nh

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ

dx(ˆc_β₀_,τ −c_β₀_,τ)µ(dτ) +nh

v(x)dx Z

(ˆc_β₀_,τ −c_β₀_,τ)²µ(dτ)

− nh

µ([0,1]) Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ

D_β(g_β₀(x) +c_β₀_,τ)dx µ(dτ)

Ω⁻¹ Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ

D_β(g_β₀(x) +c_β₀_,τ)dx µ(dτ) t

+o_p(1).

Since ˆc_β₀_,τ was defined as the minimizer ofc7→R

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c2

dx, it holds

that Z

v(x) ˆF_Y⁻¹_|X(τ|x)−gβ0(x)−ˆcβ0,τ

dx= 0 and thus

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−c_β₀_,τ dx

= Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β₀(x)−ˆc_β₀_,τ + ˆc_β₀_,τ −c_β₀_,τ dx

= Z

v(x)(ˆcβ0,τ −cβ0,τ)dx for all τ ∈supp(µ). Together withF_Y⁻¹

0|X(τ|x) =gβ0(x) +cβ0,τ, this results in T˜_n+G( ¯β)

= ˜T_n−δ_1,n+µ([0,1]) Z

v(x)∆_n(x)²dx−nh

v(x)dx Z

(ˆc_β₀_,τ −c_β₀_,τ)²µ(dτ)

− nh

µ([0,1]) Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x)

D_β(g_β₀(x) +c_β₀_,τ)dx µ(dτ)

Ω⁻¹

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹

0|X(τ|x)

Dβ(gβ0(x) +cβ0,τ)dx µ(dτ) t

+op(1)

(2.62)

= T˜_n−δ_1,n+µ([0,1]) Z

v(x)∆_n(x)²dx−nh

v(x)dx Z

(ˆc_β₀_,τ −c_β₀_,τ)²µ(dτ)

− nh

µ([0,1]) Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x)

D_β(g_β₀(x) +c_β₀_,τ)dx µ(dτ)

Ω⁻¹ Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−Fˆ_Y⁻¹

0|X(τ|x)

Dβ(gβ0(x) +cβ0,τ)dx µ(dτ) t

+op(1)

= ˜Tn−δ1,n+µ([0,1]) Z

v(x)∆n(x)²dx−µ([0,1])

R v(x)∆_n(x)dx2

R v(x)dx

− 1

µ([0,1]) 1

i=1

Z Z

v(x) f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x)f_X(x)Dβ(gβ0(x) +cβ0,τ) Khy(F_Y⁻¹

0|X(τ|x)−gβ0(Xi)−c0−εi)Khx(x−Xi)∆n(Xi)dx µ(dτ)

Ω⁻¹ 1

i=1

Z Z

v(x) f_Y₀|X(F_Y⁻¹

0|X(τ|x)|x)f_X(x)D_β(g_β₀(x) +c_β₀_,τ) K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X_i)−c₀−ε_i)K_h_x(x−X_i)∆_n(X_i)dx µ(dτ) t

+o_p(1), where (2.46) and (2.57) were applied to obtain the last equation. Let ˜κ be as in (2.61).

Then, one has 1

i=1

Z Z

v(x) f_Y₀_|X(F_Y⁻¹

0|X(τ|x)|x)f_X(x)D_β(g_β₀(x) +c_β₀_,τ) K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X_i)−c₀−ε_i)K_h_x(x−X_i)∆_n(X_i)dx µ(dτ)

= 1 n

i=1

Z Z

κ(x, τ)K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(Xi)−c0−εi)K_h_x(x−Xi)∆n(Xi)dx µ(dτ)

= Z Z

κ(x, τ)E

K_h_y(F_Y⁻¹

0|X(τ|x)−g_β₀(X₁)−c₀−ε₁)K_h_x(x−X₁)∆_n(X₁)

dx µ(dτ) +op(1)

Z Z Z Z

κ(x, τ)Khy(F_Y⁻¹

0|X(τ|x)−z)Khx(x−w)∆n(w)fY0,X(z, w)dz dw dx µ(dτ) +o_p(1)

Z Z Z Z

κ(x, τ)K(z)K(w)∆_n(x−h_xw) f_Y₀_,X(F_Y⁻¹

0|X(τ|x−h_xw)−h_yz, x−h_xw)dz dw dx µ(dτ) +o_p(1)

= Z Z

κ(x, τ)∆n(x)fY0,X(F_Y⁻¹

0|X(τ|x), x)dx µ(dτ) +op(1)

2.8. Proofs

= Z Z

v(w)Dβ(gβ0(w) +cβ0,τ)∆n(w)dw µ(dτ) +op(1), so that with (2.58)

T˜_n+G( ¯β)

= ˜T_n−δ_1,n+µ([0,1]) Z

v(x)∆_n(x)²dx−µ([0,1])

R v(x)∆n(x)dx2

R v(x)dx

− 1

µ([0,1]) Z Z

v(x)Dβ(gβ0(x) +cβ0,τ)∆n(x)dx µ(dτ)

Ω⁻¹ Z Z

v(x)D_β(g_β₀(x) +c_β₀_,τ)∆n(x)dx µ(dτ) t

+op(1)

= ˜Tn−δ1,n+µ([0,1]) Z

v(x)∆n(x)²dx−µ([0,1])

R v(x)∆n(x)dx2

R v(x)dx

−µ([0,1]) Z

v(x)∆_n(x)

D_βg_β₀(x)−

RD_βg_β₀(w)dw R v(w)dw

Ω⁻¹ Z

v(x)∆n(x)

Dβgβ0(x)−

R D_βg_β₀(w)dw R v(w)dw

+op(1)

= ˜T_n−δ_1,n+µ([0,1]) Z

v(x) ∆_n(x)−

Rv(w1)∆n(w1)dw1

R v(w2)dw2

−

Dβgβ0(x)−

R D_βg_β₀(w₃)dw₃ Rv(w₄)dw₄

Ω⁻¹ Z

v(w₅)∆_n(w₅)

D_βg_β₀(w₅)−

R D_βg_β₀(w6)dw6

Rv(w7)dw7

dw₅

t!2

dx+o_p(1)

= ˜T_n−δ_1,n+δ_n+o_p(1)

= ˜Tn+δ2,n+δ3,n+op(1),

where the third from last equality was obtained by standard calculations. Finally, Lemma 2.3.2 leads to

T_n−b−δ_n→^D Z.

2.8.4 Proof of Remark 2.3.5

δn was defined as δn=µ([0,1])

v(x) ∆n(x)−

R v(w₁)∆_n(w₁)dw₁ R v(w₂)dw₂ −

D_βg_β₀(x)−

R D_βg_β₀(w₃)dw₃ R v(w₄)dw₄

Ω⁻¹ Z

v(w₅)∆_n(w₅)

D_βg_β₀(w₅)−

R D_βg_β₀(w₆)dw₆ R v(w7)dw7

dw₅

t!2

dx.

The alternative expression for δn can be obtained by simply expanding that from above.

While doing so, the fact is used that Z

v(w₁)

Rv(w2)Dβgβ0(w2)dw2

R v(w₃)dw₃

D_βg_β₀(w₁)−

R v(w4)Dβgβ0(w4)dw4

Rv(w₅)dw₅

dw₁ = 0. (2.64) To prove the second assertion, rewrite ∆_n as

∆_n(x) = gβn(x)−gβ0(x)

c_n =D_βg_β₀(x)βn−β0

c_n +o(1) uniformly inx∈supp(v) and n∈N. Hence, the distributive law yields

δ_n=µ([0,1]) Z

v(x) D_βg_β₀(x)β_n−β₀ c_n −

R v(w1)D_βg_β₀(w1)dw1

Rv(w2)dw2

β_n−β₀ c_n

−

Dβgβ0(x)−

RD_βg_β₀(w₃)dw₃ R v(w₄)dw₄

Ω⁻¹ Z

v(w₅)∆_n(w₅)

D_βg_β₀(w₅)−

R D_βg_β₀(w6)dw6

R v(w7)dw7

dw₅

t!2

dx+o(1)

=µ([0,1]) Z

v(x)

Dβgβ0(x)−

R v(w₁)D_βg_β₀(w₁)dw₁ R v(w₂)dw₂

β_n−β₀ cn

−Ω⁻¹ Z

v(w₅)∆_n(w₅)

D_βg_β₀(w₅)−

R D_βg_β₀(w6)dw6

Rv(w7)dw7

dw₅

t!!2

+o(1).

Equation (2.64) leads to β_n−β₀

c_n −Ω⁻¹ Z

v(w₅)∆_n(w₅)

D_βg_β₀(w₅)−

R Dβgβ0(w6)dw6

R v(w7)dw7

dw₅

= β_n−β₀ cn

−Ω⁻¹ Z

v(w₅)D_βg_β₀(w₅)β_n−β₀ cn

D_βg_β₀(w₅)−

RD_βg_β₀(w6)dw6

Rv(w7)dw7

dw₅

+o(1)

= βn−β0

−Ω⁻¹ (βn−β0)^t cn

Z v(w5)

Dβgβ0(w5)−

R D_βg_β₀(w₃)dw₃ R v(w₄)dw₄

D_βg_β₀(w5)−

R Dβgβ0(w6)dw6

R v(w₇)dw₇

dw5

+o(1)

= β_n−β₀

c_n −Ω⁻¹Ωβ_n−β₀

c_n +o(1)

=o(1),

that is,δn=o(1).

2.8. Proofs 2.8.5 Proof of Theorem 2.3.6

The proof of the second part directly follows from Theorem 2.3.4 and Slutsky’s theorem, so that it remains to prove the first assertion. Since H₀ is violated in this case, one has

β∈B, c∈minR

Z Z

v(x) F_Y⁻¹_|X(τ|x)−gβ(x)−c2

dx µ(dτ)>0.

Recall

Fˆ_Y_|X(y|x)−F_Y_|X(y|x) = p(y, x)ˆ

fˆ_X(x) − p(y, x) f_X(x)

Again, the results of Hansen (2008) yield ˆp(y, x)−p(y, x) =op(1) as well as ˆf_X(x)−f_X(x) = o_p(1) uniformly on compact sets and thus

Fˆ_Y_|X(y|x)−F_Y_|X(y|x) =o_p(1)

uniformly on x ∈ supp(v) and y belonging to some compact set K ⊆ R. When choosing K= [y₁, y₂] with

y1 = inf

x∈supp(v),τ∈supp(µ)F_Y⁻¹_|X(τ, x) and y2= sup

x∈supp(v),τ∈supp(µ)

F_Y⁻¹_|X(τ, x),

assumption (2.28) ensures that the functions y 7→ F_Y_|X(y|x) are strictly increasing for all x∈supp(v), so that

Fˆ_Y⁻¹_|X(τ|x)−F_Y⁻¹_|X(τ|x) =o_p(1)

uniformly onx∈supp(v) and τ ∈supp(µ). Especially, it holds that sup

x∈supp(v),τ∈supp(µ)

|Fˆ_Y⁻¹_|X(τ|x)| ≤ sup

x∈supp(v),τ∈supp(µ)

|F_Y⁻¹_|X(τ|x)|+op(1)

and the minimization in (2.13) with respect to ccan be replaced by that over some appro-priate compact set [c1, c2]⊆Rto obtain

T_n nh

= min

β∈B,c∈[c₁,c2]

Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−g_β(x)−c2

dx µ(dτ) +o_p(1)

≥ Z Z

v(x) ˆF_Y⁻¹_|X(τ|x)−F_Y⁻¹_|X(τ|x)2

dx µ(dτ)

−2 sup

x∈supp(v),τ∈supp(µ)

Fˆ_Y⁻¹_|X(τ|x)−F_Y⁻¹_|X(τ|x) max

β∈B,c∈[c1,c2]

Z Z v(x)

F_Y⁻¹_|X(τ|x)−g_β(x)−c

dx µ(dτ)

+ min

β∈B,c∈[c1,c2]

Z Z

v(x) F_Y⁻¹_|X(τ|x)−gβ(x)−c2

dx µ(dτ) +op(1)

= min

β∈B,c∈[c1,c2]

Z Z

v(x) F_Y⁻¹_|X(τ|x)−gβ(x)−c2

dx µ(dτ) +op(1).

Hence

P(Φ(Y1, X1, ..., Yn, Xn) = 1)

T_n>ˆb+

pV uˆ 1−α

=P T_n nh

> ˆb+p V uˆ 1−α

β∈B,c∈[cmin₁,c2]

Z Z

v(x) F_Y⁻¹_|X(τ|x)−g_β(x)−c2

dx µ(dτ)> o_p(1)

= 1 +o(1).

2.8.6 Proof of Theorem 2.4.1

Note that there exists a compact intervalC= [c₁, c₂]⊆Rsuch that (F_Y^h_|X)⁻¹ τ|x

∈(h(c₁), h(c₂)) for allx∈supp(v), τ ∈supp(µ). (2.65) Similar to the case without transformations, one has

Fˆ_Y^ˆ^h_|X(y|x)−F_Y^h_|X(y|x)

= pˆ^ˆ^h(y, x)

fˆX(x) −p^h(y, x) f_X(x)

= 1

f_X(x)(p^ˆ^h(y, x)−p^h(y, x))−p^h(y, x)

f_X(x)²( ˆfX(x)−fX(x))

− fˆ_X(x)−f_X(x) fˆ_X(x)f_X(x)

p^ˆ^h(y, x)−p^h(y, x)−p^h(y, x)( ˆf_X(x)−f_X(x)) fX(x)

p^h(y, x)−p^h(y, x) =op n⁻¹⁴

(2.66) and

fˆX(x)−fX(x) =op n⁻¹⁴

uniformly inx∈supp(v) andy ∈h(C). First, the asymptotic behaviour of ˆp^ˆ^h(y, x)−pˆ^h(y, x) is examined. To this end, letδ >0. As the support ofK is compact and ˆh(y)≤ˆh(c₁−δ) = h(c1 −δ) +op(1) uniformly in y ∈ (−∞, c₁ −δ) and analogously ˆh(y) ≥ ˆh(c2 +δ) = h(c2+δ) +op(1) uniformly iny∈(c2+δ,∞) one has

∀z∈h(C), y /∈[c1−δ, c2+δ] :K_h_y z−ˆh(y)

=K_h_y z−h(y)

∈ {0,1}

→1. (2.67) (2.41) yields

1 nh^j_y

i=1

I_{Y_i_∈[c₁_−δ,c₂_+δ]}K^(j−1)

y−h(Y_i) hy

(h(Y_i)−h(Yˆ _i))^jK_h_x(x−X_i)

≤

sup

y∈[c1−δ,c2+δ]

(h(y)−ˆh(y))^j h^j−1y

1 nhy

i=1

K^(j−1)

y−h(Yi) hy

Khx(x−Xi)

2.8. Proofs

=O_p 1

√n^jh^j−1y

=o_p 1

√n for all j= 2, ..., ras well as

1 nh^r+1y

i=1

I_{Y_i_∈[c₁_−δ,c₂_+δ]}K^(r)(y_i^∗)(h(Y_i)−ˆh(Y_i))^r+1K_h_x(x−X_i)

≤

sup

y∈[c1−δ,c2+δ]

(h(y)−ˆh(y))^r+1 h^r+1y

sup

y∈R

K^(r)(y) 1 nhy

i=1

|K_h_x(x−X_i)|

=O_p

√n^r+1h^r+1y

=op

√n

Hence, one has for appropriate y^∗_i ∈R, i= 1, ..., n, ˆ

p^ˆ^h(y, x)−pˆ^h(y, x)

= 1 n

i=1

K_h_y y−ˆh(Y_i)

K_h_x(x−X_i)− 1 n

i=1

K_h_y y−h(Y_i)

K_h_x(x−X_i)

= 1 n

i=1

I_{Y_i_∈[c₁_−δ,c₂_+δ]}K_h_y y−ˆh(Y_i)

K_h_x(x−X_i)

− 1 n

i=1

I_{Y_i_∈[c₁_−δ,c₂_+δ]}K_h_y y−h(Y_i)

K_h_x(x−X_i) +o_p 1

√n

= 1 n

i=1

I_{Y_i_∈[c₁_−δ,c₂_+δ]}

j=1

h^jyj!K^(j−1)

y−h(Y_i) h_y

(h(Y_i)−ˆh(Y_i))^jK_h_x(x−X_i)

+ 1 n

i=1

I_{Y_i_∈[c₁_−δ,c₂_+δ]} 1

h^r+1y (r+ 1)!K^(r)(y)

y=y^∗_i(h(Y_i)−h(Yˆ _i))^r+1K_h_x(x−X_i) +op

√n

= 1 n²

i=1 n

k=1

ψ(Y_k, X_k, Y_i)I_{Y_i_∈[c₁_−δ,c₂_+δ]}K_h_y y−h(Y_i)

K_h_x(x−X_i) +o_p 1

√n

(2.68) and

|ˆp^ˆ^h(y, x)−pˆ^h(y, x)|

= 1 n

i=1

I{Y_i∈[c₁−δ,c₂+δ]}Khy(y−h(Yi))(h(Yi)−ˆh(Yi))Khx(x−Xi)

+op

√n

≤ sup

z∈[c1−δ,c2+δ]

|h(z)−ˆh(z)|1 n

i=1

|K_h_y(y−h(Yi))Khx(x−Xi)|+op

√n

(2.31)

= O_p 1

√n

(2.69) uniformly in y ∈ h(C), x ∈ supp(v). Due to (2.69), equation (2.66) can be extended to ˆ

p^ˆ^h(y, x)−p^h(y, x) =o_p n⁻¹⁴

, so that Fˆ_Y^ˆ^h_|X(y|x)−F_Y^h_|X(y|x) = 1

f_X(x)(ˆp^ˆ^h(y, x)−p^h(y, x))−p^h(y, x)

f_X(x)²( ˆf_X(x)−f_X(x)) +o_p

√n

= 1

f_X(x)pˆ^ˆ^h(y, x)−p^h(y, x) f_X(x)²

fˆX(x) +op

√n

=o_p n⁻¹⁴

uniformly onx∈supp(v) andy ∈h(C). A similar reasoning leads to fˆ_Y^ˆ^h_|X(y|x)−f_Y^ˆ^h_|X(y|x) =o_p n⁻¹⁴

and ∂

∂yfˆ_Y^ˆ^h_|X(y|x) =O_p(1) uniformly onx∈supp(v) andy ∈h(C), so that for an appropriate y^∗ one has

0 = ˆF_Y^ˆ^h_|X(( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)|x)−F_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x)

= ˆF_Y^ˆ^h_|X((F_Y^h_|X)⁻¹(τ|x)|x) + ˆf_Y^ˆ^h_|X((F_Y^h_|X)⁻¹(τ|x)|x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−(F_Y^h_|X)⁻¹(τ|x) + ∂

∂y

fˆ_Y^ˆ^h_|X(y^∗|x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−(F_Y^h_|X)⁻¹(τ|x)2

−F_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x)

= ˆF_Y^ˆ^h_|X((F_Y^h_|X)⁻¹(τ|x)|x)−F_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x)

+f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x) ( ˆF_Y^h^ˆ_|X)⁻¹(τ|x)−(F_Y^h_|X)⁻¹(τ|x) +op

√n

uniformly inx∈supp(v) and τ ∈supp(µ). This in turn results in (F_Y^h_|X)⁻¹(τ|x)−( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)

Fˆ_Y^ˆ^h_|X((F_Y^h_|X)⁻¹(τ|x)|x)−F_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x) f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x) +op

√n

= 1

f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x) 1

fX(x)pˆ^h^ˆ((F_Y^h_|X)⁻¹(τ|x), x)−p^h((F_Y^h_|X)⁻¹(τ|x), x) fX(x)²

fˆX(x)

+op

√n

= 1

f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x) 1 n

i=1

K_h_x(x−X_i) 1

f_X(x)K_h_y((F_Y^h_|X)⁻¹(τ|x)−ˆh(Y_i))

−p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)²

+o_p

√n

(2.70)

2.8. Proofs uniformly in x ∈supp(v) and τ ∈ supp(µ). Note that validity of H0 is assumed, that is, h(Y) here corresponds toY₀=Y in Section 2.3.2. Therefore, equation (2.45) leads to

(F_Y^h_|X)⁻¹(τ|x)−( ˆF_Y^h_|X)⁻¹(τ|x)

= 1

f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x) 1

f_X(x)pˆ^h((F_Y^h_|X)⁻¹(τ|x), x)−p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)² fˆX(x)

= 1

f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x) 1 n

i=1

Khx(x−Xi) 1

fX(x)K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Yi))

−p^h((F_Y^h_|X)⁻¹(τ|x), x) fX(x)²

+op

√n

uniformly in x∈supp(v) andτ ∈supp(µ). Hence, (2.68), (2.69) and (2.70) yield ( ˆF_Y^h_|X)⁻¹(τ|x)−( ˆF_Y^h^ˆ_|X)⁻¹(τ|x)

= (F_Y^h_|X)⁻¹(τ|x)−( ˆF_Y^h^ˆ_|X)⁻¹(τ|x)− (F_Y^h_|X)⁻¹(τ|x)−( ˆF_Y^h_|X)⁻¹(τ|x)

(2.70)

= 1

f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x)f_X(x) pˆ^ˆ^h((F_Y^h_|X)⁻¹(τ|x), x)−pˆ^h((F_Y^h_|X)⁻¹(τ|x), x) +o_p

√n

(2.68)

= 1

f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x)f_X(x)n²

i=1 n

k=1

ψ(Yk, Xk, Yi)I{Y_i∈[c₁−δ,c₂+δ]}

Khy (F_Y^h_|X)⁻¹(τ|x)−h(Yi)

Khx(x−Xi) +op

√n

(2.69)

= O_p 1

√n

(2.71) uniformly inx∈supp(v) andτ ∈supp(µ). Recall that (F_Y^h_|X)⁻¹(τ|·) =g_β₀(·)+c₀+F_ε⁻¹(τ).

Extend definitions (2.14) and (2.16) to c^h_β,τ =

R v(x)((F_Y^h_|X)⁻¹(τ|x)−g_β(x))dx R v(x)dx

ˆ c^h_β,τ =

R v(x)(( ˆF_Y^h_|X)⁻¹(τ|x)−gβ(x))dx R v(x)dx

ˆ c^h_β,τ^ˆ =

R v(x)(( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β(x))dx R v(x)dx .

Recall Theorem 2.3.4 and the definitions of δn there. Due to validity of H0 it holds that δ_n= 0. In the proof of Theorem 2.3.4 it was shown that

T_n^h= min

β∈Bnh

Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−g_β(x)−cˆ^h_β,τ2

dx µ(dτ)

=nh

Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−(F_Y^h_|X)⁻¹(τ|x)2

dx µ(dτ) +O_p(1)

=b+O_p(1). (2.72) Similar to the proof of (2.14) one can show that

c^ˆ^h_β,τ −cˆ^h_β,τ =

R v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−( ˆF_Y^h_|X)⁻¹(τ|x) R v(x)dx

(2.71)

= O_p 1

√n

(2.73) and the same calculations as in (2.57) lead to (note thatH0 was assumed, that is cn= 0)

c^ˆ^h_β,τ−c^h_β,τ = ˆc^ˆ^h_β,τ−ˆc^h_β,τ+

Rv(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−(F_Y^h_|X)⁻¹(τ|x) R v(x)dx

=O 1

√n

(2.74) uniformly inβ ∈B and τ ∈supp(µ).

Let ˆβ^h and ˆβ^h^ˆ be the minimizing values in T_n^h and T_n^ˆ^h, respectively.

Lemma 2.8.2 Let β¯^ˆ^h and β¯^h denote the minimizers of G^h^ˆ :B×R→R, G^h^ˆ(β) =−2nh

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ D_β(g_β₀(x) +c^h_β₀_,τ)dx µ(dτ)(β−β₀) +nh

x2 (β−β₀)^tΩ(β−β₀), andG^h :B×R→R,

G^h(β) =−2nh_x^dX² Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ D_β(g_β₀(x) +c^h_β₀_,τ)dx µ(dτ)(β−β0)

+nh

x2 (β−β₀)^tΩ(β−β₀).

Define

T˜_n^h :=nh

Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ2

dx µ(dτ)

=nh

Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−(F_Y^h_|X)⁻¹(τ|x)2

dx µ(dτ).

Then, one has

||βˆ^ˆ^h−β₀||=O_p n⁻¹²h⁻

x 4

, (2.75)

||βˆ^h−β0||=O_p n⁻¹²h⁻

x 4

||β¯^ˆ^h−β0||=O_p n⁻¹²

, (2.76)

||β¯^h−β0||=O_p n⁻¹² and

Z Z

v(x) ( ˆF_Y^h^ˆ_|X)⁻¹(τ|x)−gβ0(x)−c^h_β₀_,τ2

dx µ(dτ) = ˜T_n^h+op(1).

2.8. Proofs Proof: It is started with proving the last assertion. Write

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ2

dx µ(dτ)

=nh

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−( ˆF_Y^h_|X)⁻¹(τ|x) + ( ˆF_Y^h_|X)⁻¹(τ|x)−gβ0(x)−c^h_β₀_,τ2

dx µ(dτ)

=nh

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−( ˆF_Y^h_|X)⁻¹(τ|x)2

dx µ(dτ) + ˜T_n^h + 2nh

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−( ˆF_Y^h_|X)⁻¹(τ|x) ( ˆF_Y^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ

dx µ(dτ)

Thanks to (2.71) the first term is asymptotically negligible. In Lemma 2.3.2 it was shown that ˜T_n^h =O_p h⁻

x 2

. Moreover, the third term can be expressed alternatively via (2.71) and Lemma 2.8.1 as

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−( ˆF_Y^h_|X)⁻¹(τ|x) ( ˆF_Y^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ

dx µ(dτ)

=nh

Z Z v(x)

f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x)f_X(x)n²

i=1 n

k=1

ψ(Yk, Xk, Yi)

I_{Y_i_∈[c₁_−δ,c₂_+δ]}K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Y_i)

K_h_x(x−X_i) +o_p 1

√n

f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x) 1 n

i=1

Khx(x−Xi) 1

fX(x)K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Yi))

−p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)²

+O_p

√n

dx µ(dτ)

= h

n²

i=1 n

k=1 n

l=1

Z Z

κ(x, τ)ψ(Y_k, X_k, Y_i)I_{Y_i_∈[c₁_−δ,c₂_+δ]}K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Y_i) K_h_x(x−X_i)K_h_x(x−X_l)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y_l))−p^h((F_Y^h_|X)⁻¹(τ|x), x) fX(x)

dx µ(dτ) +o_p(1), where

κ(x, τ) = v(x)

f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x)²fX(x)²

has a compact support. For all compact sets C ⊆R, the function (y1, x1, y)7→ ψ(y1, x1, y) is uniformly bounded in (y₁, x₁, y)∈R^d^X⁺¹× C due to assumption (A9). The sum can be split into

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−( ˆF_Y^h_|X)⁻¹(τ|x)

( ˆF_Y^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ

dx µ(dτ)

= h

n²

i=1

Z Z

κ(x, τ)ψ(Y_i, X_i, Y_i)I_{Y_i_∈[c₁_−δ,c₂_+δ]}K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Y_i)

K_h_x(x−X_i)²

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y_i))− p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)

dx µ(dτ)

n²

i=1 n

k=1k6=i

Z Z

κ(x, τ)ψ(Yk, Xk, Yi)I{Y_i∈[c₁−δ,c₂+δ]}Khy (F_Y^h_|X)⁻¹(τ|x)−h(Yi)

K_h_x(x−X_i)K_h_x(x−X_k)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y_k))

−p^h((F_Y^h_|X)⁻¹(τ|x), x) fX(x)

dx µ(dτ)

n²

i=1 n

l=1l6=i

Z Z

κ(x, τ)ψ(Yi, Xi, Yi)I_{Y_i_∈[c₁_−δ,c₂_+δ]}K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Yi)

K_h_x(x−X_i)K_h_x(x−X_l)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y_l))

−p^h((F_Y^h_|X)⁻¹(τ|x), x) fX(x)

dx µ(dτ)

n²

i=1 n

k=1k6=i

Z Z

κ(x, τ)ψ(Y_k, X_k, Y_i)I_{Y_i_∈[c₁_−δ,c₂_+δ]}K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Y_i)

K_h_x(x−Xi)²

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Yi))− p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)

dx µ(dτ)

n²

i=1 n

k=1k6=i n

l6=i,kl=1

Z Z

κ(x, τ)ψ(Y_k, X_k, Y_i)I_{Y_i_∈[c₁_−δ,c₂_+δ]}

K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Yi)

K_h_x(x−Xi)K_h_x(x−X_l)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y_l))

−p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)

dx µ(dτ) +o_p(1)

=I+II+III+IV +V +o_p(1).

For an appropriate constantC >0, termI can be bounded by

I ≤ Ch

n²

i=1

Z Z

κ(x, τ)|K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Yi))|K_h_x(x−Xi)²dx µ(dτ)

2.8. Proofs

= C

n²h

i=1

Z Z

κ(Xi+hxx, τ)|K_h_y((F_Y^h_|X)⁻¹(τ|X_i+hxx)−h(Yi))|K(x)²dx µ(dτ), which in turn is asymptotically negligible due to

E[κ(Xi+hxx, τ)|K_h_y((F_Y^h_|X)⁻¹(τ|X_i+hxx)−h(Yi))|]

= Z

κ(w+h_xx, τ)K_h_y((F_Y^h_|X)⁻¹(τ|w+h_xx)−g_β₀(w)−c₀−e)f_X(w)f_ε(e)dw de

= Z

κ(w+h_xx, τ)K(e)f_X(w)f_ε((F_Y^h_|X)⁻¹(τ|w+h_xx)−g_β₀(w)−c₀−h_ye)dw de

=O(1)

uniformly in x∈supp(K) and τ ∈supp(µ), that isI =o_p(1). For the second term define Z_i,k^II =

Z Z

κ(x, τ)ψ(Y_k, X_k, Y_i)I_{Y_i_∈[c₁_−δ,c₂_+δ]}K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Y_i)

K_h_x(x−X_i)

K_h_x(x−X_k)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y_k))−p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)

dx µ(dτ),

so that II = ^h

dX x2

n²

Pn i=1

Pn k=1 k6=i

Z_i,k^II. For an appropriate constant C >0, the expectation of

|Z_1,2^II|can be bounded by E[|Z_1,2^II|]≤C

Z Z

κ(x, τ)E

Khy (F_Y^h_|X)⁻¹(τ|x)−h(Y1)

Khx(x−X1) E[|K_h_x(x−X₁)|]dx µ(dτ)

=O(1), that is,

n²

i=1 n

k=1k6=i

Z_i,k^II

≤ h

n²

i=1 n

k=1k6=i

|Z_i,k^II|

=o(1).

TermIII can be treated similarly to obtain III =op(1).

For the fourth term define Z_i,k^IV =

Z Z

κ(x, τ)ψ(Yk, Xk, Yi)I{Y_i∈[c₁−δ,c₂+δ]}Khy (F_Y^h_|X)⁻¹(τ|x)−h(Yi)

Khx(x−Xi)²

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Yi))−p^h((F_Y^h_|X)⁻¹(τ|x), x) fX(x)

dx µ(dτ),

that is,IV = ^h

dX2 x

n²

Pn i=1

Pn k=1k6=i

Z_i,k^IV. As before, it can be shown that ^h

dX2 x

n²

i=1Z_i,i^IV =o_p(1), so that for an appropriate constant C >0

|IV|= h

n²

i=1 n

k=1

Z_i,k^IV

+o_p(1)

≤ Ch

i=1

Z Z

κ(x, τ)

K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Yi)

K_h_x(x−Xi)²dx µ(dτ)

I_{Y_i_∈[c₁_−δ,c₂_+δ]}

1 n

k=1

ψ(Y_k, X_k, Yi)

| {z }

=O_p ^√¹

=O_p 1

√n Ch

i=1

Z Z

κ(x, τ)

K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Yi) Khx(x−Xi)²dx µ(dτ)

=O_p 1

p nh^dx^X

where the last equality follows similar to proving asymptotic negligibility ofI, II and III.

Hence,IV =op(1).

It remains to examine termV. Define Z_i,k,l^V =

Z Z

κ(x, τ)ψ(Y_k, X_k, Yi)I_{Y_i_∈[c₁_−δ,c₂_+δ]}K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Yi)

K_h_x(x−Xi)

Khx(x−Xl)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Yl))−p^h((F_Y^h_|X)⁻¹(τ|x), x) fX(x)

dx µ(dτ),

so thatV = ^h

dX2 x

n²

Pn i=1

Pn k=1k6=i

Pn l6=i,kl=1

Z_i,k,l^V . One has

E[V²] = h^d_x^X n⁴

i=1 n

k=1k6=i n

l6=i,kl=1 n

s=1 n

t=1t6=s n

u6=s,tu=1

Z_i,k,l^V Z_s,t,u^V .

Due toE[ψ(Y₂, X₂, Y₁)|Y₁] = 0 the expectation vanishes wheneverkortare occurring only once in (i, k, l, s, t, u). Only asymptotic negligibility of the summand corresponding to the case, in whichk=t and #{i, k, l, s, t, u}= 5, will be shown, since asymptotic negligibility of the remaining summands can be deduced from this case and the calculations for terms I, II, III, IV. It holds that

Z_1,2,3^V Z_4,2,5^V

=E Z Z

κ(x, τ)ψ(Y₂, X₂, Y₁)ψ(Y₂, X₂, Y₄)I_{Y₁_∈[c₁_−δ,c₂_+δ]}I_{Y₄_∈[c₁_−δ,c₂_+δ]}

K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Y₁)

K_h_x(x−X₁)K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Y₄)

K_h_x(x−X₄) K_h_x(x−X₃)K_h_x(x−X₅)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y₃))−p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y₅))− p^h((F_Y^h_|X)⁻¹(τ|x), x) fX(x)

dx µ(dτ)

2.8. Proofs

=E Z Z

κ(x, τ)ψ(Y₂, X₂, Y₁)ψ(Y₂, X₂, Y₄)I_{Y₁_∈[c₁_−δ,c₂_+δ]}I_{Y₄_∈[c₁_−δ,c₂_+δ]}

K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Y₁)

K_h_x(x−X₁)K_h_y (F_Y^h_|X)⁻¹(τ|x)−h(Y₄)

K_h_x(x−X₄) E

K_h_x(x−X3)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y3))−p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)

dx µ(dτ)

As in (2.48) the inner expectation can be bounded via E

Khx(x−X3)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y3))− p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)

= Z

K(w) Z

K(u)p^h((F_Y^h_|X)⁻¹(τ|x)−hyu, x−hxw)dz du

−p^h((F_Y^h_|X)⁻¹(τ|x), x)f_X(x−h_xw) fX(x)

=o_p 1

√n

(2.77) uniformly in x∈supp(v) andτ ∈supp(µ). By the same reasoning as before this results in E

Z_1,2,3^V Z_4,2,5^V

=o ¹_n

and thus E[V²] = h^d_x^X

n⁴

i=1 n

k=1k6=i n

l6=i,kl=1 n

s6=i,k,ls=1 n

u6=i,k,l,su=1

Z_i,k,l^V Z_s,k,u^V

=o(1).

Finally, this leads to V =o_p(1), that is nh

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ2

dx µ(dτ) = ˜T_n^h+o_p(1).

Treatment of ˆβ^h,βˆ^ˆ^h,β¯^h and ¯β^ˆ^h For treating ˆβ^h and ˆβ^ˆ^h note that

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β(x)−ˆc^ˆ^h_β,τ2

dx µ(dτ)

= Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−gβ(x)−cˆ^h_β,τ2

dx µ(dτ) +op(1)

= Z Z

v(x) (F_Y^h_|X)⁻¹(τ|x)−gβ(x)−c^h_β,τ2

dx µ(dτ) +op(1) uniformly in β∈B with

sup

β∈B,||β−β0||>δ

Z Z

v(x) (F_Y^h_|X)⁻¹(τ|x)−g_β(x)−c^h_β,τ2

dx µ(dτ)>0

for allδ >0, which leads because of (A7) to||βˆ^h−β₀||=o_p(1) and||βˆ^ˆ^h−β₀||=o_p(1). Due to (2.72), one hasT_n^h−b=O_p(1) and a Taylor expansion ofβ 7→ ( ˆF_Y^ˆ^h_|X)⁻¹−g_β(x)−c^h_β,τ2

(compare (2.59)) yields T_n^h−b

(2.72)

= nh

Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ2

dx µ(dτ)−b+O_p(1)

=nh

Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−g_β_ˆ^ˆh(x)−c^h_ˆ

β^ˆ^h,τ

dx µ(dτ)−b

−2nh

Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−g_β_ˆh^ˆ(x)−c^h_ˆ

β^h^ˆ,τ

Dβ g_β_ˆ^ˆh(x) +c^h_ˆ

β^h^ˆ,τ

dx µ(dτ)(β0−βˆ^ˆ^h) + (β0−βˆ^ˆ^h)^tnh

x2 Ω(β0−βˆ^h^ˆ) +op nh

x2 ||β₀−βˆ^ˆ^h||²

+O_p(1)

≥T_n^h−b+nh

x2 (β0−βˆ^ˆ^h)^tΩ(β0−βˆ^ˆ^h)−2nh

Z Z v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−g_β_ˆh^ˆ(x)−cˆ^ˆ^h_ˆ

β^h^ˆ,τ

Dβ g_β_ˆh^ˆ(x) + ˆc^ˆ^h_ˆ

β^h^ˆ,τ

dx µ(dτ)(β0−βˆ^ˆ^h) +O_p

nh^dx^X||β₀−βˆ^ˆ^h||

+o_p nh

x2 ||β₀−βˆ^h^ˆ||²

+O_p(1)

= (β0−βˆ^ˆ^h)^tnh

x2 Ω(β0−βˆ^ˆ^h) +O_p q

nh^dx^X||β₀−βˆ^ˆ^h||

+op nh

x2 ||β₀−βˆ^ˆ^h||²

+O_p(1).

(2.78) Here, the second to last inequality, where ˆc^h_ˆ

β^ˆ^h,τ was replaced withc^ˆ^h_ˆ

β^ˆ^h,τ, follows from (com-pare (2.58) and (2.74))

Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−g_β_ˆˆh(x)−c^h_ˆ

β^h^ˆ,τ

D_β g_β_ˆˆh(x) +c^h_ˆ

β^h^ˆ,τ

dx µ(dτ)

=nh

Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−g_β_ˆˆh(x)−ˆc^ˆ^h_ˆ

β^h^ˆ,τ

D_β g_β_ˆˆh(x) + ˆc^ˆ^h_ˆ

β^h^ˆ,τ

dx µ(dτ) +O_p

q nh^d_x^X

=nh

Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−( ˆF_Y^ˆ^h_|X)⁻¹(τ|x) + ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β_ˆˆh(x)−ˆc^ˆ^h_ˆ

β^ˆ^h,τ

D_β g_β_ˆˆh(x) + ˆc^ˆ^h_ˆ

β^h^ˆ,τ

dx µ(dτ) +O_p q

nh^dx^X

(2.71)

= nh

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β_ˆˆh(x)−cˆ^ˆ^h_ˆ

β^ˆ^h,τ

D_βg_β_ˆˆh(x) + ˆc^ˆ^h_ˆ

β^ˆ^h,τ

dx µ(dτ) +O_p

q nh^d_x^X

The last equality in (2.78) follows from the definition of ˆβ^ˆ^h as the minimizer of T_n^h, which implies

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β_ˆˆh(x)−cˆ^h^ˆ_ˆ

β^ˆ^h,τ

D_β g_β_ˆˆh(x) + ˆc^h^ˆ_ˆ

β^ˆ^h,τ

dx µ(dτ) = 0.

Since Ω is positive definite, equation (2.78) leads to||βˆ^ˆ^h−β0||=O_p n⁻¹²h⁻

x 4

. The same assertion for ˆβ^h was already shown in the proof of Theorem 2.3.4.

As the minimizer ofG^ˆ^h(β), ¯β^ˆ^h it is determined by 0 =DβG^ˆ^h(β)

2.8. Proofs

=−2nh_x^dX² Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ

D_β(g_β₀(x) +c^h_β₀_,τ)dx µ(dτ) + 2nh

x2 (β−β0)^tnh

x2 Ω and consequently can be expressed as

β¯^ˆ^h =β₀+ Ω⁻¹ Z Z

v(x) ( ˆF_Y^h^ˆ_|X)⁻¹(τ|x)−(F_Y^h_|X)⁻¹(τ|x)

D_β(g_β₀(x) +c^h_β₀_,τ)^tdx µ(dτ)

=β₀+ Ω⁻¹ Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−(F_Y^h_|X)⁻¹(τ|x)

D_β(g_β₀(x) +c^h_β₀_,τ)^tdx µ(dτ) +O_p

√n

(2.79)

=β0−Ω⁻¹1 n

i=1

Z Z

v(x)

f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x)f_X(x)D_β(g_β₀(x) +c^h_β₀_,τ)^tK_h_x(x−Xi)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Yi))−p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)

dx µ(dτ) +O_p 1

√n

=β₀− 1 n

i=1

Z Z

κ(x)K_h_x(x−X_i)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y_i))

−p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)

dx µ(dτ) +O_p 1

√n

, where

κ(x) = Ω⁻¹ v(x)

f_Y^h_|X((F_Y^h_|X)⁻¹(τ|x)|x)f_X(x)Dβ(gβ0(x) +c^h_β₀_,τ)^t

is a (multidimensional) function with compact support. To show ||β¯^h^ˆ −β₀||=O_p ^√¹_n it is sufficient to prove

E 1

i=1

Z Z

κ_k(x)K_h_x(x−X_i)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y_i))− p^h((F_Y^h_|X)⁻¹(τ|x), x) fX(x)

dx µ(dτ) 2

=O 1

. for each component ˜κ_k of ˜κ,k= 1, ..., d_B. This in turn leads to analysing

Z Z

κk(x)Khx(x−X1)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y1))−p^h((F_Y^h_|X)⁻¹(τ|x), x) f_X(x)

dx µ(dτ) 2 and

E Z Z

κ_k(x)K_h_x(x−X₁)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y₁))−p^h((F_Y^h_|X)⁻¹(τ|x), x) fX(x)

dx µ(dτ) 2

For some sufficiently largeC >0 the first expectation can be bounded by E

Z Z

κ_k(x)K_h_x(x−X₁)

K_h_y((F_Y^h_|X)⁻¹(τ|x)−h(Y₁))

−p^h((F_Y^h_|X)⁻¹(τ|x), x) fX(x)

dx µ(dτ) 2

≤CE Z

|K_h_x(x−X₁)|dx 2

≤C Z

|K(x)|dx 2

while the second expectation can be treated as in (2.48). Finally,||β¯^ˆ^h−β₀||=O_p ^√¹

has been proven. Additionally, due to (2.79) it was shown that

||β¯^h−β0||

Ω⁻¹ Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−(F_Y^h_|X)⁻¹(τ|x)

D_β g_β₀(x) +c^h_β₀_,τt

dx µ(dτ)

=O_p 1

√n

Putting Things together

Let β = (β_n)n∈N be a sequence in B with β−β₀ = O_p n⁻¹²h⁻

x 4

. Then, as in (2.59) a Taylor expansion of β 7→ ( ˆF_Y^ˆ^h_|X)⁻¹ −gβ(x)−cβ,τ

and the binomial formula yield for someβ^∗ between ˆβ and β0

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β(x)−ˆc^ˆ^h_β,τ2

dx µ(dτ)

=nh

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β₀(x)−ˆc^ˆ^h_β₀_,τ2

dx µ(dτ)

−2nh

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−gβ0(x)−ˆc^ˆ^h_β₀_,τ Dβ gβ0(x) + ˆc^ˆ^h_β₀_,τ

dx µ(dτ)(β−β0) +nh

x2 (β−β0)^tΩ(β−β0) +op(1)

=nh

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ2

dx µ(dτ)

+nh

Z Z

v(x)(c^h_β₀_,τ −ˆc^ˆ^h_β₀_,τ)²dx µ(dτ) + 2nh

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β₀(x)−c^h_β₀_,τ

(c^h_β₀_,τ −ˆc^ˆ^h_β₀_,τ)dx µ(dτ)

−2nh

Z Z

v(x) ( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−gβ0(x)−c^h_β₀_,τ

2.8. Proofs

D_β g_β₀(x) +c^h_β₀_,τ

dx µ(dτ)(β−β0) +nh

x2 (β−β0)^tΩ(β−β0) +op(1)

= ˜T_n^h+G^ˆ^h(β) +op(1) (2.80)

and nh

Z Z

v(x) ( ˆF_Y^h_|X)⁻¹(τ|x)−gβ(x)−c^h_β,τ2

dx µ(dτ) = ˜T_n^h+G^h(β) +op(1).

Note that in contrast to the proof of Theorem 2.3.4, equation (2.74) leads to asymptotic negligibility of the terms containingc^h_β

0,τ−ˆc^ˆ^h_β

0,τ =O_p n⁻¹²

. Due to (2.75) and (2.76), one has

T_n^ˆ^h = ˜T_n^h+G^ˆ^h( ˆβ^ˆ^h) +op(1)

≥T˜_n^h+G^ˆ^h( ¯β^ˆ^h) +op(1)

=nh

Z Z v(x)

( ˆF_Y^ˆ^h_|X)⁻¹(τ|x)−g_β_¯hˆ(x)−cˆ^ˆ^h_¯

β^ˆ^h,τ

dx µ(dτ) +op(1)

≥T_n^ˆ^h+op(1)

so that it suffices to consider the minimum ofβ 7→T˜_n^h+G^ˆ^h(β) in (2.80). Therefore, T_n^ˆ^h is asymptotically equivalent to

T˜_n^h+G^ˆ^h( ¯β^ˆ^h)

= ˜T_n^h−2nh

Z Z

v(x) ( ˆF_Y^h^ˆ_|X)⁻¹(τ|x)−gβ0(x)−c^h_β₀_,τ Dβ(gβ0(x) +c^h_β₀_,τ)dx µ(dτ)( ¯β^ˆ^h−β0) +nh

x2 ( ¯β^ˆ^h−β0)^tΩ( ¯β^ˆ^h−β0)

= ˜T_n^h−nh

x2 ( ¯β^ˆ^h−β0)^tΩ( ¯β^ˆ^h−β0)

(2.76)

= T˜_n^h+O_p h

x2 .

Recall that H₀ was assumed, that is δ_n = δ_1,n = δ_2,n =δ_3,n = 0 with δ_1,n, δ_2,n, δ_3,n from Remark 2.3.5. Hence, a similar reasoning to that from above for T_n^h (with ¯β^h instead of β¯^ˆ^h) leads to T_n^h = ˜T_n^h+o_p(1), so that

T_n^ˆ^h = ˜T_n^h+op(1) =T_n^h+op(1).

2.8.7 Proof of Theorem 2.4.3

It will be started with the first assertion. ˆF_Y^ˆ^h_|X(y|x) was defined as Fˆ_Y^ˆ^h_|X(y|x) = pˆ^ˆ^h(y, x)

fˆ_X(x) with pˆ^ˆ^h(y, x) = 1 n

i=1

K_h_y(y−h(Yˆ i))Khx(x−Xi).

LetK= [k1, k2] be compact and δ >0. One has sup

y∈[k1−δ,k2+δ]

|h(y)ˆ −h(y)|=op(1).

Letδn&0 be the monotonic sequence from Lemma 1.5.1 with sup

y∈[k1−δ,k2+δ]

|ˆh(y)−h(y)|=op(δn).

Then, the results of Hansen (2008) can be adjusted as later in (4.6.10) to obtain sup

x∈supp(v),y∈K

|ˆp^ˆ^h(y, x)−pˆ^h(y, x)|

= sup

x∈supp(v),y∈K

1 n

i=1

K_h_y(y−h(Yi) +h(Yi)−ˆh(Yi))− K_h_y(y−h(Yi))Khx(x−Xi)

≤ sup

x∈supp(v),y∈K

1 n

i=1

K_h_y(y−h(Yi) +h(Yi)−ˆh(Yi))− K_h_y(y−h(Yi))

|K_h_x(x−Xi)|

≤ sup

x∈supp(v),y∈K

1 n

i=1

Z ^y+δn−h(_hy ^Yi⁾

y−δn−h(Yi) hy

|K(u)|du|K_h_x(x−X_i)|+o_p(1)

= sup

x∈supp(v),y∈K

Z ^y+δn−h(Y_hy ¹⁾

y−δn−h(Y1) hy

|K(u)|du|K_h_x(x−X₁)|

+o_p(1)

= sup

x∈supp(v),y∈K

Z Z Z

y+δn−gβ0(w)−c0−e hy y−δn−gβ0(w)−c0−e

|K(u)|du|K_h_x(x−w)|f_X(w)f_ε(e)de dw+o_p(1)

= sup

x∈supp(v),y∈K

Z Z Z _y+δ_n−g_β₀(w)−c0−hyu y−δ_n−g_β

0(w)−c₀−h_yu

f_ε(e)de|K(u)|du|K_h_x(x−w)|f_X(w)dw +o_p(1)

= sup

x∈supp(v),y∈K

Z Z

F_ε(y+δ_n−g_β₀(x−h_xw)−c₀−h_yu)

−F_ε(y−δ_n−g_β₀(x−h_xw)−c₀−h_yu)

|K(u)|du|K(w)|f_X(x−h_xw)dw+o_p(1)

=op(1) and thus

sup

x∈supp(v),y∈K

Fˆ_Y^h^ˆ_|X(y|x)−Fˆ_Y^h_|X(y|x)

=o_p(1).

The rest of the proof of the first assertion was already given in the proof of Theorem 2.3.6.

With the reasoning from above the proof of the second part directly follows from Theorem

2.4.1 and Slutsky’s theorem.

2.8.8 Proof of Theorem 2.5.1

Since the support ofv is compact, the results of Hansen (2008) yield fˆ_X(x)−f_X(x) =O_p

slog(n) nh^dx^X

2.8. Proofs uniformly in x∈supp(v). Due tofX(x)>0 for all x∈supp(v), this leads to

1 n

i=1

v(X_i) fˆ_X(X_i)² = 1

i=1

v(X_i)

fX(Xi)² +O_p

slog(n) nh^dx^X

v(X₁) fX(X1)²

+O_p

slog(n) nh^dx^X

Z v(w)

f_X(w)dw+o h

and analogously to (2.33) and _n¹Pn

i=1v(Xi) =R

v(w)fX(w)dw+o h

. The numerator in (2.35) can be treated similarly to the proof of Lemma 2.8.1. To this end, recall

(F_Y^h_|X)⁻¹ 1

2|X_i

−h(Yi) =F_ε⁻¹ 1

−εi. The results of Hansen (2008) imply

1 n

i=1

v(X_i)K_h_ε

(F_Y^h_|X)⁻¹1 2|X_i

−h(Y_i)

= 1 n

i=1

v(X_i)K_h_ε

F_ε⁻¹1 2

−ε_i

=fε

F_ε⁻¹

1 2

v(w)fX(w)dw+op h

, that is, is suffices to show

1 n

i=1

v(Xi)Khε

( ˆF_Y^ˆ^h_|X)⁻¹ 1

2|X_i

−ˆh(Yi)

= 1 n

i=1

v(Xi)Khε

(F_Y^h_|X)⁻¹ 1

2|X_i

−h(Yi)

+op h

Since the set n

( ˆF_Y^h_|X)⁻¹ 1

2|x

: x ∈ supp(v) o

is bounded, there exists a compact set K, such that

P 1 n

i=1

v(X_i)K_h_ε

( ˆF_Y^ˆ^h_|X)⁻¹1 2|X_i

−h(Yˆ _i)

= 1 n

i=1

v(X_i)I_{Y_i_∈K}K_h_ε

( ˆF_Y^ˆ^h_|X)⁻¹1 2|X_i

−h(Yˆ _i) !

→1 and

P 1 n

i=1

v(X_i)K_h_ε

(F_Y^h_|X)⁻¹1 2|X_i

−h(Y_i)

= 1 n

i=1

v(X_i)I_{Y_i_∈K}K_h_ε

(F_Y^h_|X)⁻¹1 2|X_i

−h(Y_i) !

→1.

Hence, one has for some appropriate C >0, y^∗_i ∈R

1 n

i=1

v(Xi)Kh_ε

( ˆF_Y^ˆ^h_|X)⁻¹1 2|Xi

−ˆh(Yi)

− 1 n

i=1

v(Xi)Kh_ε

(F_Y^h_|X)⁻¹1 2|Xi

−h(Yi)

= 1 n

i=1

v(Xi)I_{Y_i_∈K}

Kh_ε

( ˆF_Y^ˆ^h_|X)⁻¹1 2|Xi

−ˆh(Yi)

−Kh_ε

(F_Y^h_|X)⁻¹1 2|Xi

−h(Yi)

+op h

≤ 1 n

i=1 r−1

j=1

v(X_i)I_{Y_i_∈K}

∂^j

∂y^jK_h_ε(y) _y=(Fh

Y|X)⁻¹(¹₂|X_i)−h(Y_i)

( ˆF_Y^ˆ^h_|X)⁻¹1 2|Xi

−( ˆF_Y^h_|X)⁻¹1 2|Xi

+h(Yi)−ˆh(Yi) ^j

+ 1 n

i=1

v(Xi)I_{Y_i_∈K}

∂^r

∂y^rK_h_ε(y) _y=y∗

( ˆF_Y^ˆ^h_|X)⁻¹1 2|Xi

−(F_Y^h_|X)⁻¹1 2|Xi

+h(Y_i)−ˆh(Y_i) ^r

+o_p hx^dX²

≤

r−1

j=1

C h^jε

sup

x∈supp(v)

( ˆF_Y^ˆ^h_|X)⁻¹1 2|x

−(F_Y^h_|X)⁻¹1 2|x

+ sup

y∈K

|ˆh(y)−h(y)|^j

1 nh^jε

i=1

v(X_i)

∂^j

∂y^jK(y)

_y=^Fε⁻¹( 12)−εi hε

+ C

h^r+1ε

sup

x∈supp(v)

( ˆF_Y^ˆ^h_|X)⁻¹1 2|x

−(F_Y^h_|X)⁻¹1 2|x

+ sup

y∈K

|ˆh(y)−h(y)|^r

+o_p hx^dX²

=Op

nh⁴_ε⁻¹₄

+ n^rh^4(r+1)_ε ⁻¹₄

=o_p hx^dX² .

3

Identification in a Fully

Nonparametric Transformation Model with Heteroscedasticity

The underlying question of this Chapter can be formulated quite easily: Given some real valued random variableY and someR^d^X-valued random variableX fulfilling the heterosce-dastic transformation model

h(Y) =g(X) +σ(X)ε (3.1)

with some error termεfulfillingε⊥X, E[ε] = 0 and Var(ε) = 1, are the model components h:R→R, g:R^d^X →R, σ:R^d^X →(0,∞) and the error distribution uniquely determined if the joint distribution of (Y, X) is known? This uniqueness is called identification of a model.

Already Box and Cox (1964), Bickel and Doksum (1981) and Zellner and Revankar (1969) introduced some parametric classes of transformation models. Horowitz (1996) proved for a linear regression function g and homoscedastic errors that the model is identified, when h(y₀) = 0 is assumed for somey₀ ∈Rand the regression parameter is standardized so that the first component, which is different from zero, is equal to one. Later, the ideas of Horowitz (1996) were extended by Ekeland et al. (2004) to general smooth regression functionsg. The arguably most general identification results so far were provided by Chiappori et al. (2015) and Vanhems and Van Keilegom (2019), who considered general regression functions and homoscedastic errors as well, but allowed endogenous regressors. Linton et al. (2008) used similar ideas to obtain identifiability of a model with parametric transformation functions as a special case. Results allowing heteroscedasticity are rare. Zhou et al. (2009) showed identifiability in some kind of single-index model with a linear regression function g and a known variance function σ. Neumeyer et al. (2016) assumed identifiability implicitly by their assumption (a7).

In contrast to the approaches mentioned above, it is tried here to avoid any parametric assumption onh, gorσ, which to the author’s knowledge has not been done in the literature so far. Note that the validity of the model is unaffected by linear transformations. This means that for arbitrary constantsa >0, b∈Requation (3.1) still holds when replacingh,

gand σ by

˜h(y) =ah(y) +b,

g(x) =ag(x) +b,

σ(x) =aσ(x).

Of course, one could have chosen an arbitrary a ∈ R as well, but as in Section 1.4 the transformation functionhwill be restricted to be strictly increasing. Nevertheless, at least two conditions for fixing a and b are needed. Referring to the fact that these conditions will determine the linear transformation they are sometimes called location and scale con-straints.

This chapter is organized as follows. First, some differences to the homoscedastic case (that is,σ ∈R is constant) are pointed out, before the main identification result for heterosce-dastic transformation models as in (3.1) is presented. The chapter is completed by a brief discussion in 3.3. The proof of the main result is given in 3.5 and some additional remarks are postponed to 3.6.

Im Dokument Nonparametric Transformation Models (Seite 54-108)