Supplement to “Inference for Subvectors and Other Functions of
Partially Identified Parameters in Moment Inequality Models”∗
Federico A. Bugni
Ivan A. Canay
Department of Economics
Department of Economics
Duke University
Northwestern University
[email protected]
[email protected]
Xiaoxia Shi
Department of Economics
University of Wisconsin, Madison
[email protected]
February 12, 2016
Abstract
This document provides auxiliary lemmas and their proofs for the authors’ paper “Inference for
Subvectors and Other Functions of Partially Identified Parameters in Moment Inequality Models”.
Keywords: Partial Identification, Moment Inequalities, Subvector Inference, Hypothesis Testing.
JEL Classification: C01, C12, C15.
∗ We thank Frank Schorfheide and two anonymous referees whose valuable suggestions helped greatly improve the paper.
We also thank Francesca Molinari, Elie Tamer and Azeem Shaikh for helpful comments. Bugni and Canay thank the National
Science Foundation for research support via grants SES-1123771 and SES-1123586, respectively. Takuya Ura and Zhonglin Li
provided excellent research assistance. Any and all errors are our own. First version: 27 January 2014, CeMMAP Working
Paper CWP05/14.
S.1
Notation
Throughout the Appendix we employ the notation defined in Bugni et al. (2016, Section A). For the reader’s
convenience, we restate the main elements in the table below.
P0
{F ∈ P : ΘI (F ) 6= ∅}
ΣF (θ)
VarF (m(W, θ))
DF (θ)
diag(ΣF (θ))
√
S( nEF [m(W, θ)] , ΣF (θ))
QF (θ)
ΘIln κn (F )
{θ ∈ Θ : QF (θ) ≤ ln κn }
Γn,F (λ)
{(θ, `) ∈ Θ(λ) × Rk : ` =
ΓSS
bn ,F (λ)
{(θ, `) ∈ Θ(λ) × R : ` =
ΓPR
n,F (λ)
{(θ, `) ∈ Θ(λ) × Rk : ` =
ΓDR
n,F (λ)
vn,j (θ)
ΩF (θ, θ0 )[j1 ,j2 ]
k
√
√
−1/2
nDF
(θ)EF [m(Wi , θ)]}
−1/2
bn DF (θ)EF [m(W, θ)]}
√ −1/2
κ−1
nDF (θ)EF [m(Wi , θ)]}
n
√ −1/2
κn
k
−1
{(θ, `) ∈ Θln
nDF (θ)EF [m(W, θ)]}
I,F (λ) × R : ` = κn
Pn
−1/2 −1
n
σF,j (θ) i=1 (mj (Wi , θ) − EF [mj (Wi , θ)]), j = 1, . . . , k
i
h
mj2 (W,θ 0 )−EF [mj2 (W,θ 0 )]
mj1 (W,θ)−EF [mj1 (W,θ)]
EF
0
σF,j (θ)
σF,j (θ )
1
2
Table 1: Important Notation
S.2
Auxiliary Theorems
PR
Theorem S.2.1. Let ΓPR
n,F (λ) be as in Table 1 and Tn (λ) be as in (4.6). Let {(λn , Fn ) ∈ L0 }n≥1 be a
u
(sub)sequence of parameters such that for some (ΓPR , Ω) ∈ S(Θ × Rk[±∞] ) × C(Θ2 ): (i) ΩFn → Ω and (ii)
H
PR
ΓPR
. Then, there exists a further subsequence {un }n≥1 of {n}n≥1 such that, along {Fun }n≥1 ,
n,Fn (λn ) → Γ
d
{TuPnR (λun )|{Wi }ni=1 } → J(ΓPR , Ω) ≡
inf
(θ,`)∈ΓPR
S(vΩ (θ) + `, Ω(θ)), a.s. ,
where vΩ : Θ → Rk is a tight Gaussian process with covariance (correlation) kernel Ω.
DR
PR
Theorem S.2.2. Let ΓPR
n,F (λ) and Γn,F (λ) be as in Table 1. Let Tn (λ) be as in (4.6) and define
T̃nDR (λ) ≡
inf
κn
θ∈Θ(λ)∩Θln
(F )
I
√ −1/2
nD̂n (θ)m̄n (θ)), Ω̂n (θ)) ,
S(vn∗ (θ) + ϕ∗ (κ−1
n
(S.1)
κn
where vn∗ (θ) is as in (2.8), ϕ∗ (·) as in Assumption A.1, and Θln
(F ) as in Table 1. Let {(λn , Fn ) ∈ L0 }n≥1
I
u
be a (sub)sequence of parameters such that for some (ΓDR , ΓPR , Ω) ∈ S(Θ × Rk[±∞] )2 × C(Θ2 ): (i) ΩFn → Ω,
H
H
DR
PR
(ii) ΓDR
, and (iii) ΓPR
. Then, there exists a further subsequence {un }n≥1 of
n,Fn (λn ) → Γ
n,Fn (λn ) → Γ
{n}n≥1 such that, along {Fun }n≥1 ,
d
(λun ), TuPnR (λun )}|{Wi }ni=1 } → J(ΓMR , Ω) ≡
{min{T̃uDR
n
1
inf
(θ,`)∈ΓMR
S(vΩ (θ) + `, Ω(θ)), a.s. ,
where vΩ : Θ → Rk is a tight Gaussian process with covariance (correlation) kernel Ω,
PR
ΓMR ≡ ΓDR
∗ ∪Γ
and
ΓDR
≡ {(θ, `) ∈ Θ × Rk[±∞] : ` = ϕ∗ (`0 ) for some (θ, `0 ) ∈ ΓDR } .
∗
(S.2)
SS
Theorem S.2.3. Let ΓSS
bn ,F (λ) be as in Table 1 and Tbn (λ) be the subsampling test statistic. Let {(λn , Fn ) ∈
u
L0 }n≥1 be a (sub)sequence of parameters such that for some (ΓSS , Ω) ∈ S(Θ × Rk[±∞] ) × C(Θ2 ): (i) ΩFn → Ω
H
SS
and (ii) ΓSS
bn ,Fn (λn ) → Γ . Then, there exists a further subsequence {un }n≥1 of {n}n≥1 such that, along
{Fun }n≥1 ,
d
{TuSS
(λun )|{Wi }ni=1 } → J(ΓSS , Ω) ≡
n
inf
(θ,`)∈ΓSS
S(vΩ (θ) + `, Ω(θ, θ)), a.s. ,
where vΩ : Θ → Rk is a tight Gaussian process with covariance (correlation) kernel Ω.
Theorem S.2.4. Let Γn,F (λ) be as in Table 1 and Tn (λ) be as in (4.1). Let {(λn , Fn ) ∈ L0 }n≥1 be a
u
(sub)sequence of parameters such that for some (Γ, Ω) ∈ S(Θ × Rk[±∞] ) × C(Θ2 ): (i) ΩFn → Ω and (ii)
H
Γn,Fn (λn ) → Γ. Then, there exists a further subsequence {un }n≥1 of {n}n≥1 such that, along {Fun }n≥1 ,
d
Tun (λun ) → J(Γ, Ω) ≡ inf S(vΩ (θ) + `, Ω(θ)), as n → ∞ ,
(θ,`)∈Γ
where vΩ : Θ → Rk is a tight Gaussian process with zero-mean and covariance (correlation) kernel Ω.
S.3
Auxiliary Lemmas
u
Lemma S.3.1. Let {Fn ∈ P}n≥1 be a (sub)sequence of distributions s.t. ΩFn → Ω for some Ω ∈ C(Θ2 ).
Then,
d
1. vn → vΩ in l∞ (Θ), where vΩ : Θ → Rk is a tight zero-mean Gaussian process with covariance (correlation) kernel Ω. In addition, vΩ is a uniformly continuous function, a.s.
p
2. Ω̂n → Ω in l∞ (Θ).
−1/2
1/2
p
1/2
p
3. DFn (·)D̂n (·) − Ik → 0k×k in l∞ (Θ).
−1/2
4. D̂n
(·)DFn (·) − Ik → 0k×k in l∞ (Θ).
p
∞
5. For any arbitrary sequence {an ∈ R++ }n≥1 s.t. an → ∞, a−1
n vn → 0k in l (Θ).
p
∞
6. For any arbitrary sequence {an ∈ R++ }n≥1 s.t. an → ∞, a−1
n ṽn → 0k in l (Θ).
d
7. {vn∗ |{Wi }ni=1 } → vΩ in l∞ (Θ) a.s., where vΩ is the tight Gaussian process described in part 1.
d
8. {ṽbSS
|{Wi }ni=1 } → vΩ in l∞ (Θ) a.s., where
n
ṽbSS
(θ)
n
bn
1 X
−1/2
D
(θ)(m(WiSS , θ) − m̄n (θ)) ,
≡√
bn i=1 Fn
(S.1)
n
{WiSS }bi=1
is a subsample of size bn from {Wi }ni=1 , and vΩ is the tight Gaussian process described in
part 1.
2
−1/2
p
−1/2
SS
SS
n
∞
9. For Ω̃SS
bn (θ) ≡ DFn (θ)Σ̂bn (θ)DFn (θ), {Ω̃bn |{Wi }i=1 } → Ω in l (Θ) a.s.
Lemma S.3.2. For any sequence {(λn , Fn ) ∈ L}n≥1 there exists a subsequence {un }n≥1 of {n}n≥1 s.t.
u
H
H
H
PR
DR
ΩFun → Ω, Γun ,Fun (λun ) → Γ, ΓPR
, and ΓDR
, for some (Γ, ΓDR , ΓPR , Ω) ∈
un ,Fun (λun ) → Γ
un ,Fun (λun ) → Γ
PR
S(Θ × Rk[±∞] )3 × C(Θ2 ), where Γn,Fn (λ), ΓDR
n,Fn (λ), and Γn,Fn (λ) are defined in Table 1.
Lemma S.3.3. Let {Fn ∈ P}n≥1 be an arbitrary (sub)sequence of distributions and let Xn (θ) : Ω → l∞ (Θ)
p
be any stochastic process such that Xn → 0 in l∞ (Θ). Then, there exists a subsequence {un }n≥1 of {n}n≥1
a.s.
such that Xun → 0 in l∞ (Θ).
Lemma S.3.4. Let the set A be defined as follows:
A≡
x∈
Rp[+∞]
×R
k−p
: max
max {[xj ]− },
j=1,...,p
max
s=p+1,...,k
{|xs |}
=1 .
(S.2)
Then, inf (x,Ω)∈A×Ψ S(x, Ω) > 0.
Lemma S.3.5. If S(x, Ω) ≤ 1 then there exist a constant $ > 0 such that xj ≥ −$ for all j ≤ p and
|xj | ≤ $ for all j > p.
Lemma S.3.6. The function S satisfies the following properties: (i) x ∈ (−∞, ∞]p ×Rk−p implies supΩ∈Ψ S(x, Ω) <
∞, (ii) x 6∈ (−∞, ∞]p × Rk−p implies inf Ω∈Ψ S(x, Ω) = ∞.
u
H
Lemma S.3.7. Let (Γ, ΓDR , Ω) ∈ S(Θ × Rk[±∞] )2 × C(Θ2 ) be such that ΩFn → Ω, Γn,Fn (λn ) → Γ, and
H
DR
ΓDR
, for some {(λn , Fn ) ∈ L0 }n≥1 . Then, Assumptions A.1 and A.3 imply that for all
n,Fn (λn ) → Γ
DR
˜ ∈ Γ with `˜j ≥ ϕ∗ (`j ) for j ≤ p and `˜j = `j ≡ 0 for j > p, where ϕ∗ (·) is
(θ, `) ∈ Γ
there exists (θ, `)
j
defined in Assumption A.1.
u
H
Lemma S.3.8. Let (Γ, ΓPR , Ω) ∈ S(Θ × Rk[±∞] )2 × C(Θ2 ) be such that ΩFn → Ω, Γn,Fn (λn ) → Γ, and
H
PR
, for some {(λn , Fn ) ∈ L0 }n≥1 . Then, Assumption A.3 implies that for all (θ, `) ∈ ΓPR
ΓPR
n,Fn (λn ) → Γ
˜ ∈ Γ with `˜j ≥ `j for j ≤ p and `˜j = `j for j > p.
with ` ∈ Rp[+∞] × Rk−p , there exists (θ, `)
Lemma S.3.9. Let Assumptions A.3-A.5 hold. For λ0 ∈ Γ and {λn ∈ Γ}n≥1 as in Assumption A.5,
u
H
H
H
H
PR
SS
PR
PR
assume that ΩFn → Ω, Γn,Fn (λ0 ) → Γ, ΓPR
, ΓSS
n,Fn (λ0 ) → Γ
bn ,Fn (λ0 ) → Γ , Γn,Fn (λn ) → ΓA , and
H
SS
SS
PR
PR
k
5
2
ΓSS
, ΓSS
A , ΓA , Ω) ∈ S(Θ × R[±∞] ) × C(Θ ). Then,
bn ,Fn (λn ) → ΓA for some (Γ, Γ , Γ
c(1−α) (ΓPR , Ω) ≤ c(1−α) (ΓSS , Ω) .
Lemma S.3.10. Let Assumptions A.3-A.7 hold. Then,
R
SS
lim inf (EFn [φP
n (λ0 )] − EFn [φn (λ0 )]) > 0 .
n→∞
S.4
Proofs of Theorems in Section S.2
PR
Proof of Theorem S.2.1. Step 1. To simplify expressions, let ΓPR
n ≡ Γn,Fn (λn ). Consider the following deriva-
tion,
TnP R (λn ) =
inf
θ∈Θ(λn )
√ −1/2
S vn∗ (θ) + µn,1 (θ) + µn,2 (θ)0 κ−1
nDFn (θ)EFn [m(W, θ)], Ω̂n (θ)
n
3
=
inf
(θ,`)∈ΓPR
n
S vn∗ (θ) + µn,1 (θ) + µn,2 (θ)0 `, Ω̂n (θ) ,
−1
k
where µn (θ) = (µn,1 (θ), µn,2 (θ)), µn,1 (θ) ≡ κ−1
n ṽn (θ), µn,2 (θ) ≡ {σ̂n,j (θ)σFn ,j (θ)}j=1 , and ṽn (θ) ≡
√ −1
−1/2
1/2
nD̂n (θ)(m̄n (θ) − EF [m(W, θ)]). Note that D̂n (θ) and DFn (θ) are both diagonal matrices.
d
an
}→
Step 2. We now show that there is a subsequence {an }n≥1 of {n}n≥1 s.t. {(va∗n , µan , Ω̂an )|{Wi }i=1
d
(vΩ , (0k , 1k ), Ω) in l∞ (θ) a.s. By part 8 in Lemma S.3.1, {vn∗ |{Wi }ni=1 } → vΩ in l∞ (θ). Then the result
n
} → ((0k , 1k ), Ω) in
would follow from finding a subsequence {an }n≥1 of {n}n≥1 s.t. {(µan , Ω̂an )|{Wi }ai=1
l∞ (θ) a.s. Since (µn , Ω̂n ) is conditionally non-random, this is equivalent to finding a subsequence {an }n≥1
a.s.
of {n}n≥1 s.t. (µan , Ω̂an ) → ((0k , 1k ), Ω) in l∞ (θ). In turn, this follows from step 1, part 5 of Lemma S.3.1,
and Lemma S.3.3.
Step 3. Since ΘI (Fn , λn ) 6=
√ −1
nσFn ,j (θn )EFn [mj (W, θn )],
∅, there is a sequence {θn
∈
Θ(λn )}n≥1 s.t.
for `n,j
≡
κ−1
n
lim sup `n,j ≡ `¯j ≥ 0,
for j ≤ p,
and
n→∞
lim |`n,j | ≡ `¯j = 0,
n→∞
for j > p .
(S.1)
¯ →0
By compactness of (Θ × Rk[±∞] , d), there is a subsequence {kn }n≥1 of {an }n≥1 s.t. d((θkn , `kn ), (θ̄, `))
p
¯ ∈ Θ×R
for some (θ̄, `)
+,∞ × 0k−p . By step 2, lim(vkn (θkn ), µkn (θkn ), Ωkn (θkn )) = (vΩ (θ̄), (0k , 1k ), Ω(θ̄)),
and so
¯ Ω(θ̄)) ,
TkPnR (λkn ) ≤ S(vkn (θkn ) + µkn ,1 (θkn ) + µkn ,2 (θkn )0 `kn , Ωkn (θkn )) → S(vΩ (θ̄) + `,
(S.2)
where the convergence occurs because by the continuity of S(·) and the convergence of its argument. Since
k−p
¯ Ω(θ̄)) ∈ Rp
¯ Ω(θ̄)) is bounded.
(vΩ (θ̄) + `,
× Ψ, we conclude that S(vΩ (θ̄) + `,
[+∞] × R
Step 4. Let D denote the space of functions that map Θ onto Rk × Ψ and let D0 be the space of uniformly
continuous functions that map Θ onto Rk × Ψ. Let the sequence of functionals {gn }n≥1 with gn : D → R
given by
gn (v(·), µ(·), Ω(·)) ≡
inf
S(v(θ) + µ1 (θ) + µ2 (θ)0 `, Ω(θ)) .
inf
S(v(θ) + µ1 (θ) + µ2 (θ)0 `, Ω(θ)) .
(θ,`)∈ΓPR
n
(S.3)
Let the functional g : D0 → R be defined by
g(v(·), µ(·), Ω(·)) ≡
(θ,`)∈ΓPR
We now show that if the sequence of (deterministic) functions {(vn (·), µn (·), Ωn (·)) ∈ D}n≥1 satisfies
lim sup ||(vn (θ), µn (θ), Ωn (θ)) − (v(θ), (0k , 1k ), Ω(θ))|| = 0 ,
n→∞ θ∈Θ
(S.4)
for some (v(·), Ω(·)) ∈ D0 , then limn→∞ gn (vn (·), µn (·), Ωn (·)) = g(v(·), (0k , 1k ), Ω(·)). To prove this we
show that lim inf n→∞ gn (vn (·), µn (·), Ωn (·)) ≥ g(v(·), (0k , 1k ), Ω(·)). Showing the reverse inequality for the
lim sup is similar and therefore omitted. Suppose not, i.e., suppose that ∃δ > 0 and a subsequence {an }n≥1
of {n}n≥1 s.t. ∀n ∈ N,
gan (van (·), µan (·), Ωan (·)) < g(v(·), (0k , 1k ), Ω(·)) − δ .
4
(S.5)
By definition, ∃ {(θan , `an )}n≥1 that approximates the infimum in (S.3), i.e., ∀n ∈ N, (θan , `an ) ∈ ΓPR
an and
|gan (van (·), µan (·), Ωan (·)) − S(van (θan ) + µ1 (θan ) + µ2 (θan )0 `an , Ωan (θan ))| ≤ δ/2 .
(S.6)
k
k
Since ΓPR
an ⊆ Θ × R[±∞] and (Θ × R[±∞] , d) is a compact metric space, there exists a subsequence {un }n≥1 of
{an }n≥1 and (θ∗ , `∗ ) ∈ Θ × Rk[±∞] s.t. d((θun , `un ), (θ∗ , `∗ )) → 0. We first show that (θ∗ , `∗ ) ∈ ΓPR . Suppose
not, i.e. (θ∗ , `∗ ) 6∈ ΓPR , and consider the following argument
PR
d((θun , `un ), (θ∗ , `∗ )) + dH (ΓPR
) ≥
un , Γ
≥
d((θun , `un ), (θ∗ , `∗ )) +
inf
(θ,`)∈ΓPR
inf
(θ,`)∈ΓPR
d((θ, `), (θun , `un ))
d((θ, `), (θ∗ , `∗ )) > 0 ,
where the first inequality follows from the definition of Hausdorff distance and the fact that (θun , `un ) ∈ ΓPR
un
, and the second inequality follows by the triangular inequality. The final strict inequality follows from
the fact that ΓPR ∈ S(Θ × Rk[±∞] ), i.e., it is a compact subset of (Θ × Rk[±∞] , d), d((θ, `), (θ∗ , `∗ )) is a
continuous real-valued function, and Royden (1988, Theorem 7.18). Taking limits as n → ∞ and using that
H
PR
R
, we reach a contradiction.
d((θun , `un ), (θ∗ , `∗ )) → 0 and ΓP
un → Γ
We now show that `∗ ∈ Rp[+∞] × Rk−p . Suppose not, i.e., suppose that ∃j = 1, . . . , k s.t. lj∗ = −∞
or ∃j > p s.t. `∗j = ∞. Let J denote the set of indices j = 1, . . . , k s.t. this occurs. For any ` ∈ Rk[±∞]
k
define Ξ(`) ≡ maxj∈J ||`j ||. By definition of ΓPR
un ,Fun , `un ∈ R and thus, Ξ(`un ) < ∞. By the case under
consideration, lim Ξ(`un ) = Ξ(`∗ ) = ∞. Since (Θ, || · ||) is a compact metric space, d((θun , `un ), (θ∗ , `∗ )) → 0
implies that θun → θ∗ . Then,
||(vun (θun ), µun (θun ), Ωun (θun )) − (v(θ∗ ), (0k , 1k ), Ω(θ∗ ))||
≤ ||(vun (θun ), µun (θun ), Ωun (θun )) − (v(θun ), (0k , 1k ), Ω(θun ))|| + ||(v(θun ), Ω(θun )) − (v(θ∗ ), Ω(θ∗ ))||
≤ sup ||(vun (θ), µun (θ), Ωun (θ)) − (v(θ), (0k , 1k ), Ω(θ))|| + ||(v(θun ), Ω(θun )) − (v(θ∗ ), Ω(θ∗ ))|| → 0 ,
θ∈Θ
where the last convergence holds by (S.4), θun → θ∗ , and (v(·), Ω(·)) ∈ D0 .
Since (v(·), Ω(·)) ∈ D0 , the compactness of Θ implies that (v(θ∗ ), Ω(θ∗ ) is bounded. Since lim Ξ(`un ) =
Ξ(`∗ ) = ∞ and lim vun (θun ) = v(θ∗ ) ∈ Rk , it then follows that lim Ξ(`un )−1 ||vun (θun )|| = 0. By construction,
{Ξ(`un )−1 `un }n≥1 is s.t. lim Ξ(`un )−1 [`un ,j ]− = 1 for some j ≤ p or lim Ξ(`un )−1 |`un ,j | = 1 for some j > p.
By this, it follows that {Ξ(`un )−1 (vun (θun ) + `un ), Ωun (θun )}n≥1 with lim Ωun (θun ) = Ω(θ∗ ) ∈ Ψ and
lim Ξ(`un )−1 [vun ,j (θun ) + `un ,j ]− = 1 for some j ≤ p or lim Ξ(hun )−1 |vun ,j (θun ) + `un ,j | = 1 for some j > p.
This implies that,
S(vun (θun ) + `un , Ωun (θun )) = Ξ(`un )χ S(Ξ(`un )−1 (vun (θun ) + `un ), Ωun (θun )) → ∞ .
Since {(θun , `un )}n≥1 is a subsequence of {(θan , `an )}n≥1 that approximately achieves the infimum in (S.3),
gn (vn (·), µn (·), Σn (·)) → ∞ .
However, (S.7) violates step 3 and is therefore a contradiction.
5
(S.7)
We then know that d((θan , `an ), (θ∗ , `∗ )) → 0 with `∗ ∈ Rp[+∞] × Rk−p . By repeating previous arguments,
we conclude that lim(vun (θun ), µun (θun ), Ωun (θun )) = (v(θ∗ ), (0k , 1k ), Ω(θ∗ )) ∈ Rk × Ψ. This implies that
lim(vun (θun ) + µun ,1 (θun ) + µun ,2 (θun )0 `un , Ωun (θun )) = (v(θ∗ ) + `∗ , Ω(θ∗ )) ∈ (Rk[±∞] × Ψ), i.e., ∃N ∈ N s.t.
∀n ≥ N ,
||S(vun (θun ) + µun ,1 (θun ) + µun ,2 (θun )0 `un , Ωun (θun )) − S(v(θ∗ ) + `∗ , Ω(θ∗ ))|| ≤ δ/2 .
(S.8)
By combining (S.6), (S.8), and the fact that (θ∗ , `∗ ) ∈ ΓPR , it follows that ∃N ∈ N s.t. ∀n ≥ N ,
gun (vun (·), µun (·), Ωun (·)) ≥ S(vΩ (θ∗ ) + `∗ , Ω(θ∗ )) − δ ≥ g(v(·), (0k , 1k ), Ω(·)) − δ ,
which is a contradiction to (S.5).
Step 5. The proof is completed by combining the representation in step 1, the convergence result in step
2, the continuity result in step 4, and the extended continuous mapping theorem (see, e.g., van der Vaart
and Wellner, 1996, Theorem 1.11.1). In order to apply this result, it is important to notice that parts 1 and
5 in Lemma S.3.1 and standard convergence results imply that (v(·), Ω(·)) ∈ D0 a.s.
PR
DR
Proof of Theorem S.2.2. Step 1. To simplify expressions let ΓPR
≡ ΓDR
n ≡ Γn,Fn (λn ), Γn
n,Fn (λn ), and con-
sider the following derivation,
min{T̃nDR (λn ), TnP R (λn )}
)
(
√ −1/2
nD̂n (θ)m̄n (θ)), Ω̂n (θ)),
inf θ∈Θ(λn )∩Θln κn (Fn ) S(vn∗ (θ) + ϕ∗ (κ−1
n
I
= min
√ −1/2
inf θ∈Θ(λn ) S(vn∗ (θ) + κ−1
nD̂n (θ)m̄n (θ), Ω̂n (θ))
n
(
)
√
−1/2
DFn (θ) n(EFn m(W, θ))), Ω̂n (θ)),
inf θ∈Θ(λn )∩Θln κn (Fn ) S(vn∗ (θ) + ϕ∗ (µn,1 (θ) + µn,2 (θ)0 κ−1
n
I
= min
√
−1/2
inf θ∈Θ(λn ) S(vn∗ (θ) + µn,1 (θ) + µn,2 (θ)0 κ−1
n DFn (θ) n(EFn m(W, θ)), Ω̂n (θ))
= min
inf S(vn∗ (θ) + ϕ∗ (µn,1 (θ) + µn,2 (θ)0 `), Ω̂n (θ)), inf S(vn∗ (θ) + µn,1 (θ) + µn,2 (θ)0 `, Ω̂n (θ))
(θ,`)∈ΓDR
n
(θ,`)∈ΓPR
n
−1/2
where µn (θ) ≡ (µn,1 (θ), µn,2 (θ)), µn,1 (θ) ≡ κ−1
n D̂n
√
(θ) n(m̄n (θ) − EFn m(W, θ)) ≡ κ−1
n ṽn (θ), and
−1/2
−1/2
−1
µn,2 (θ) ≡ {σn,j
(θ)σFn ,j (θ)}kj=1 . Note that we used that DFn (θ) and D̂n
(θ) are both diagonal ma-
trices.
n
Step 2. There is a subsequence {an }n≥1 of {n}n≥1 s.t. {(v̂a∗n , µan , Ω̂an )|{Wi }ai=1
} →d (vΩ , (0k , 1k ), Ω) in
l∞ (Θ) a.s. This step is identical to Step 2 in the proof of Theorem S.2.1.
Step 3. Let D denote the space of bounded functions that map Θ onto R2k × Ψ and let D0 be the space of
bounded uniformly continuous functions that map Θ onto R2k × Ψ. Let the sequence of functionals {gn }n≥1 ,
{gn1 }n≥1 , {gn2 }n≥1 with gn : D → R, gn1 : D → R, and gn2 : D → R be defined by
gn (v(·), µ(·), Ω(·))
≡ min gn1 (v(·), µ(·), Ω(·)), gn2 (v(·), µ(·), Ω(·))
gn1 (v(·), µ(·), Ω(·)) ≡
gn2 (v(·), µ(·), Ω(·)) ≡
inf
S(vn∗ (θ) + ϕ∗ (µn,1 (θ) + µn,2 (θ)0 `), Ω(θ))
inf
S(vn∗ (θ) + µn,1 (θ) + µn,2 (θ)0 `, Ω(θ)) .
(θ,`)∈ΓDR
n
(θ,`)∈ΓPR
n
6
Let the functional g : D0 → R, g 1 : D0 → R, and g 2 : D0 → R be defined by:
≡ min g 1 (v(·), µ(·), Ω(·)), g 2 (v(·), µ(·), Ω(·))
g(v(·), µ(·), Ω(·))
g 1 (v(·), µ(·), Ω(·))
≡
g 2 (v(·), µ(·), Ω(·)) ≡
inf
S(vΩ (θ) + ϕ∗ (µ1 (θ) + µ2 (θ)0 `), Ω(θ))
inf
S(vΩ (θ) + µ1 (θ) + µ2 (θ)0 `, Ω(θ)) .
(θ,`)∈ΓDR
(θ,l)∈ΓPR
If the sequence of deterministic functions {(vn (·), µn (·), Ωn (·))}n≥1 with (vn (·), µn (·), Ωn (·)) ∈ D for all n ∈ N
satisfies
lim sup ||(vn (θ), µn (θ), Ωn (θ)) − (vΩ (θ), (0k , 1k ), Ω(θ))|| = 0 ,
n→∞ θ∈Θ
for some (v(·), (0k , 1k ), Ω(·)) ∈ D0 then limn→∞ ||gns (vn (·), µn (·), Ωn (·)) − g s (v(·), (0k , 1k ), Ω(·))|| = 0 for
s = 1, 2, respectively. This follows from similar steps to those in the proof of Theorem S.2.1, step 4. By
continuity of the minimum function,
lim ||gn (vn (·), µn (·), Ωn (·)) − g(v(·), (0k , 1k ), Ω(·))|| = 0 .
n→∞
Step 4. By combining the representation of min{T̃nDR (λn ), TnP R (λn )} in step 1, the convergence results
in steps 2 and 3, Theorem S.2.1, and the extended continuous mapping theorem (see, e.g., Theorem 1.11.1
of van der Vaart and Wellner (1996)) we conclude that
d
PR
{min{T̃nDR (λn ), TnP R (λn )}|{Wi }ni=1 } → min J(ΓDR
, Ω) a.s.,
∗ , Ω), J(Γ
where
J(ΓDR
∗ , Ω) ≡
inf
(θ,`)∈ΓDR
∗
S(vΩ (θ) + `, Ω(θ)) =
inf
(θ,`0 )∈ΓDR
S(vΩ (θ) + ϕ∗ (`0 ), Ω(θ)) .
(S.9)
The result then follows by noticing that,
min
PR
J(ΓDR
, Ω)
∗ , Ω), J(Γ
=
=
min
inf
(θ,`)∈ΓDR
∗
inf
PR
(θ,`)∈ΓDR
∗ ∪Γ
S(vΩ (θ) + `, Ω(θ)),
inf
(θ,`)∈ΓPR
S(vΩ (θ) + `, Ω(θ))
S(vΩ (θ) + `, Ω(θ)) = J(ΓMR , Ω) .
This completes the proof.
Proof of Theorem S.2.3. This proof is similar to that of Theorem S.2.1. For the sake of brevity, we only
provide a sketch that focuses on the main differences. From the definition of TbSS
(λn ), we can consider the
n
following derivation,
TbSS
(λn ) ≡
n
=
inf
θ∈Θ(λn )
inf
QSS
bn (θ) =
(θ,`)∈ΓSS
bn
inf
θ∈Θ(λn )
S(
p
SS
bn m̄SS
bn (θ), Σ̂bn (θ))
S(ṽbSS
(θ) + µn (θ) + `, Ω̃SS
bn (θ)) ,
n
√
−1/2
bn DFn (θ)(m̄n (θ) − EFn [m(W, θ)]), ṽbSS
(θ)
n
−1/2
−1/2
SS
DFn (θ)Σ̂bn (θ)DFn (θ). From here, we can repeat the arguments
where µn (θ)
≡
7
is as in (S.1), and Ω̃SS
n (θ)
≡
used in the proof of Theorem S.2.1.
The main difference in the argument is that the reference to parts 2 and 7 in Lemma S.3.1 need to be
replaced by parts 9 and 8, respectively.
Proof of Theorem S.2.4. The proof of this theorem follows by combining arguments from the proof of Theorem S.2.1 with those from Bugni et al. (2014, Theorem 3.1). It is therefore omitted.
S.5
Proofs of Lemmas in Section S.3
We note that Lemmas S.3.2-S.3.5 correspond to Lemmas D3-D7 in Bugni et al. (2014) and so we do not
include the proofs of those lemmas in this paper.
Proof of Lemma S.3.1. The proof of parts 1-7 follow from similar arguments to those used in the proof of
Bugni et al. (2014, Theorem D.2). Therefore, we now focus on the proof of parts 8-9.
Part 9. By the argument used to prove Bugni et al. (2014, Theorem D.2 (part 1)), M(F ) ≡
−1/2
{DF (θ)m(·, θ)
: W → Rk } is Donsker and pre-Gaussian, both uniformly in F ∈ P. Thus, we can
extend the arguments in the proof of van der Vaart and Wellner (1996, Theorem 3.6.13 and Example 3.6.14)
to hold under a drifting sequence of distributions {Fn }n≥1 along the lines of van der Vaart and Wellner
(1996, Section 2.8.3). From this, it follows that:
r
n
d
{Wi }ni=1
→ vΩ (θ)
ṽbSS
(θ)
(n − bn ) n
in l∞ (Θ)
a.s.
(S.1)
To conclude the proof, note that,
s
r
n
bn /n
sup = sup kṽbSS
(θ)k
ṽbSS
(θ) − ṽbSS
(θ)
.
n
n
n
(n − bn )
(1 − bn /n)
θ∈Θ
θ∈Θ
In order to complete the proof, it suffices to show that the RHS of the previous equation is op (1) a.s. In
turn, this follows from bn /n = o(1) and (S.1) as they imply that {supθ∈Θ ||ṽbSS
(θ)|||{Wi }ni=1 } = Op (1) a.s.
n
Part 10. This result follows from considering the subsampling analogue of the arguments used to prove
Bugni et al. (2014, Theorem D.2 (part 2)).
Proof of Lemma S.3.6. Part 1. Suppose not, that is, suppose that supΩ∈Ψ S(x, Ω) = ∞ for x ∈ (−∞, ∞]p ×
Rk−p . By definition, there exists a sequence {Ωn ∈ Ψ}n≥1 s.t. S(x, Ωn ) → ∞. By the compactness of Ψ, there
exists a subsequence {kn }n≥1 of {n}n≥1 s.t. Ωkn → Ω∗ ∈ Ψ. By continuity of S on (−∞, ∞]p × Rk−p × Ψ it
then follows that lim S(x, Ωkn ) = S(x, Ω∗ ) = ∞ for (x, Ω∗ ) ∈ (−∞, ∞]p × Rk−p × Ψ, which is a contradiction
to S : (−∞, ∞]p × Rk−p → R+ .
Part 2. Suppose not, that is, suppose that supΩ∈Ψ S(x, Ω) = B < ∞ for x 6∈ (−∞, ∞]p × Rk−p . By
definition, there exists a sequence {Ωn ∈ Ψ}n≥1 s.t. S(x, Ωn ) → ∞. By the compactness of Ψ, there exists
a subsequence {kn }n≥1 of {n}n≥1 s.t. Ωkn → Ω∗ ∈ Ψ. By continuity of S on Rk[±∞] × Ψ it then follows
that lim S(x, Ωkn ) = S(x, Ω∗ ) = B < ∞ for (x, Ω∗ ) ∈ Rk[±∞] × Ψ. Let J ∈ {1, . . . k} be set of coordinates
s.t. xj = −∞ for j ≤ p or |xj | = ∞ for j > p . By the case under consideration, there is at least one
such coordinate. Define M ≡ max{maxj6∈J,j≤p [xj ]− , maxj6∈J,j>p |xj |} < ∞. For any C > M , let x0 (C) be
8
defined as follows. For j 6∈ J, set x0j (C) = xj and for j ∈ J, set x0j (C) as follows x0j (C) = −C for j ≤ p and
|x0j (C)| = C for j > p . By definition, limC→∞ x0 (C) = x and by continuity properties of the function S,
0
limC→∞ S(x (C), Ω∗ ) = S(x, Ω∗ ) = B < ∞. By homogeneity properties of the function S and by Lemma
S.3.4, we have that
S(x0 (C), Ω∗ ) = C χ S(C −1 x0 (C), Ω∗ ) ≥ C χ
inf
S(x, Ω) > 0,
(x,Ω)∈A×Ψ
where A is the set in Lemma S.3.4. Taking C → ∞ the RHS diverges to infinity, producing a contradiction.
Proof of Lemma S.3.7. The result follows from similar steps to those in Bugni et al. (2014, Lemma D.10)
and is therefore omitted.
Proof of Lemma S.3.8. Let (θ, `) ∈ ΓPR with ` ∈ Rp[+∞] × Rk−p . Then, there is a subsequence {an }n≥1
√ −1/2
nDFn (θn )EFn [m(W, θn )],
of {n}n≥1 and a sequence {(θn , `n )}n≥1 such that θn ∈ Θ(λn ), `n ≡ κ−1
n
u
limn→∞ `an = `, and limn→∞ θan = θ. Also, by ΩFn → Ω we get ΩFn (θn ) → Ω(θ). By continuity of
S(·) at (`, Ω(θ)) with ` ∈ Rp[+∞] × Rk−p ,
√
−1
χ/2
κ−χ
QFan (θan ) = S(κ−1
an an
an an σFan ,j (θan )EFan [mj (W, θan )], ΩFan (θan )) → S(`, Ω(θ)) < ∞ .
(S.2)
Hence QFan (θan ) = O(κχan an −χ/2 ). By this and Assumption A.3(a), it follows that
O(κχan an −χ/2 ) = c−1 QFan (θan ) ≥ min{δ,
inf
θ̃∈ΘI (Fan ,λan )
||θan − θ̃||}χ
⇒
√
||θan − θ̃an || ≤ O(κan / an ) ,
(S.3)
for some sequence {θ̃an ∈ ΘI (Fan , λan )}n≥1 . By Assumptions A.3(b)-(c), the intermediate value theorem
implies that there is a sequence {θn∗ ∈ Θ(λn )}n≥1 with θn∗ in the line between θn and θ̃n such that
√ −1/2
√
√ −1/2
κ−1
nDFn (θn )EFn [m(W, θn )] = GFn (θn∗ )κ−1
n(θn − θ̃n ) + κ−1
nDFn (θ̃n )EFn [m(W, θ̃n )] .
n
n
n
−1
−1
Define θ̂n ≡ (1 − κ−1
n )θ̃n + κn θn or, equivalently, θ̂n − θ̃n ≡ κn (θn − θ̃n ). We can write the above equation
as
√
√ −1/2
√ −1/2
GFn (θn∗ ) n(θ̂n − θ̃n ) = κ−1
nDFn (θn )EFn [m(W, θn )] − κ−1
nDFn (θ̃n )EFn [m(W, θ̃n )] .
n
n
By convexity of Θ(λn ) and κ−1
n → 0, {θ̂n ∈ Θ(λn )}n≥1 and by (S.3),
intermediate value theorem again, there is a sequence
{θn∗∗
√
(S.4)
an ||θ̂an − θ̃an || = O(1). By the
∈ Θ(λn )}n≥1 with θn∗∗ in the line between θ̂n and
θ̃n such that
√
√
√ −1/2
−1/2
nDFn (θ̂n )EFn [m(W, θ̂n )] = GFn (θn∗∗ ) n(θ̂n − θ̃n ) + nDFn (θ̃n )EFn [m(W, θ̃n )]
√
√ −1/2
= GFn (θn∗ ) n(θ̂n − θ̃n ) + nDFn (θ̃n )EFn [m(W, θ̃n )] + 1,n ,
(S.5)
√
where the second equality holds by 1,n ≡ (GFn (θn∗∗ ) − GFn (θn∗ )) n(θ̂n − θ̃n ). Combining (S.4) with (S.5)
9
we get
√
√ −1/2
−1/2
nDFn (θ̂n )EFn [m(W, θ̂n )] = κ−1
nDFn (θn )EFn [m(W, θn )] + 1,n + 2,n ,
n
(S.6)
√ −1/2
−1
→ 0, it
where 2,n ≡ (1 − κ−1
n ) nDFn (θ̃n )EFn [m(W, θ̃n )]. From {θ̃an ∈ ΘI (Fan , λan )}n≥1 and κn
follows that 2,an ,j ≥ 0 for j ≤ p and 2,an ,j = 0 for j > p. Moreover, Assumption A.3(c) implies that
||GFan (θa∗∗n ) − GFan (θa∗n )|| = o(1) for any sequence {Fan ∈ P0 }n≥1 whenever ||θa∗n − θa∗∗n || = o(1). Using
√
an ||θ̂an − θ̃an || = O(1), we have
√
||1,an || ≤ ||GFan (θa∗∗n ) − GFan (θa∗n )|| an ||θ̂an − θ̃an || = o(1) .
√
(S.7)
Finally, since (Rk[±∞] , d) is compact, there is a further subsequence {un }n≥1 of {an }n≥1 s.t.
√
−1/2
−1/2
un DFun (θ̂un )EFun [m(W, θ̂un )] and κ−1
un un DFun (θun )EFun [m(W, θun )] converge. Then, from (S.6), (S.7),
and the properties of 2,an we conclude that
√
√
−1
(θ̂un )EFun [mj (W, θ̂un )] ≥ lim κ−1
lim `˜un ,j ≡ lim un σF−1
un un σFun ,j (θun )EFun [mj (W, θun )],
un ,j
n→∞
n→∞
√
√
−1
(θ̂un )EFun [mj (W, θ̂un )] = lim κ−1
lim `˜un ,j ≡ lim un σF−1
un un σFun ,j (θun )EFun [mj (W, θun )],
un ,j
n→∞
n→∞
n→∞
n→∞
for j ≤ p ,
for j > p ,
which completes the proof, as {(θ̂un , `˜un ) ∈ Γun ,Fun (λun )}n≥1 and θ̂un → θ.
Proof of Lemma S.3.9. We divide the proof into four steps.
Step 1. We show that inf (θ,`)∈ΓSS S(vΩ (θ) + `, Ω(θ)) < ∞ a.s. By Assumption A.5, there exists a sequence
{θ̃n ∈ ΘI (Fn , λn )}n≥1 , where dH (Θ(λn ), Θ(λ0 )) = O(n−1/2 ). Then, there exists another sequence {θn ∈
√
Θ(λ0 )}n≥1 s.t. n||θn − θ̃n || = O(1) for all n ∈ N. Since Θ is compact, there is a subsequence {an }n≥1
√
an (θan − θ̃an ) → λ ∈ Rdθ , and θan → θ∗ and θ̃an → θ∗ for some θ∗ ∈ Θ. For any n ∈ N, let
s.t.
p
(θan )EFan [mj (W, θan )] for j = 1, . . . , k, and note that
`an ,j ≡ ban σF−1
an ,j
`an ,j =
p
ban σF−1
(θ̃an )EFan [mj (W, θ̃an )] + ∆an ,j
an ,j
(S.8)
by the intermediate value theorem, where θ̂an lies between θan and θ̃an for all n ∈ N, and
p
ban
ba
√
∗ √
≡ √ (GFan ,j (θ̂an ) − GFan ,j (θ )) an (θan − θ̃an ) + √ n GFan ,j (θ∗ ) an (θan − θ̃an ) .
an
an
p
∆an ,j
Letting ∆an = {∆an ,j }kj=1 , it follows that
p
p
ban
ba
√
√
∗
||∆an || ≤ √ ||GFan (θ̂an )−GFan (θ )||×|| an (θan − θ̃an )||+|| √ n GFan (θ∗ )||×|| an (θan − θ̃an )|| = o(1) ,
an
an
(S.9)
p
√
∗ √
∗
∗
where bn /n → 0, an (θan − θ̃an ) → λ, ban GFan (θ )/ an = o(1), θ̂an → θ , and ||GFan (θ̂an )−GFan (θ )|| =
o(1) for any sequence {Fan ∈ P0 }n≥1 by Assumption A.3(c). Thus, for all j ≤ k,
lim `an ,j ≡ lim
n→∞
n→∞
p
p
∗
ban σF−1
(θ
)E
[m
(W,
θ
)]
=
`
≡
lim
ban σF−1
(θ̃an )EFan [mj (W, θ̃an )] .
a
F
j
a
j
n
a
n
,j
n
an
an ,j
n→∞
Since {θ̃n ∈ ΘI (Fn , λn )}n≥1 , `∗j ≥ 0 for j ≤ p and `∗j = 0 for j > p. Let `∗ ≡ {`∗j }kj=1 . By definition,
10
∗ ∗
∗ ∗
SS
{(θan , `an ) ∈ ΓSS
ban ,Fan (λ0 )}n≥1 and d((θan , `an ), (θ , ` )) → 0, which implies that (θ , ` ) ∈ Γ . From here,
we conclude that
inf
(θ,`)∈ΓSS
S(vΩ (θ) + `, Ω(θ)) ≤ S(vΩ (θ∗ ) + `∗ , Ω(θ∗ )) ≤ S(vΩ (θ∗ ), Ω(θ∗ )) ,
where the first inequality follows from (θ∗ , `∗ ) ∈ ΓSS , the second inequality follows from the fact that `∗j ≥ 0
for j ≤ p and `∗j = 0 for j > p and the properties of S(·). Finally, the RHS is bounded as vΩ (θ∗ ) is bounded
a.s.
k−p
¯ ∈ ΓSS with `¯ ∈ Rp
Step 2. We show that if (θ̄, `)
, ∃(θ̄, `∗ ) ∈ ΓPR where `∗j ≥ `¯j for j ≤ p and
[+∞] × R
`∗j = `¯j for j > p. As an intermediate step, we use the limit sets under the sequence {(λn , Fn )}n≥1 , denoted
PR
by ΓSS
A and ΓA in the statement of the lemma.
H
¯ ∈ ΓSS . Since ΓSS (λ0 ) → ΓSS , there exist a subsequence {(θa , `a ) ∈
We first show that (θ̄, `)
n
n
A
bn ,Fn
p
−1/2
¯
(θ
)E
[m(W,
θ
)]
→
`.
To
show
that
b
D
(λ
)}
,
θ
→
θ̄,
and
`
≡
ΓSS
0
n≥1
a
a
a
F
a
a
n
n
n
an
n
n
ban ,Fan
Fan
0
0
SS
0
0
SS
¯
(θ̄, `) ∈ ΓA , we now find a subsequence {(θan , `an ) ∈ Γban ,Fan (λn )}n≥1 , θan → θ̄, and `an ≡
p
−1/2
¯ Notice that {(θa , `a ) ∈ ΓSS
ban DFan (θa0 n )EFan [m(W, θa0 n )] → `.
n
n
ban ,Fan (λ0 )}n≥1 implies that {θan ∈
Θ(λ0 )}n≥1 . This and dH (Θ(λn ), Θ(λ0 )) = O(n−1/2 ) implies that there is {θa0 n ∈ Θ(λan )}n≥1 s.t.
√
an ||θa0 n − θan || = O(1) which implies that θa0 n → θ̄. By the intermediate value theorem there exists a
sequence {θn∗ ∈ Θ}n≥1 with θn∗ in the line between θn and θn0 such that
`0an
≡
p
−1/2
ban DFan (θa0 n )EFan [m(W, θa0 n )] =
p
−1/2
ban DFan (θan )EFan [m(W, θan )] +
p
ban GFan (θa∗n )(θa0 n − θan )
= `an + ∆an → `¯ ,
where we have defined ∆an ≡
p
ban GFan (θa∗n )(θa0 n − θan ) and ∆an = o(1) holds by similar arguments to
¯ ∈ ΓSS .
those in (S.9). This proves (θ̄, `)
A
∗
∗
¯
¯
We now show that ∃(θ̄, `∗ ) ∈ ΓPR
A where `j ≥ `j for j ≤ p and `j = `j for j > p. Using similar arguments
−χ/2
to those in (S.2) and (S.3) in the proof of Lemma S.3.8, we have that QFan (θa0 n ) = O(ban
p
is a sequence {θ̃n ∈ ΘI (Fn , λn )}n≥1 s.t. ban ||θa0 n − θ̃an || = O(1).
) and that there
Following similar steps to those leading to (S.4) in the proof of Lemma S.3.8, it follows that
p
p
√
−1/2
−1/2
κ−1
nGFn (θn∗ )(θ̂n − θ̃n ) = bn DFn (θn0 )EFn [m(W, θn0 )] − bn DFn (θ̃n )EFn [m(W, θ̃n )] ,
n
where {θn∗ ∈ Θ(λn )}n≥1 lies in the line between θn0 and θ̃n , and θ̂n ≡ (1 − κn
p
bn /n)θ̃n + κn
(S.10)
p
bn /nθn0 .
By Assumption A.4, θ̂n is a convex combination of θ̃n and θn0 for n sufficiently large. Note also that
p
ban ||θ̂an − θ̃an || = o(1). By doing yet another intermediate value theorem expansion, there is a sequence
{θn∗∗ ∈ Θ(λn )}n≥1 with θn∗∗ in the line between θ̂n and θ̃n such that
√ −1/2
√
√ −1/2
κ−1
nDFn (θ̂n )EFn [m(W, θ̂n )] = κ−1
nGFn (θn∗∗ )(θ̂n − θ̃n ) + κ−1
nDFn (θ̃n )EFn [m(W, θ̃n )] .
n
n
n
Since
(S.11)
p
p
p
ban ||θa∗n − θ̃an || = O(1) and ban ||θ̃an − θa∗∗n || = o(1), it follows that ban ||θa∗n − θa∗∗n || = O(1). Next,
√ −1/2
√
√ −1/2
κ−1
nDFn (θ̂n )EFn [m(W, θ̂n )] = κ−1
nGFn (θn∗ )(θ̂n − θ̃n ) + κ−1
nDFn (θ̃n )EFn [m(W, θ̃n )] + ∆n,1
n
n
n
p
−1/2 0
0
= bn DFn (θn )EFn [m(W, θn )] + ∆n,1 + ∆n,2 ,
(S.12)
11
√
where the first equality follows from (S.11) and ∆n,1 ≡ κ−1
n(GFn (θn∗∗ ) − GFn (θn∗ ))(θ̂n − θ̃n ) , and the
n
p
√
−1/2
n(1 − κn bn /n)DFn (θ̃n )EFn [m(W, θ̃n )]. By similar arguments to
second holds by (S.10) and ∆n,2 ≡ κ−1
n
those in the proof of Lemma S.3.8, ||∆an ,1 || = o(1). In addition, Assumption A.4 and {θ̃n ∈ ΘI (Fn , λn )}n≥1
imply that ∆n,2,j ≥ 0 for j ≤ p and n sufficiently large, and that ∆n,2,j = 0 for j > p and all n ≥ 1.
√
−1/2
k
Now define `00an ≡ κ−1
an an DFan (θ̂an )EFn [m(W, θ̂an )] so that by compactness of (R[±∞] , d) there is a fur√
−1/2
ther subsequence {un }n≥1 of {an }n≥1 s.t. `00un = κ−1
un un DFun (θ̂un )EFun [m(W, θ̂un )] and ∆un ,1 converges.
We define `∗ ≡ limn→∞ `00un . By (S.12) and properties of ∆n,1 and ∆n,2 , we conclude that
p
√
−1
bun σF−1
lim `00un ,j = lim κ−1
(θu0 n )EFun [mj (W, θu0 n )] = `¯j , for j ≤ p ,
un un σFun ,j (θ̂un )EFun [mj (W, θ̂un )] ≥ lim
un ,j
n→∞
n→∞
n→∞
p
√
−1
lim `00un ,j = lim κ−1
bun σF−1
(θu0 n )EFun [mj (W, θu0 n )] = `¯j , for j > p ,
un un σFun ,j (θ̂un )EFun [mj (W, θ̂un )] = lim
un ,j
n→∞
n→∞
n→∞
00
∗
∗
∗
¯
¯
Thus, {(θ̂un , `00un ) ∈ ΓPR
un ,Fun (λn )}n≥1 , θ̂un → θ̄, and `un → ` where `j ≥ `j for j ≤ p and `j = `j for j > p,
and (θ̄, `∗ ) ∈ ΓPR
A .
We conclude the step by showing that (θ̄, `∗ ) ∈ ΓPR . To this end, find a subsequence {(θu† n , `†un ) ∈
√
−1/2 †
†
∗
00
PR
Γbun ,Fun (λ0 )}n≥1 , θu† n → θ̄, and `†un ≡ κ−1
un un DFun (θun )EFun [m(W, θun )] → ` . Notice that {(θ̂un , `un ) ∈
−1/2
ΓPR
) implies that
un ,Fun (λn )}n≥1 implies that {θ̂un ∈ Θ(λun )}n≥1 . This and dH (Θ(λn ), Θ(λ0 )) = O(n
√
†
†
†
there is {θun ∈ Θ(λ0 )}n≥1 s.t. un ||θ̂un − θun || = O(1) which implies that θun → θ̄. By the intermediate
value theorem there exists a sequence {θn∗∗∗ ∈ Θ}n≥1 with θn∗∗∗ in the line between θ̂n and θn† such that
`†un
√
−1/2 †
−1/2 †
†
−1 √
†
−1 √
∗∗∗
†
≡ κ−1
un un DFun (θun )EFun [m(W, θun )] = κun un DFun (θun )EFun [m(W, θun )] + κun un GFun (θun )(θun − θ̂un )
= `00un + ∆un → `∗ ,
√
†
∗∗∗
where we have define ∆un ≡ κ−1
un un GFun (θun )(θun − θ̂un ) and ∆un = o(1) holds by similar arguments to
those used before. By definition, this proves that (θ̄, `∗ ) ∈ ΓPR .
Step 3. We show that inf (θ,`)∈ΓSS S(vΩ (θ) + `, Ω(θ)) ≥ inf (θ,`)∈ΓPR S(vΩ (θ) + `, Ω(θ)) a.s. Since vΩ is a
tight stochastic process, there is a subset of the sample space W, denoted A1 , s.t. P (A1 ) = 1 and ∀ω ∈ A1 ,
supθ∈Θ ||vΩ (ω, θ)|| < ∞. By step 1, there is a subset of W, denoted A2 , s.t. P (A2 ) = 1 and ∀ω ∈ A2 ,
inf
(θ,`)∈ΓSS
S(vΩ (ω, θ) + `, Ω(θ)) < ∞ .
Define A ≡ A1 ∩ A2 and note that P (A) = 1. In order to complete the proof, it then suffices to show that
∀ω ∈ A,
inf
(θ,`)∈ΓSS
S(vΩ (ω, θ) + `, Ω(θ)) ≥
inf
(θ,`)∈ΓPR
S(vΩ (ω, θ) + `, Ω(θ)) .
(S.13)
Fix ω ∈ A arbitrarily and suppose that (S.13) does not occur, i.e.,
∆≡
inf
(θ,`)∈ΓPR
S(vΩ (ω, θ) + `, Ω(θ)) −
inf
(θ,`)∈ΓSS
S(vΩ (ω, θ) + `, Ω(θ)) > 0 .
(S.14)
¯ ∈ ΓSS s.t. inf (θ,`)∈ΓSS S(vΩ (ω, θ) + `, Ω(θ)) + ∆/2 ≥ S(vΩ (ω, θ̄) + `,
¯ Ω(θ̄)),
By definition of infimum, ∃(θ̄, `)
12
and so, from this and (S.14) it follows that
¯ Ω(θ̄)) ≤
S(vΩ (ω, θ̄) + `,
inf
(θ,`)∈ΓPR
S(vΩ (ω, θ) + `, Ω(θ)) − ∆/2 .
(S.15)
We now show that `¯ ∈ Rp[+∞] × Rk−p . Suppose not, i.e., suppose that `¯j = −∞ for some j < p and
|`¯j | = ∞ for some j > p. Since ω ∈ A ⊆ A1 , ||vΩ (ω, θ̄)|| < ∞. By part 2 of Lemma S.3.6 it then follows
¯ Ω(θ̄)) = ∞. By (S.15), inf (θ,`)∈ΓSS S(vΩ (ω, θ) + `, Ω(θ)) = ∞, which is a contradiction
that S(vΩ (ω, θ̄) + `,
to ω ∈ A2 .
Since `¯ ∈ Rp[+∞] × Rk−p , step 2 implies that ∃(θ̄, `∗ ) ∈ ΓPR where `∗j ≥ `¯j for j ≤ p and `∗j = `¯j for j > p.
By properties of S(·),
¯ Ω(θ̄)) .
S(vΩ (ω, θ̄) + `∗ , Ω(θ̄)) ≤ S(vΩ (ω, θ̄) + `,
(S.16)
Combining (S.14), (S.15), (S.16), and (θ̄, `∗ ) ∈ ΓPR , we reach the following contradiction,
0 < ∆/2 ≤
≤
inf
¯ Ω(θ̄))
S(vΩ (ω, θ) + `, Ω(θ)) − S(vΩ (ω, θ̄) + `,
inf
S(vΩ (ω, θ) + `, Ω(θ)) − S(vΩ (ω, θ̄) + `∗ , Ω(θ̄)) ≤ 0 .
(θ,`)∈ΓPR
(θ,`)∈ΓPR
Step 4. Suppose the conclusion of the lemma is not true. This is, suppose that c(1−α) (ΓPR , Ω) >
c(1−α) (ΓSS , Ω). Consider the following derivation
α < P (J(ΓPR , Ω) > c(1−α) (ΓSS , Ω))
≤ P (J(ΓSS , Ω) > c(1−α) (ΓSS , Ω)) + P (J(ΓPR , Ω) > J(ΓSS , Ω)) = 1 − P (J(ΓSS , Ω) ≤ c(1−α) (ΓSS , Ω)) ≤ α ,
where the first strict inequality holds by definition of quantile and c(1−α) (ΓPR , Ω) > c(1−α) (ΓSS , Ω), the last
equality holds by step 3, and all other relationships are elementary. Since the result is contradictory, the
proof is complete.
SS
R
Proof of Lemma S.3.10. By Theorem 4.3, lim inf(EFn [φP
n (λ0 )] − EFn [φn (λ0 )]) ≥ 0. Suppose that the
desired result is not true. Then, there is a further subsequence {un }n≥1 of {n}n≥1 s.t.
R
SS
lim EFun [φP
un (λ0 )] = lim EFun [φun (λ0 )] .
(S.17)
This sequence {un }n≥1 will be referenced from here on. We divide the remainder of the proof into steps.
Step 1. We first show that there is a subsequence {an }n≥1 of {un }n≥1 s.t.
d
n
{TaSS
(λ0 )|{Wi }ai=1
} → S(vΩ (θ∗ ) + (g, 0k−p ), Ω(θ∗ )), a.s.
n
(S.18)
Conditionally on {Wi }ni=1 , Assumption A.7(c) implies that
p
SS
SS SS
TnSS (λ0 ) = S( bn DF−1
(θ̂nSS )m̄SS
bn (θ̂n ), Ω̃bn (θ̂n )) + op (1), a.s.
n
13
(S.19)
By continuity of the function S, (S.18) follows from (S.19) if we find a subsequence {an }n≥1 of {un }n≥1 s.t.
p
an
∗
SS
{Ω̃SS
an (θ̂an )|{Wi }i=1 } → Ω(θ ), a.s.
p
d
−1/2
an
∗
SS
)m̄SS
{ ban DFan (θ̂aSS
an (θ̂an )|{Wi }i=1 } → vΩ (θ ) + (g, 0k−p ), a.s.
n
(S.20)
(S.21)
To show (S.20), note that
SS
∗
SS
SS
∗
||Ω̃SS
n (θ̂n ) − Ω(θ )|| ≤ sup ||Ω̃n (θ, θ) − Ω(θ, θ)|| + ||Ω(θ̂n ) − Ω(θ )|| .
θ∈Θ
the first term on the RHS is conditionally op (1) a.s. by Lemma S.3.1 (part 5) and the second term is
p
conditionally op (1) a.s. by Ω ∈ C(Θ2 ) and {θ̂nSS |{Wi }ni=1 } → θ∗ a.s. Then, (S.20) holds for the original
sequence {n}n≥1 .
To show (S.21), note that
p
−1/2
SS
SS ∗
bn DFn (θ̂nSS )m̄SS
n (θ̂n ) = ṽn (θ ) + (g, 0k−p ) + µn,1 + µn,2 ,
where
µn,1
p
≡ ṽn (θ̂nSS ) bn /n
µn,2
p
−1/2
−1/2
SS
SS ∗
≡ (ṽbSS
bn (DFn (θ̂nSS )EFn [m(W, θ̂nSS )] − DFn (θ̃nSS )EFn [m(W, θ̃nSS )])
(
θ̂
)
−
ṽ
(θ
))
+
n
b
n
n
p
−1/2
+( bn DFn (θ̃nSS )EFn [m(W, θ̃nSS )] − (g, 0k−p )) .
d
Lemma S.3.1 (part 9) implies that {ṽbSS
(θ∗ )|{Wi }ni=1 } → vΩ (θ∗ ) a.s. and so, (S.21) follows from
n
n
{µan ,1 |{Wi }ai=1
}
= op (1), a.s.
(S.22)
n
{µan ,2 |{Wi }ai=1
}
= op (1), a.s.
(S.23)
p
By Lemma S.3.1 (part 7), supθ∈Θ ||ṽn (θ)|| bn /n = op (1), and by taking a further subsequence {an }n≥1 of
p
{n}n≥1 , supθ∈Θ ||ṽan (θ)|| ban /an = oa.s. (1). Since ṽn (·) is conditionally non-stochastic, this result implies
(S.23).
By Assumption A.7(c), (S.23) follows from showing that {ṽnSS (θ∗ ) − ṽnSS (θ̂nSS )|{Wi }ni=1 } = op (1) a.s.,
which we now show. Fix µ > 0 arbitrarily, it suffices to show that
lim sup PFn (||ṽnSS (θ∗ ) − ṽnSS (θ̂nSS )|| > ε|{Wi }ni=1 ) < µ a.s.
(S.24)
Fix δ > 0 arbitrarily. As a preliminary step, we first show that
lim PFn (ρFn (θ∗ , θ̂nSS ) ≥ δ|{Wi }ni=1 ) = 0 a.s. ,
where ρFn is the intrinsic variance semimetric in (A-1). Then, for any j = 1, . . . , k,
VFn (σF−1
(θ̂nSS )mj (W, θ̂nSS ) − σF−1
(θ∗ )mj (W, θ∗ )) = 2(1 − ΩFn (θ∗ , θ̂nSS )[j,j] ) .
n ,j
n ,j
14
(S.25)
By (A-1), this implies that
PFn (ρFn (θ∗ , θ̂nSS ) ≥ δ|{Wi }ni=1 ) ≤
Xk
j=1
PFn (1 − ΩFn (θ∗ , θ̂nSS )[j,j] ≥ δ 2 2−1 k −1 |{Wi }ni=1 ) .
(S.26)
For any j = 1, . . . , k, note that
PFn (1 − ΩFn (θ∗ , θ̂nSS )[j,j] ≥ δ 2 2−1 k −1 |{Wi }ni=1 ) ≤ PFn (1 − Ω(θ∗ , θ̂nSS )[j,j] ≥ δ 2 2−2 k −1 |{Wi }ni=1 ) + o(1)
≤ PFn (||θ∗ − θ̂nSS || > δ̃|{Wi }ni=1 ) + o(1) = oa.s. (1) ,
u
where we have used that ΩFn → Ω and so supθ,θ0 ∈Θ ||Ω(θ, θ0 )[j,j] − ΩFn (θ, θ0 )[j,j] || < δ 2 2−2 k −1 for all sufficiently large n, that Ω ∈ C(Θ2 ) and so ∃δ̃ > 0 s.t. ||θ∗ − θ̂nSS || ≤ δ̃ implies that 1 − Ω(θ∗ , θ̂nSS )[j,j] ≤ δ 2 2−2 k −1 ,
p
and that {θ̂nSS |{Wi }ni=1 } → θ∗ a.s. Combining this with (S.26), (S.25) follows.
Lemma S.3.1 (part 1) implies that {ṽnSS (·)|{Wi }ni=1 } is asymptotically ρF -equicontinuous uniformly in
F ∈ P (a.s.) in the sense of van der Vaart and Wellner (1996, page 169). Then, ∃δ > 0 s.t.
lim sup PF∗n (
n→∞
sup
ρFn (θ,θ 0 )<δ
||ṽnSS (θ) − ṽnSS (θ0 )|| > ε|{Wi }ni=1 ) < µ a.s.
(S.27)
Based on this choice, consider the following argument:
PF∗n (||ṽnSS (θ∗ ) − ṽnSS (θ̂nSS )|| > ε|{Wi }ni=1 ) ≤ PF∗n (
sup
ρFn (θ,θ 0 )<δ
||ṽnSS (θ∗ ) − ṽnSS (θ̂n )|| > ε|{Wi }ni=1 )
+ PF∗n (ρFn (θ∗ , θ̂n ) ≥ δ|{Wi }ni=1 ) .
From this, (S.25), and (S.27), (S.24) follows.
Step 2. For arbitrary ε > 0 and for the subsequence {an }n≥1 of {un }n≥1 in step 1 we want show that
∗
lim PFan (|cSS
an (λ0 , 1 − α) − c(1−α) (g, Ω(θ ))| ≤ ε) = 1 ,
(S.28)
where c(1−α) (g, Ω(θ∗ )) denotes the (1 − α)-quantile of S(vΩ (θ∗ ) + (g, 0k−p ), Ω(θ∗ )). By our maintained
assumptions and Assumption A.7(b.iii), it follows that c(1−α) (g, Ω(θ∗ )) > 0.
Fix ε̄ ∈ (0, min{ε, c(1−α) (g, Ω(θ∗ ))}). By our maintained assumptions, c(1−α) (g, Ω(θ∗ )) − ε̄ and
c(1−α) (g, Ω(θ∗ )) + ε̄ are continuity points of the CDF of S(vΩ (θ∗ ) + (g, 0k−p ), Ω(θ∗ )). Then,
∗
∗
∗
n
lim PFan (TaSS
(λ0 ) ≤ c(1−α) (g, Ω(θ∗ )) + ε̄|{Wi }a
i=1 ) = P (S(vΩ (θ ) + (g, 0k−p ), Ω(θ )) ≤ c(1−α) (g, Ω(θ )) + ε̄) > 1 − α , (S.29)
n
where the equality holds a.s. by step 1, and the strict inequality holds by ε̄ > 0. By a similar argument,
∗
∗
∗
lim PFan (TaSS
(λ0 ) ≤ c(1−α) (g, Ω(θ∗ )) − ε̄|{Wi }n
i=1 ) = P (S(vΩ (θ ) + (g, 0k−p ), Ω(θ )) ≤ c(1−α) (g, Ω(θ )) − ε̄) < 1 − α , (S.30)
n
where, as before, the equality holds a.s. by step 1. Next, notice that
∗
n
{lim PFan (TaSS
(λ0 ) ≤ c(1−α) (g, Ω(θ∗ ))+ε̄|{Wi }ai=1
) > 1−α} ⊆ {lim inf{cSS
an (λ0 , 1−α) < c(1−α) (g, Ω(θ ))+ε̄}} ,
n
with the same result holding with −ε̄ replacing +ε̄. By combining this result with (S.29) and (S.30), we get
∗
{lim inf{|cSS
an (λ0 , 1 − α) − c(1−α) (g, Ω(θ ))| ≤ ε̄}} a.s.
15
From this result, ε̄ < ε, and Fatou’s Lemma, (S.28) follows.
Step 3. For an arbitrary ε > 0 and for a subsequence {wn }n≥1 of {an }n≥1 in step 2 we want to show that
R
lim PFwn (c(1−α) (π, Ω(θ∗ )) + ε ≥ cP
wn (λ0 , 1 − α)) = 1 ,
(S.31)
where c(1−α) (π, Ω(θ∗ )) denotes the (1 − α)-quantile of S(vΩ (θ∗ ) + (π, 0k−p ), Ω(θ∗ )) and π ∈ Rp[+,+∞] is a
parameter to be determined that satisfies π ≥ g and πj > gj for some j = 1, . . . , p.
The arguments required to show this are similar to those used in steps 1-2. For any θ ∈ Θ(λ0 ), define
√ −1/2
≡ S(vn∗ (θ) + κ−1
nD̂n (θ)m̄n (θ), Ω̂n (θ)). We first show that there is a subsequence {wn }n≥1 of
n
T̃nP R (θ)
{an }n≥1 s.t.
d
wn
SS
{T̃wPnR (θ̂w
} → S(vΩ (θ∗ ) + (π, 0k−p ), Ω(θ∗ )) a.s.
)|{Wi }i=1
n
(S.32)
Consider the following derivation:
T̃nP R (θ̂nSS )
√ −1/2 SS
nD̂n (θ̂n )m̄n (θ̂nSS ), Ω̂n (θ̂nSS ))
= S(vn∗ (θ̂nSS ) + κ−1
n
√ −1/2 SS
−1/2
= S(DFn (θ̂nSS )D̂n1/2 (θ̂nSS )vn∗ (θ̂nSS ) + κ−1
nDFn (θ̂n )m̄n (θ̂nSS ), Ω̃n (θ̂nSS )) .
n
By continuity of the function S, (S.31) would follow from if we find a subsequence {wn }n≥1 of {an }n≥1 s.t.
p
SS
∗
n
{Ω̃wn (θ̂w
)|{Wi }w
i=1 } → Ω(θ ), a.s.
n
√
d
−1/2 SS
−1/2 SS
wn
SS
∗
1/2 SS ∗
SS
)D̂w
(θ̂wn )vwn (θ̂w
) + κ−1
{DFwn (θ̂w
wn wn DFwn (θ̂wn )m̄wn (θ̂wn )|{Wi }i=1 } → vΩ (θ ) + (π, 0k−p ), a.s.
n
n
n
The first statement is shown as in (S.20) in step 1. To show the second statement, note that
√
−1/2 SS
−1/2 SS
1/2 SS ∗
SS
SS
∗
∗
DFwn (θ̂w
)D̂w
(θ̂wn )vwn (θ̂w
) + κ−1
wn wn DFwn (θ̂wn )m̄wn (θ̂wn ) = vwn (θ ) + (π, 0k−p ) + µwn ,3 + µwn ,4 ,
n
n
n
where
µwn ,3
µwn ,4
−1/2
SS
1/2 SS ∗
SS
∗
SS
= DFwn (θ̂w
)D̂w
(θ̂wn )vwn (θ̂w
) − vw
(θ∗ ) + κ−1
wn ṽwn (θ̂wn )
n
n
n
n
√
−1/2 SS
−1/2 SS
SS
SS
+κ−1
wn wn (DFwn (θ̂wn )EFwn [m(W, θ̂wn )] − DFwn (θ̃wn )EFwn [m(W, θ̃wn )]),
p
p
−1/2 SS
SS
= (κ−1
wn /bwn ) bwn DFwn (θ̃w
)EFwn [m(W, θ̃w
)] − (π, 0k−p ).
wn
n
n
d
∗
By Lemma S.3.1 (part 9), we have {vw
(θ∗ )|{Wi }ni=1 } → vΩ (θ∗ ) a.s. By the same arguments as in step
n
p
−1
n
wn /bwn →
1, we have that {µwn ,3 |{Wi }w
i=1 } = op (1), a.s. By possibly considering a subsequence, κwn
√
−1/2 SS
wn
−1
SS
K
∈ (1, ∞] by Assumption A.7(d) and { bn DFwn (θ̃wn )EFwn [m(W, θ̃wn )]|{Wi }i=1 } = (g, 0k−p ) + op (1)
a.s. by Assumption A.7(c,iii), with g ∈ Rp[+,+∞] by step 1. By combining these two and by possibly considerp
√
−1/2 SS
wn
SS
wn /bwn ) bn DFwn (θ̃w
)EFwn [m(W, θ̃w
)]|{Wi }i=1
}=
ing a further subsequence, we conclude that {(κ−1
wn
n
n
(π, 0k−p ) + op (1) a.s. where π ∈ Rp[+,+∞] . Since K −1 > 1, πj ≥ gj ≥ 0 for all j = 1, . . . , p. By Assumption A.7(c,iii), there is j = 1, . . . , p, s.t. gj ∈ (0, ∞) and so πj = K −1 gj > gj . From this, we have that
n
{µwn ,4 |{Wi }w
i=1 } = op (1), a.s.
R
PR
Let c̃P
n (θ, 1 − α) denote the conditional (1 − α)-quantile of T̃n (θ). On the one hand, (S.32) and the
R SS
∗
arguments in step 2 imply that lim PFwn (|c̃P
wn (θ̂wn , 1 − α) − c(1−α) (π, Ω(θ ))| ≤ ε) = 1 for any ε > 0.
16
R SS
On the other hand, TnP R (λ0 ) = inf θ∈Θ(λ0 ) T̃nP R (θ) and {θ̂nSS ∈ Θ(λ0 )}n≥1 imply that c̃P
n (θ̂n , 1 − α) ≥
R
cP
n (λ0 , 1 − α). By combining these, (S.31) follows.
We conclude by noticing that by c(1−α) (g, Ω(θ∗ )) > 0 (by step 2) and π ≥ g with πj > gj for some
j = 1, . . . , p, our maintained assumptions imply that c(1−α) (g, Ω(θ∗ )) > c(1−α) (π, Ω(θ∗ )).
Step 4. We now conclude the proof. By Assumption A.7(a) and arguments similar to step 1 we deduce
that
d
Twn (λ0 ) → S(vΩ (θ∗ ) + (λ, 0k−p ), Ω(θ∗ )) .
(S.33)
Fix ε ∈ (0, min{c(1−α) (g, Ω(θ∗ )), (c(1−α) (g, Ω(θ∗ )) − c(1−α) (π, Ω(θ∗ )))/2} (possible by steps 2-3), and note
that
∗
SS
∗
PFwn (Twn (λ0 ) ≤ cSS
wn (λ0 , 1 − α)) ≤ PFwn (Twn (λ0 ) ≤ c(1−α) (g, Ω(θ )) + ε) + PFwn (|cwn (λ0 , 1 − α) − c(1−α) (g, Ω(θ ))| > ε) ,
By (S.28), (S.33), and our maintained assumptions, it follows that
∗
∗
∗
lim sup PFwn (Twn (λ0 ) ≤ cSS
wn (λ0 , 1 − α)) ≤ P (S(vΩ (θ ) + (λ, 0k−p ), Ω(θ )) ≤ c(1−α) (g, Ω(θ )) + ε) , (S.34)
∗
∗
∗
lim inf PFwn (Twn (λ0 ) ≤ cSS
wn (λ0 , 1 − α)) ≥ P (S(vΩ (θ ) + (λ, 0k−p ), Ω(θ )) ≤ c(1−α) (g, Ω(θ )) − ε) . (S.35)
Since (S.34), (S.35), c(1−α) (g, Ω(θ∗ )) > 0, and our maintained assumptions,
∗
∗
∗
lim EFwn [φSS
wn (λ0 )] = P (S(vΩ (θ ) + (λ, 0k−p ), Ω(θ )) > c(1−α) (g, Ω(θ ))) .
(S.36)
We can repeat the same arguments to deduce an analogous result for the Penalize Resampling Test. The main
difference is that for Test PR we do not have a characterization of the minimizer, which is not problematic
as we can simply bound the asymptotic rejection rate using the results from step 3. This is,
R
∗
∗
∗
lim EFwn [φP
wn (λ0 )] ≥ P (S(vΩ (θ ) + (λ, 0k−p ), Ω(θ )) > c(1−α) (π, Ω(θ ))) .
(S.37)
By our maintained assumptions, c(1−α) (g, Ω(θ∗ )) > c(1−α) (π, Ω(θ∗ )), (S.36), and (S.37), we conclude that
R
lim EFwn [φP
wn (λ0 )] ≥
>
P (S(vΩ (θ∗ ) + (λ, 0k−p ), Ω(θ∗ )) > c(1−α) (π, Ω(θ∗ )))
P (S(vΩ (θ∗ ) + (λ, 0k−p ), Ω(θ∗ )) > c(1−α) (g, Ω(θ∗ ))) = lim EFwn [φSS
wn (λ0 )] .
Since {wn }n≥1 is a subsequence of {un }n≥1 , this is a contradiction to (S.17) and concludes the proof.
References
Bugni, F. A., I. A. Canay, and X. Shi (2014): “Specification Tests for Partially Identified Models defined
by Moment Inequalities,” Accepted for publication in Journal of Econometrics on October 6, 2014.
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——— (2016): “Inference for Subvectors and Other Functions of Partially Identified Parameters in Moment
Inequality Models,” accepted for publication at Quantitative Economics.
Royden, H. L. (1988): Real Analysis, Prentice-Hall.
van der Vaart, A. W. and J. A. Wellner (1996): Weak Convergence and Empirical Processes, SpringerVerlag, New York.
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