Statistical inference for expectile-based risk measures

0.45 MB·

by  Volker Krätschmer

Statistical inference for expectile-based
risk measures
Volker Kr¨atschmer Henryk Za¨hle
∗ †
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p
e
S
0 Abstract
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Expectiles were introduced by Newey and Powell [43] in the context of linear

]
T regression models. Recently, Bellini et al. [6] revealed that expectiles can also
S be seen as reasonable law-invariant risk measures. In this article, we show that
.
h the corresponding statistical functionals are continuous w.r.t. the 1-weak topol-
t
a ogy and suitably functionally differentiable. By means of these regularity results
m
we can derive several properties such as consistency, asymptotic normality, boot-
[
strap consistency, and qualitative robustness of the corresponding estimators in

3 nonparametric and parametric statistical models.
v
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6 Keywords: Expectile-based risk measure; 1-weak continuity; Quasi-Hadamard dif-
2
ferentiability; Statistical estimation; Weak dependence; Strong consistency; Asymptotic
5
0 normality; Bootstrap consistency; Qualitative robustness; Functional delta-method
.
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0
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v
i
X
r
a
∗Faculty of Mathematics, University of Duisburg–Essen;[email protected]
†Department of Mathematics, Saarland University; [email protected]
1
1. Introduction
Let (Ω, ,P) be an atomless probability space and use Lp = Lp(Ω, ,P) to denote the
F F
usual Lp-space. The α-expectile of X L2, with α (0,1), can uniquely be defined by
∈ ∈
ρ (X) := argmin αE[((X m)+)2]+(1 α)E[((m X)+)2]
α m∈R{ − − − }
= argmin E[V (X m)] (1)
m∈R α −
(see Proposition 1 and Example 4 in [6]), where
αx2 , x 0
V (x) := ≥ , x R.
α (1 α)x2 , x < 0 ∈
(cid:26) −
Expectiles were introduced by Newey and Powell [43] in the context of linear regression
models. On the one hand, (1) generalizes the expectation of X which coincides with
ρ (X) when specifically α = 1/2. On the other hand, (1) is similar to the α-quantile of
α
X which can be obtained by replacing x2 by x in the definition of V . This motivates
α
| |
the name α-expectile.
For every X L2 the mapping m E[V (X m)] is convex and differentiable with
α
∈ 7→ −
derivative given by m 2U (X)(m), where
α
7→ −
U (X)(m) := E[U (X m)], m R (2)
α α
− ∈
with
αx , x 0
U (x) := ≥ , x R.
α
(1 α)x , x < 0 ∈
(cid:26) −
Moreover, for X L1 the mapping m U (X)(m) is well defined and bijective; cf.
α
∈ 7→
Lemma A.1 (Appendix A). These observations together imply that for X L2 the
∈
α-expectile admits the representation
ρ (X) = U (X) 1(0), (3)
α α −
where U (X) 1 denotes the inverse function of U (X). In particular, (3) can be used to
α − α
define a map ρ : L1 R which is compatible with (1). For every X L1 the value in
α
→ ∈
(3) will be called the corresponding α-expectile.
Recently, Bellini et al. [6] revealed that expectiles can be also seen as reasonable
risk measures when 1/2 α < 1. In Proposition 6 in [6], they prove that the map
≤
ρ : L2 R provides a coherent risk measure if (and only if) 1/2 α < 1. Recall that
α
→ ≤
a map ρ : R, with a subspace of L0, is said to be a coherent risk measure if it is
X → X
monotone: ρ(X ) ρ(X ) for all X ,X with X X ,
1 2 1 2 1 2
• ≤ ∈ X ≤
cash-invariant: ρ(X +m) = ρ(X)+m for all X and m R,
• ∈ X ∈
subadditive: ρ(X +X ) ρ(X )+ρ(X ) for all X ,X ,
1 2 1 2 1 2
• ≤ ∈ X
2
positively homogenous: ρ(λX) = λρ(X) for all X and λ 0.
• ∈ X ≥
It is shown in the Appendix A (Proposition A.2) that even the map ρ : L1 R
α
→
provides a coherent risk measure if (and only if) 1/2 α < 1. For 0 < α < 1/2 the
≤
map ρ : L1 R is at least monotone, cash-invariant, and positively homogeneous. For
α
→
this reason we will henceforth refer to ρ : L1 R as expectile-based risk measure at
α
→
level α (0,1). It is worth mentioning that ρ already appeared implicitly in an earlier
α
∈
paper by Weber [49]. As Ziegel [51] pointed out that ρ satisfies a particularly desirable
α
property of risk measures in the context of backtesting, ρ attracted special attention
α
in the field of monetary risk measurement in the last few years [1, 5, 6, 22, 24, 51]. For
pros and cons of expectile-based risk measures and of other standard risk measures see,
for instance, the discussions by Acerbi and Szekely [1], Bellini and Di Bernardino [5],
and Emmer et al. [24].
This article is concerned with the statistical estimation of expectile-based risk mea-
sures. The goal is the estimation of ρ (X) for some X L1 with unknown distri-
α
∈
bution function F. Let F be the class of all distribution functions on R satisfying
1
x dF(x) < . Note that F coincides with the set of the distribution functions of all
1
| | ∞
´elements of L1, because the underlying probability space was assumed to be atomless.
Also note that F ∈ F1 if and only if 0 F(x)dx < ∞ and 0∞(1−F(x))dx < ∞ hold.
Since ρ is law-invariant (i.e. ρ (X ´)−=∞ ρ (X ) when P X´ 1 = P X 1), we may
α α 1 α 2 1− 2−
◦ ◦
associate with ρ a statistical functional : F R via
α α 1
R →
(F ) := ρ (X), X L1, (4)
α X α
R ∈
where F denotes the distribution function of X. That is,
X
(F) = (F) 1(0) for all F F , (5)
α α − 1
R U ∈
where
(F)(m) := U (x m)dF(x), m R. (6)
Uα ˆ α − ∈
Then, if F is a reasonable estimator for F, the plug-in estimator (F ) is typically a
n α n
R
reasonable estimator for ρ (X) = (F).
α α
R
Inanonbparametricframework, acanonicalexampleforF istheempirbicaldistribution
n
function
n
1 b
F := (7)
n n 1[Xi,∞)
i=1
X
of n identically distributed randombvariables X ,...,X drawn according to F. In this
1 n
case we have
n
(F ) = (F ) 1(0) = unique solution in m of U (X m) = 0. (8)
α n α n − α i
R U −
i=1
X
b b
3
That is, the plug-in estimator is nothing but a simple Z-estimator (M-estimator). For
Z-estimators (M-estimators) there are several results concerning consistency and the
asymptotic distribution in the literature. A classical reference is Huber’s seminal paper
[30]; see also standard textbooks as [31, 46, 47, 48]. Recently Holzmann and Klar [29]
used results of Arcones [3] and Van der Vaart [47] to derive asymptotic properties of the
Z-estimator in (8). They restricted their attention to i.i.d. observations but allowed for
observations without finite second moment.
On the other hand, even in the nonparametric setting the estimator F may differ
n
from the empirical distribution function so that the plug-in estimator need not be a
Z-estimator. See, for instance, Section 3 in [7] for estimators F being dbifferent from
n
the empirical distribution function. Also, in a parametric setting the estimator F will
n
hardly be the empirical distribution function. For these reasobns, we will consider a
suitable linearization of the functional in order to be in the position to derive sbeveral
α
R
asymptotic properties of the plug-in estimator (F ) in as many as possible situations.
α n
R
Linearizations of Z-functionals have been considered before, for instance, by Clarke
[17, 18]. However, these results do not cover the parbticular Z-functional , because the
α
R
function U is unbounded. By using the concept of quasi-Hadamard differentiability as
α
well asthecorrespondingfunctionaldelta-methodintroducedbyBeutner andZa¨hle[7,8]
we will overcome the difficulties with the unboundedness of U . Quasi-Hadamard differ-
α
entiability of the functional will in particular admit some bootstrap results for the
α
R
plug-in estimator (F ) when F is the empirical distribution function of X ,...,X .
α n n 1 n
R
It is worth mentioning that Heesterman and Gill [28] also considered a linearization
approach to Z-estimatbors. Howebver (boiled down to our setting) they did not consider
a linearization of the functional (to be evaluated at the estimator F of F) but only
α n
R
of the functional that provides the unique zero of a strictly decreasing and continuous
function tending to as its argument tends to (as the function b (F )). To some
α n
±∞ ∓∞ U
extent this approach is less flexible than our approach. Especially parametric estimators
cannot be handled by this approach without further ado. b
The rest of this article is organized as follows. In Section 2 we will establish a certain
continuity andthe above-mentioned differentiability of the functional . In Sections 3–
α
R
4 we will apply the results of Section 2 to the nonparametric and parametric estimation
of (F). In Section 5 we will prove the main result of Section 2, and in Section 6 we
α
R
will verify two examples and a lemma presented in Sections 3–4. The Appendix provides
some auxiliary results. In particular, in Section B of the Appendix we formulate a slight
generalization of the functional delta-method in the form of Beutner and Za¨hle [8].
2. Regularity of the functional
α
R
In this section we investigate the functional : F R defined in (5) for continuity
α 1
R →
and differentiability. We equip F with the 1-weak topology. This topology is defined
1
4
to be the coarsest topology for which the mappings µ f dF, f , are continuous,
1
7→ ∈ C
where is the set of all continuous functions f : R R´ with f(x) C (1+ x ) for
1 f
C → | | ≤ | |
all x R and some finite constant C > 0. A sequence (F ) F converges 1-weakly to
f n 1
∈ ⊆
some F F if and only if f dF f dF for all f ; cf. Lemma 3.4 in [33]. The
0 1 n 0 1
∈ → ∈ C
set F can obviously be iden´tified with´the set of all Borel probability measures µ on R
1
satisfying x µ(dx) < . In this context the 1-weak topology is sometimes referred to
| | ∞
as ψ -weak´topology; see, for instance, [34]. But for our purposes it is more convenient
1
to work with the F -terminology.
1
Let L be the space of all Borel measurable functions v : R R modulo the equiv-
0
→
alence relation of ℓ-almost sure identity. Note that F L , and let L L be the
1 0 1 0
⊆ ⊆
subspace of all v L for which
0
∈
v := v(x) ℓ(dx) (9)
k k1,ℓ ˆ | |
is finite. Here, and henceforth, ℓ stands for the Borel Lebesgue measure on R. Note
that F F L for F ,F F . It is well-known that : L R provides a
1 2 1 1 2 1 1,ℓ 1 +
− ∈ ∈ k· k →
complete and separable norm on L and that
1
d (F ,F ) := F F
W,1 1 2 1 2 1,ℓ
k − k
defines the Wasserstein-1 metric d : F F R on F . Also note that d
W,1 1 1 + 1 W,1
× →
metrizes the 1-weak topology on F ; cf. Remark 2.9 in [34].
1
2.1. Continuity
Since the Wasserstein-1 metric d metrizes the 1-weak topology on F , the following
W,1 1
theorem is an immediate consequence of a recent result by Bellini et al. [6, Theorem 10].
Theorem 2.1 The functional : F R is continuous for the 1-weak topology.
α 1
R →
Theorem 2.1 can also be obtained by combining Theorem 4.1 in [16] with the Repre-
sentation theorem 3.5 in [34]. Indeed, these two theorems together imply that the risk
functional associated with any law-invariant coherent risk measure on L1 is 1-weakly
continuous. For 1/2 α < 1 the functional itself is derived from a law-invariant
α
≤ R
coherent risk measure (see Proposition A.2 in Appendix A). So it is 1-weakly continu-
ous. For 0 < α < 1/2 the map ρˇ : L1 R defined by ρˇ (X) := ρ ( X) provides a
α α α
→ − −
law-invariant coherent risk measure (cf. Proposition A.2 in Appendix A), so that the as-
sociated statistical functional ˇ : F R, ˇ (F) = (Fˇ), is 1-weakly continuous.
α 1 α α
R 7→ R −R
Here Fˇ stands for the distribution function derived from F via Fˇ(x) := 1 F(( x) ).
− − −
Sinceforanysequence (Fn)n∈N0 ⊆ F1,Fn → F0 1-weaklyifandonlyifFˇn → Fˇ0 1-weakly,
it follows that also the functional is 1-weakly continuous.
α
R
By the 1-weak continuity of we are in the position to easily derive strong consis-
α
R
tency of the plug-in estimator (F ) for (F) in several situations; see Sections 3.1
α n α
R R
and 4.1.
b
5
2.2. Differentiability
We will use the notion of quasi-Hadamard differentiability introduced in [7, 8]. Quasi-
Hadamard differentiability is a slight (but useful) generalization of the conventional
tangential Hadamard differentiability. The latter is commonly acknowledged to be a
suitable notion of differentiability in the context of the functional delta-method (see e.g.
the bottom of p.166 in [28]), and it was shown in [7, 8] that the former is still strong
enough to obtain a functional delta-method. Let L be equipped with the norm .
1 1,ℓ
k·k
Definition 2.2 Let : F R be a map and L0 be a subset of L . Then is said
R 1 → 1 1 R
to be quasi-Hadamard differentiable at F F tangentially to L0 L if there exists a
∈ 1 1h 1i
continuous map ˙ : L0 R such that
RF 1 →
(F +ε v ) (F)
lim ˙ (v) R n n −R = 0 (10)
F
n→∞(cid:12)R − εn (cid:12)
(cid:12) (cid:12)
holds for each triplet (v,(cid:12)(v ),(ε )), with v L0, (ε ) (cid:12) (0, ) satisfying ε 0,
n n ∈ 1 n ⊆ ∞ n →
(v ) L satisfying v v 0 as well as (F +ε v ) F . In this case the map
n 1 n 1,ℓ n n 1
⊆ k − k → ⊆
˙ is called quasi-Hadamard derivative of at F tangentially to L0 L .
RF R 1h 1i
Notethat even when L0 = L , quasi-Hadamard differentiability of at F tangentially
1 1 R
to L L is not the same as Hadamard differentiability of at F tangentially to L
1 1 1
h i R
(with L regarded as the basic linear space containing both F and L ). Indeed,
0 1 1 1,ℓ
k·k
does not impose a norm on all of L (but only on L ), so that Hadamard differentiability
0 1
w.r.t. the norm is not defined.
1,ℓ
k·k
Theorem 2.3 Let F F and assume that it is continuous at (F). Then the func-
1 α
∈ R
tional : F R is quasi-Hadamard differentiable at F tangentially to L L with
α 1 1 1
R → h i
linear quasi-Hadamard derivative ˙ : L R given by
α;F 1
R →
(1 α) v(x+ (F))ℓ(dx) + α v(x+ (F))ℓ(dx)
˙α;F(v) := − ´(−∞,0) Rα ´(0,∞) Rα . (11)
R − (1 2α)F( (F)) + α
α
− R
Note that (1 2α)F( (F)) + α = (1 α)F( (F))+α(1 F( (F))) > 0 holds
α α α
− R − R − R
so that the denominator in (11) is strictly positive. Also note that quasi-Hadamard dif-
ferentiability is already known form Theorem 2.4 in [35]. However, in [35] the derivative
was not specified explicitly. The proof of Theorem 2.3 can be found in Section 5.
Remark 2.4 As a direct consequence of Theorem 2.3 we obtain that the functional
α
R
is also quasi-Hadamard differentiable at F (being continuous at (F)) tangentially to
α
R
any subspace of L that is equipped with a norm being at least as strict as the norm
1
✸
.
1,ℓ
k·k
6
Example 2.5 To illustrate Remark 2.4, let φ : R [1, ) be a continuous function
→ ∞
that is non-increasing on ( ,0] and non-decreasing on [0, ). Let F be the set
φ
−∞ ∞
of all distribution functions F on R for which F < , where v :=
[0, ) φ φ
sup v(x) φ(x). Let D be the space of all bouknde−d 1c`ad∞l`agk funct∞ions on Rkankd D
x∈R| | φ
be the subspace of all v D satisfying v < and lim v(x) = 0. If C :=
φ x φ
∈ k k ∞ | |→∞| |
1/φdℓ < , then D L and F F . On the space D the norm is stricter
φ 1 φ 1 φ φ
∞ ⊆ ⊆ k·k
´than , because
1,ℓ
k·k
v = v(x) ℓ(dx) C v for every v D . (12)
k k1,ℓ ˆ | | ≤ φk kφ ∈ φ
Therefore is also quasi-Hadamard differentiable at F tangentially to D D with
α φ φ
R h i
linear quasi-Hadamard derivative ˙ : D R given by (11) restricted to v D ,
α;F φ φ
R → ∈
where D is equipped with the norm . ✸
φ φ
k·k
The established quasi-Hadamard differentiability of brings us in the position to
α
R
easily derive results on the asymptotics of (F ); see Sections 3.2–3.3 and 4.2. In
α n
R
Section 3.2 we combine Theorem 2.3 with a central limit theorem (by Dede [19]; cf.
Theorem C.3 below) for the empirical process in tbhe space (L , ) in order to obtain
1 1,ℓ
k·k
the asymptotic distribution of (F ) in a rather general nonparametric setting. In
α n
R
view of Example 2.5 one can alternatively use central limit theorems for the empirical
process in the space (D , ) to obbtain the asymptotic distribution of (F ). See,
φ φ α n
k·k R
for instance, Examples 4.4–4.5 in [8] as well as references cited there.
b
3. Nonparametric estimation of (F)
α
R
In thissection we consider nonparametric statistical models. Wewill always assume that
the sequence of observations (X ) is a strictly stationary sequence of real-valued random
i
variables. In addition we will mostly assume that (X ) is ergodic; see Section 6.1 and
i
6.7 in [14] for the definition of a strictly stationary and ergodic sequence. Recall that
every sequence of i.i.d. random variables is strictly stationary and ergodic. Moreover a
strictly stationary sequence is ergodic when it is mixing in the ergodic sense, and it is
mixing in the ergodic sense when it is α-mixing; see Section 2.5 in [13]. For illustration,
also note that many GARCH processes are strictly stationary and ergodic; cf. [11, 42].
Throughout this section the estimator for the marginal distribution function F of
(X ) is assumed to be the empirical distribution function F of X ,...,X as defined
i n 1 n
in (7). Note that the mapping Ω F , ω F (ω, ), is ( , (F ))-measurable for
1 n 1
→ 7→ · F B
the Borel σ-algebra (F ) on (F ,d ), because the mappinbg Rn F , (x ,...,x )
1 1 W,1 1 1 n
B → 7→
1 n , is( ,d )-continuous. Hence bybcontinuity of w.r.t. d , it follows
tnhati=11([Fxi,∞))is a rkea·kl-vaWlu,1ed random variable on (Ω, ,P). Rα W,1
P Rα n F
b
7
3.1. Strong consistency
For 1/2 α < 1 the following theorem is a direct consequence of Theorem 2.6 in [34].
≤
In the general case, Theorem 2.1 ensures that one can follow the lines in the proof of
Theorem 2.6 in [34] to obtain the assertion of Theorem 3.1; we omit the details.
Theorem 3.1 Let (X ) be a strictly stationary and ergodic sequence of L1-random vari-
i
ables on some probability space (Ω, ,P), and denote by F the distribution function of
F
the X . Let F be the empirical distribution function of X ,...,X as defined in (7).
i n 1 n
Then the plug-in estimator (F ) is strongly consistent for (F) in the sense that
α n α
R R
b
(F ) (F) P-a.s.
αb n α
R → R
If X ,X ,... are i.i.d. random variables, then strong consistency can also be obtained
1 2 b
from classical results on Z-estimators as, for example, Lemma A in Section 7.2.1 of [46].
Moreover, it was shown recently by Holzmann and Klar [29, Theorem 2] that in the i.i.d.
case one even has sup (F ) (F) 0 P-a.s. for any α ,α (0,1) with
α∈[αℓ,αu]|Rα n → Rα | → ℓ u ∈
α < α .
ℓ u
b
3.2. Asymptotic distribution
Dedecker and Prieur [20] introduced the following dependence coefficients for a strictly
stationary sequence of real-valued random variables (Xi) (Xi)i N on some probability
space (Ω, ,P): ≡ ∈
F
φ(n) := sup sup P[X ( ,x] k]( ) P[X ( ,x]] , (13)
k N x R k n+k ∈ −∞ |F1 · − n+k ∈ −∞ k∞
∈ ∈
α(n) := sup sup P[X ( ,x] k]( ) P[X ( ,x]] . (14)
e k N x R k n+k ∈ −∞ |F1 · − n+k ∈ −∞ k1
∈ ∈
Here ke:= σ(X ,...,X ) and denotes the usual Lp-norm on Lp = Lp(Ω, ,P),
F1 1 k k · kp F
p [1, ]. Note that by Proposition 3.22 in [12] the usual φ- and α-mixing coefficients
∈ ∞
φ(n) and α(n) can be represented as in (13)–(14) with sup and ( ,x] replaced by
x∈R −∞
sup and A, respectively. In particular, φ(n) φ(n) and α(n) α(n). It is worth
A∈B(R) ≤ ≤
mentioning that in [20] the starting point is actually a strictly stationary sequence of
random variables indexed by Z (rather than Ne) and that therefeore the definitions of the
above dependence coefficients are slightly different. However, it is discussed in detail in
the Appendix D that any strictly stationary sequence (Xi) (Xi)i N can be extended
≡ ∈
to a strictly stationary sequence (Yi)i Z satisfying φ(n) = φ(n) and α(n) = α(n), where
∈
φ(n) and α(n) are the dependence coefficients of (Yi)i Z as originally introduced in [20].
It is also discussed in the Appendix D that if (Xi) eis i∈n addition ergoedic, then (Yi)i Z is
∈
ergodic too.
Let us denote by Q the c`adl`ag inverse of the tail function x P[ X > x]. Let us
|X1| 7→ | 1|
write N for the centered normal distribution with variance s2. Moreover, let us use
0,s2
❀
to denote convergence in distribution.
8
Theorem 3.2 Let (X ) be a strictly stationary and ergodic sequence of real-valued ran-
i
dom variables on some probability space (Ω, ,P). Denote by F the distribution function
F
of the X , and assume that F is continuous at (F) and that F(1 F)dℓ <
i α
R − ∞
(in particular F F ). Let F be the empirical distribution func´tion of X ,...,X as
∈ 1 n p 1 n
defined in (7). Finally assume that one of the following two conditions holds:
b
n 1/2φ(n)1/2 < , (15)
−
∞
n N
X∈
e
n 1/2 Q (u)u 1/2ℓ(du) < . (16)
n N − ˆ(0,αe(n)) |X1| − ∞
X∈
Then
√n( (F ) (F)) ❀ Z in (R, (R))
α n α F
R −R B
for Z N with
F ∼ 0,s2 b
s2 = s2 := f (t )C (t ,t )f (t )(ℓ ℓ)(d(t ,t )), (17)
α,F ˆ α,F 0 F 0 1 α,F 1 ⊗ 0 1
R2
where
1
f (t) := (1 α) (t)+α (t) , (18)
α,F (1 2α)F( α(F))+α − 1(−∞,Rα(F)] 1(Rα(F),∞)
− R (cid:16) (cid:17)
1
∞
C (t ,t ) := F(t t )(1 F(t t ))+ Cov( , ). (19)
F 0 1 0 ∧ 1 − 0 ∨ 1 1{X1≤ti} 1{Xk≤ti−1}
i=0 k=2
XX
Proof Theorem 2.3 shows that is quasi-Hadamard differentiable at F tangentially
α
R
to L L (w.r.t. the norm ) with quasi-Hadamard derivative ˙ given by (11).
1 1 1,ℓ α;F
h i k·k R
The functional delta-methodin theformof Theorem B.3(i) and Theorem C.3 thenimply
that √n( (F ) (F)) converges in distribution to ˙ (B ), where B is an L -
α n α α;F F F 1
R −R R
valued centered Gaussian random variable with covariance operator Φ given by (60).
BF
Now, ˙ (Bb) = f (x)B (x)ℓ(dx) for the L -function f given by (18). Since
α;F F α,F F α,F
R − ∞
B is a centered Gau´ssian random element of L , and since f represents a continuous
F 1 α,F
linear functional on L , the random variable ˙ (B ) is normally distributed with zero
1 α;F F
R
mean and variance Var[ ˙ (B )] = E[ ˙ (B )2] = Φ (f ,f ), and the latter
Rα;F F Rα;F F BF α,F α,F
✷
expression is equal to the right-hand side in (17).
Note that when X ,X ,... are i.i.d. random variables, then (15) and (16) are clearly
1 2
satisfied and the expression for the variance s2 in (17) simplifies insofar as the sum
1i=0 ∞k=2(···)in(19)vanishes, sothats2 = E[Uα(X1−Rα(F))2]/dF(α)2 withdF(α) :=
(1 2α)F( (F))+α. The latter may be seen by applying Hoeffding’s variance formula
P− P Rα
(cf., e.g., Lemma 5.24 in [40]) to calculate Var[(X (F))+] and Var[(X (F)) ]
1 α 1 α −
−R −R
(take into account that by (3) we have E[U (X (F))2] = Var[U (X (F))],
α 1 α α 1 α
− R − R
and obviously Cov((X (F))+,(X (F)) ) = 0). But even in this case the
1 α 1 α −
− R − R
9
variance s2 depends on the unknown distribution function F in a fairly complex way.
So, for the derivation of asymptotic confidence intervals the bootstrap results of Section
3.3 are expected to lead to a more efficient method than the method that is based on
the nonparametric estimation of s2 = s2 .
α,F
Remark 3.3 In the i.i.d. case Theorem 3.2 can also be obtained from classical results
on Z-estimators as, for example, Theorem A in Section 7.2.2 of [46]. Recently Holzmann
and Klar [29, Theorem 7] showed that, still in the i.i.d. case, continuity of F at (F)
α
R
is even necessary in order to obtain a normal limit. It is also worth mentioning that the
integrability condition on F in Theorem 3.2 is slightly stronger than needed, at least in
the i.i.d. case. Holzmann and Klar [29, Corollary 4] only assumed that F possesses a
finite second absolute moment which is slightly weaker than assuming our integrability
✸
condition.
Remark 3.4 The following assertions illustrate the assumptions of Theorem 3.2.
(i) The integrability condition F(1 F)dℓ < holds if φ2dF < for some
− ∞ ∞
continuous function φ : R ´ [1, ) satisfying 1/φdℓ <´ and being strictly
→p ∞ ∞
decreasing and strictly increasing on R and R ´, respectively.
+
−
(ii) Condition (15) holds if φ(n) = (n b) for some b > 1.
−
O
(iii) Condition (16) implies condition (15) with φ(n) replaced by α(n).
e
(iv) Condition (16) is equivalent to
e
e
n 1/2 α(n)1/2 P[ X > x]1/2ℓ(dx) < .
− ˆ ∧ | 1| ∞
Xn∈N (0,∞)
e
(v) Condition (16) holds if F(1 F)dℓ < and α(n) = (n b) for some b > 1.
−
− ∞ O
´
p ✸
See Section 6.1 for the proofs of these assertions.
e
3.3. Bootstrap consistency
In this section we present two results on bootstrap consistency in the setting of Theorem
3.2. In the following Theorem 3.5 we will assume that the random variables X ,X ,...
1 2
are i.i.d. In Theorem 3.6 ahead we will assume that the sequence (X ) is β-mixing. We
i
will use ̺ to denote the bounded Lipschitz metric on the set of all Borel probability
BL
measures on R; see the Appendix B for the definition of the bounded Lipschitz metric.
By P we will mean the law of a random variable ξ under P, and as before N refers
′ξ ′ 0,s2
to the centered normal distribution with variance s2.
10