Characteristic function-based goodness-of-fit tests
under weak dependence
Anne Leucht
Friedrich-Schiller-Universität Jena
Institut für Stochastik
Ernst-Abbe-Platz 2
D-07743 Jena
Germany
E-mail: [email protected]
Phone: ++49 (0)3641 946 273
Fax: ++49 (0)3641 946 252
Abstract
In this article we propose two consistent hypothesis tests of L2 -type for weakly dependent
observations based on the empirical characteristic function. We consider a symmetry test
and a goodness-of-fit test for the marginal distribution of a time series. The asymptotic
behaviour under the null as well as under fixed and certain local alternatives are investigated. Since the limit distributions of the test statistics depend on unknown parameters
in a complicated way, we suggest to apply certain parametric bootstrap methods in order
to determine critical values of the tests.
2010 Mathematics Subject Classification. 62G10, 62E20, 62F40.
Keywords and Phrases. Bootstrap; empirical characteristic function; goodness-offit; symmetry test; time series; V -statistics.
Short title. Hypothesis tests under dependence.
2
1. Introduction and Motivation
Time series models are well-established tools to describe various real-life phenomena, e.g. in finance and biometrics. In the present paper we provide consistent
testing procedures that allow for checking adequacy of a certain model with respect to symmetry and parametric structure of the marginal distribution of the
time series.
Let X1 , . . . , Xn be Rd -valued observations of a stationary weakly dependent process with marginal distribution PX . We consider problems of the following structure:
H0 : gX (·) = fX (·, θ0 ) for some θ0 ∈ Θ ⊆ Rp against
H1 : gX (·) 6= fX (·, θ)
∀ θ ∈ Θ.
Besides supremum-type tests, L2 -tests are the most convenient ones
in matheR
matical statistics. We apply a test statistic of L2 -type, Tbn = n Rd [ψ(cn )(t) −
fX (t, θbn )]2 w(t)dt. Here, θbn is a consistent estimator of θ0 and cn denotes the emPn
0
pirical characteristic function, i.e. cn (t) = n−1 k=1 eit Xk . The function ψ is chosen such that ψ(cn ) forms an appropriate nonparametric estimate of gX = ψ(cX ),
where cX is the characteristic function of X1 . We specify the functions ψ and fX
for the concrete test problems in the Sections 2 and 3. There are two basic approaches in order to derive the asymptotic null distribution of Tbn . One can either
invoke empirical process theory or approximate the test statistic by a degenerate
V -statistic and apply asymptotic results on these quantities. Here, we employ a
recent result on degenerate V -statistics by Leucht [22] that allows for the derivation
of the limit distributions under easily verifiable assumptions. In order to study the
behaviour of our test statistics under local alternatives, this approach is carried
over to a special kind of triangular schemes of random variables; cf. Section 4.2.
In Section 2, we extend the symmetry test which was initially proposed by
Feuerverger and Mureika [15] for independent and identically distributed (i.i.d.)
data and a known center of symmetry to the case of weakly dependent observations
with unknown location parameter. Answering the question whether a distribution
is symmetric or not is interesting for several reasons. First, there is a pure statistical
interest since presence or absence of symmetry is important for deciding what parameter to estimate. If the underlying distribution is symmetric, the point of symmetry is the only reasonable location parameter, whereas in the non-symmetric case
there is no longer only one measure of location; cf. Antille, Kersting and Zucchini [1].
Moreover, robust estimators of location, e.g. trimmed means, and robust tests for
location parameters assume the underlying observations to arise from a symmetric
distribution; see for instance Staudte and Sheather [29]. Consequently it is important to check this assumption before applying those methods. Rejecting the hypothesis of symmetry also has substantial impact on model selection. In this case, fitting
data to nonlinear AR(p) processes with skew symmetric regression functions and
symmetric innovations is inappropriate; cf. Pemberton and Tong [26]. Finally, symmetry plays a central role in analyzing and modeling business circles since time reversibility of a process (Xt )t∈Z , i.e. PXt1 ,...,Xtk = PXtk ,...,Xt1 , t1 , . . . , tk ∈ Z, k ∈ N,
implies symmetry of the distributions of the differences Yt,k = Xt − Xt−k , k ∈ N,
3
about the origin. For a detailed discussion of this topic see Ramsey and Rothman
[27] and Chen et al. [8].
A great variety of symmetry tests for i.i.d. random variables are available in the
literature. Detailed lists of references are given by Lee [21] as well as by Henze, Klar
and Meintanis [18]. Also in the context of time series numerous tests of symmetry
have been employed. There are several moment-based tests, e.g. by Ramsey and
Rothman [27] or Bai and Ng [2]. However, both approaches are inconsistent against
alternatives whose third moments vanish but still are not symmetric. Lee [21] built
an M test for the composite hypothesis of symmetry around an unknown location
parameter and derived asymptotic normality under the null and certain local alternatives. However, a consistency result is not stated. Fan and Ullah [14] considered
a consistent L2 -test for the simple hypothesis based on kernel density estimates
with vanishing bandwidth under the assumption that the sample arises from a certain absolute regular process. The test for the simple hypothesis by Chen et al. [8]
employs the fact that a distribution is symmetric if and only if the imaginary part of
its characteristic function =c(t) = E sin(t0 X) vanishes. No moment restrictions on
the marginal distributions of the underlying data are required. They investigated
the asymptotics under the null and certain local alternatives but they did not derive consistency of their test. We also establish a characteristic function-based test
for symmetry that is asymptotically unbiased against Pitman local alternatives.
In contrast to Chen et al. [8], we consider the composite hypothesis of symmetry
around an unknown center and obtain consistency of our approach.
In Section 3, a goodness-of-fit test for the marginal distribution of a time series
is constructed. So far, the normality assumption is still dominating the literature
in many fields, however, not for reasons of empirical evidence but for theoretical
simplicity. Within the last decades the literature on non-Gaussian time series has
addressed the topic of finding distributions of the innovations corresponding to
specified marginals of a certain model. Surveys of those results are given by Jose,
Lishamol and Sreekumar [20] as well as by Block, Langberg and Stoffer [6].
While there is a great variety of tests concerning the problem whether the distribution of a sample belongs to a parametric class of distributions if the underlying
observations are i.i.d., the number of consistent tests is limited in the dependent
case. Recently, Ignaccolo [19] and Munk et al. [23] developed results for α-mixing
variables generalizing Neyman’s smooth test for a simple null hypothesis. Tests for
the composite hypothesis based on the L2 -difference between a smoothed version
of the parametric density estimate and a nonparametric estimator were considered
by Fan and Ullah [14] as well as by Neumann and Paparoditis [24]. The bandwidths involved in the estimators were supposed to be asymptotically vanishing. A
drawback resulting from the latter assumption is the loss of power against Pitman
local alternatives compared to approaches with fixed bandwidths; see Ghosh and
Huang [16]. Motivated by the fact that tests based on kernel density estimators
with decreasing smoothing parameter are very sensitive to the choice of the bandwidth, Fan [13] established a characteristic-function based test for the composite
hypothesis in the i.i.d. case that evaluates the difference between the characteristic
function and its fixed-kernel estimate, the empirical characteristic function. To the
4
author’s best knowledge this approach has not been considered if the observations
are dependent. Here, we extend the L2 -test of Fan [13] to the time series setting.
This type of test is useful in particular when the density has a complicated structure while the characteristic function is of simple form. A typical example of such
a distribution is the normal inverse Gaussian distribution that is widely used in
mathematical finance, see e.g. Barndorff-Nielsen [3].
For the above-named tests, we derive their limit distributions under the respective null hypotheses and under certain local alternatives. It turns out that already in
the case of i.i.d. observations the asymptotic distributions depend on unknown parameters in a complicated way. The bootstrap offers a convenient way to determine
critical values of the tests. However, a naive application of these resampling methods fails. We obtain bootstrap consistency after a modification of the test statistics
on the bootstrap side. It is well-known that model-based bootstrap counterparts
of the underlying observations can often not proved to be mixing even though the
original process satisfies some mixing condition. However, other concepts of weak
dependence can be verified, see e.g. Bickel and Bühlmann [4] or Doukhan and Neumann [12]. Throughout the paper, the underlying process is assumed to meet the
following definition, which is due to Dedecker and Prieur [10].
Definition 1.1. Let (Ω, A, P ) be a probability space and (Xk )k∈N be a stationary sequence of integrable Rd -valued random variables. The process is called τ dependent if
1
sup
{τ (σ(X1 , . . . , Xi ), (Xk1 , . . . , Xkl ))} −→ 0,
τr = sup
r→∞
l∈N l r+i≤k1 ,...,kl
where
τ (M, X) = E
sup
f ∈Λ1 (Rp )
Z
Rp
Z
f (x)dPX|M (x) −
Rp
!
f (x)dPX (x) .
Here, M is a sub-σ-algebra of A, PX|M denotes the conditional distribution of the
Rp -valued random variable X given M, and Λ1 (Rp ) denotes the set of 1-Lipschitz
functions from Rp to R.
e that is independent of
If Ω is rich enough, there exists a random variable X
d
e
M = σ(X1 , . . . , Xi ) such that X = X and
e ,
τ (M, X) = EX − X
(1.1)
l1
Pp
where kxkl1 = j=1 |xj |, x ∈ Rp . Actually, τ (M, X) is the minimal distance bed
tween X and any Y = X that is independent of M. This L1 -coupling property of
the τ -coefficient was verified by Dedecker and Prieur [9] and will be essential for all
our proofs below. Dedecker and Prieur [10] discussed relations of the τ -coefficient to
ordinary mixing coefficients. Additionally, they provided an extensive list of examples for τ -dependent processes including causal linear and functional autoregressive
processes. Note that also ARMA processes with weakly dependent innovations and
GARCH processes are τ -dependent; see Shao and Wu [28]. We conclude this section mentioning that all proofs and asymptotic results on V -statistics under local
alternatives a deferred to a final Section 4.
5
2. Testing symmetry
2.1. The test statistic and its asymptotics. Suppose that X1 , . . . , Xn are Rd valued random variables with distribution PX and characteristic function cX . We
establish a test for the problem
H0 : PX−µ0 = Pµ0 −X
for some µ0 ∈ Rd
vs.
H1 : PX−µ 6= Pµ−X
∀ µ ∈ Rd
that can be reformulated in terms of the imaginary part of the corresponding characteristic functions:
H0 : =(cX−µ0 ) ≡ 0
for some µ0 ∈ Rd
vs.
H1 : =(cX−µ ) 6= 0 ∀ µ ∈ Rd .
Inspired by Feuerverger and Mureika [15], who tested for symmetry around the origin in the case of i.i.d. univariate observations, and Henze, Klar and Meintanis [18],
who investigated the composite hypothesis for i.i.d. multivariate random variables,
we suggest the test statistic
Z
Sbn = n
2
0
=(e−it µbn cn (t)) w(t) dt =
Rd
Rd
Here, µ
bn denotes a
tions:
Z
√
"
#2
n
1 X
0
√
sin(t (Xk − µ
bn )) w(t) dt.
n
k=1
n-consistent estimator of µ0 . We make the following assump-
(S1)(i) (Xn )n∈N is a sequence of τ -dependent random variables with values in Rd
P∞
2
and with r=1 r(τr )δ < ∞ for some δ ∈ (0, 1).
(ii) EkX1 k1+ε
< ∞ and all ε ∈ (0, 1/(3 − 2δ)).
l1
(iii) The sequence of estimators µ
bn admits the expansion
n
1X
l(Xk , µ0 ) + oP (n−1/2 ),
µ
bn − µ0 =
n
k=1
where l : Rd ×Rd → R is a Lipschitz continuous function satisfying El(X1 , µ0 )
= 0d and Ekl(X1 , µ0 )k2ν
l1 < ∞ for some ν > (2 − δ)/(1 − δ). (Here, 0d denotes the d-dimensional null vector.)
(iv) The function w is a measurable, positive
almost everywhere w.r.t. the
R
Lebesgue measure on Rd and satisfies Rd (1 + ktk2ν
l1 ) w(t) dt < ∞.
Remark 2.1.
(i) The assumption of l being Lipschitz continuous can be weakened at the expense of additional moment constraints. However, if one
simply uses the arithmetic mean to estimate µ0 , (S1)(ii) is satisfied if
EkX1 k2ν
l1 < ∞.
(ii) Note that it does not suffice to postulate strict positivity of w on some
interval around the centre of symmetry since there are non-symmetric distributions whose characteristic functions are real in a certain neighbourhood
of the origin. For an example, see Ushakov [31], Ex. 21, Appendix A.
6
Under these weak constraints Sbn can be approximated by a degenerate V statistic (multiplied by n) with fixed parameter µ0 ,
n Z
1 X
Sn =
[sin(t0 (Xj − µ0 )) − cX−µ0 (t) t0 l(Xj , µ0 )]
n
Rd
j,k=1
× [sin(t0 (Xk − µ0 )) − cX−µ0 (t) t0 l(Xk , µ0 )] w(t) dt.
Lemma 2.1. Suppose that assumption (S1) holds. Then, under H0 ,
Sbn − Sn = oP (1).
Due to the above approximation, the asymptotic distributions of the test statistic
and the degenerate V -statistic Sn coincide. We obtain the limit of the latter variable
by applying Theorem 2.2 of Leucht [22].
Theorem 2.1. Suppose that assumption (S1) holds. Then, under the null hy(c)
pothesis, there are families of constants (γj;k1 ,k2 (µ0 ))c,j,k1 ,k2 , (Cj;k1 ,k2 )j,k1 ,k2 and
centered, jointly normal random variables (Zj;k )j,k such that
d
Sbn −→ Z1 := lim
∞
X
c→∞
X
(c)
γj;k1 ,k2 (µ0 ) Zj;k1 Zj;k2 + Cj;k1 ,k2
j=1 k1 ,k2 ∈Zd
Z
+E
2
[sin(t0 (X1 − µ0 )) − cX−µ0 (t) t0 l(X1 , µ0 )] w(t) dt.
Rd
Here, the r.h.s. converges in the L2 -sense.
Remark 2.2. The limit distribution of the test statistic is of a complicated structure.
Therefore, problems arise as soon as (asymptotic) critical values of the test have to
be determined. In the following subsection we propose the application of certain
bootstrap methods in order to circumvent these difficulties.
Next, the behaviour of the test statistic under fixed alternatives is considered.
Lemma 2.2. Suppose that (S1)(i) and (iii) hold. Moreover, assume that there
P
exists a constant µ0 ∈ Rd such that µ
bn −→ µ0 . Then, under H1 ,
P Sbn > K −→ 1, ∀ K < ∞.
n→∞
Finally, we study the asymptotic behaviour of Sbn under Pitman local alternatives. More precisely, we consider alternatives
H1,n :
dPXn,1
gn
= 1+ √
dPX1
n
with kgn −gk∞ −→ 0 and =cXn,1 −µ (t) 6= 0, ∀µ ∈ Rd ,
n→∞
where g is supposed to be a measurable bounded function. To this end, we assume
(S2)(i) The triangular scheme (Xn,k )nk=1 , n ∈ N, of row-wise stationary and Rd P∞
δ2
valued random variables fulfils r=1 r (τ̄r ) < ∞ for some δ ∈ (0, 1). Here,
the sequence (τ̄r )r∈N is defined as τ̄r := supn>r τr,n with
1
τr,n := sup
sup
τ (σ(Xn,1 , . . . , Xn,i ), (Xn,k1 , . . . , Xn,kl )) .
l∈N l r+i≤k1 ,...,kl
7
(ii) There exists a sequence of random variables (Xn )n∈N satisfying assump0 d
P∞
δ2
0
0
tion (S1)(i) with
< ∞ and such that Xn,t
, Xn,t
−→
r=1 r (τr )
1
2
0
Xt01 , Xt02 , 1 ≤ t1 , t2 ≤ n, n ∈ N. The Radon Nikodym derivatives
are given by dPXn,1 /dPX1 = 1 + n−1/2 gn , where (gn )n∈N is a sequence of
functions with kgn − gk∞ −→n→∞ 0 for some bounded measurable function g.
In Subsection 4.2 we establish asymptotic results on V -statistics under local
alternatives for dependent random variables. Invoking these findings, we obtain
the subsequent assertion.
Proposition 2.1. Suppose that the conditions (S1) and (S2) are satisfied. AddiPn
tionally, assume that µ
bn (Xn,1 , . . . , Xn,n ) − µ0 = n−1 k=1 l(Xn,k , µ0 ) + oP (n−1/2 ).
Then, under H1,n ,
d
Sbn −→ Z1,loc := lim
∞
X
c→∞
X
(c)
γj;k1 ,k2 (µ0 ) {Zj;k1 + Dj;k1 }{Zj;k2 + Dj;k2 } + Cj;k1 ,k2
j=1 k1 ,k2 ∈Zd
Z
+E
2
[sin(t0 (X1 − µ0 )) − cX−µ0 (t) t0 l(X1 , µ0 )] w(t) dt.
Rd
with a family of constants (Dj;k )j,k . If additionally var(Z1 ) > 0, then
h
i
lim inf PH1,n Sbn > x − PH0 Sbn > x ≥ 0, ∀ x ∈ R.
n→∞
2.2. Bootstrap validity. The foregoing results suggest to reject the hypothesis
of symmetry at asymptotic significance level α if Sbn > tα . Here, tα denotes the
(1 − α)-quantile of the distribution of Z1 . The latter random variable depends on
the unknown parameter and the dependence structure of the underlying process
(Xt )t in a complicated way and thus (asymptotic) critical values can hardly be
derived analytically or be tabulated. Therefore, we consider a parametric bootstrap
procedure to approximate these quantities.
Given n := (X10 , . . . , Xn0 )0 , let X ∗ and Y ∗ denote vectors of bootstrap random
variables with values in Rd1 and Rd2 , respectively. To describe the dependence
structure of the bootstrap sample, we define, in analogy to Definition 1.1,
X
τ ∗ (Y ∗ , X ∗ , xn ) :=
Z
Z
E
sup f (x)PX ∗ |Y ∗ (dx) −
d1
d
f ∈Λ1 (R
1)
Rd1
R
f (x)PX ∗ (dx) Xn = xn
!
,
X
provided that E(kX ∗ kl1 | n = xn ) < ∞. We make the following assumption concerning the bootstrap sample (Xk∗ )k and the corresponding estimator µ∗n (X1∗ , . . . , Xn∗ )
of the location parameter:
(S3)(i) The sequence of bootstrap variables is stationary with probability tending
d
to one. Additionally, (Xt∗01 , Xt∗02 )0 −→ (Xt01 , Xt02 )0 , ∀ t1 , t2 ∈ N, holds true in
probability.
8
X
(ii) Conditionally on n , the random variables (Xk∗ )k∈Z are τ -weakly depenP∞
2
dent, i.e. there exist a sequence of coefficients (τ̄r )r∈N with r=1 r(τ̄r )δ <
∞ for some δ ∈ (0, 1) and a sequence of sets (Xn )n∈N with P ( n ∈
Xn )−→n→∞ 1 such that the following property holds: For any sequence
(xn )n∈N with xn ∈ Xn , supk∈N E(kXk∗ kl1 | n = xn ) < ∞ and
∗
1
τr∗ (xn ) := sup
sup
τ (X1∗ , . . . , Xi∗ ), (Xk∗01 , . . . , Xk∗0l )0 , xn ≤ τ̄r .
l∈N l r+i≤k1 ,...,kl
X
X
1
(iii) E∗ (kX1∗ k1+ε
l1 ) = OP (1) for all ε ∈ (0, 1/(3 − 2δ)).
∗
(iv) The sequence of estimators µ
bn admits the expansion
n
1X
l(Xk∗ , µ
bn ) + oP ∗ (n−1/2 ),
µ
b∗n − µ
bn =
n
k=1
∗
l(X1∗ , µ
bn )
2
with E
δ)/(1 − δ).
= 0d and E∗ kl(X1∗ , µ
bn )k2ν
l1 = OP (1) for some ν > (2 −
Remark 2.3.
(i) We do not assume the bootstrap variables to be stationary
for all sufficiently large n. The reason for using the weaker assumption
above is that for instance the ordinary ARCH(p)-bootstrap method with
estimated parameters would violate this assumption; see also Neumann and
Paparoditis [25].
(ii) Neumann and Paparoditis [25] proved that in case of stationary Markov
chains of finite order, the key for convergence of the finite-dimensional distributions is convergence of the conditional distributions, cf. their Lemma 4.2.
In particular, they showed that parametric AR(p)- and ARCH(p)-bootstrap
methods yield samples that satisfy (S2)(i). Also sieve bootstrap methods
for linear processes exhibit this property; see Bühlmann [7].
Intuitively, one might consider to use the bootstrap counterpart of Sbn as the
bootstrap test statistic. Unfortunately, it turns out that the bootstrap counterpart
of the approximating V -statistic Sn is no longer degenerate in general. This is
however not surprising since a similar problem occurs when Efron’s bootstrap is
applied to V -type statistics of i.i.d. data. Dehling and Mikosch [11] proposed to
degenerate the kernel on the bootstrap side artificially to overcome this difficulty.
We will proceed analogously, cf. the proof of Proposition 2.2 below. For this reason
the additional term −=[c∗ (t)] has to be included in the integrand of the bootstrap
statistic Sbn∗ . Here, c∗ is the characteristic function of X1∗ − µ
bn , conditionally on
n . More precisely, we propose the following bootstrap algorithm:
X
(1) Determine µ
bn such that (S1)(iii) is satisfied.
(2) Generate X1∗ , . . . , Xn∗ such that (S3)(i) to (iii) hold.
(3) Determine µ
b∗n such that (S3)(iv) is satisfied.
1Throughout the paper the expressions P ∗ and E∗ denote the bootstrap distribution and
bootstrap expectation conditionally on X1 , . . . , Xn .
P
2R∗ = o ∗ (a ) if P ∗ (kR k /|a | > ε) −→
0, ∀ ε > 0.
n
n l1
n
P
n
9
(4) Compute the bootstrap version of our test statistic:
)2
Z (
n
1 X
∗
0
∗
∗
∗
b
√
w(t) dt.
Sn :=
sin(t (Xk − µ
bn )) − =[c (t)]
n
Rd
k=1
(5) Define the critical value t∗α as the (1 − α)-quantile of the (conditional)
distribution of Sbn∗ . Reject H0 if Sbn > t∗α .
(In practice, steps (1) to (4) will be repeated B times, for some large B.
The critical value will then be approximated by the (1 − α)-quantile of the
∗
∗
empirical distribution associated with Sbn,1
, . . . , Sbn,B
.)
Employing results of bootstrap for V -statistics, we can verify consistency of the
algorithm above. So far, bootstrap for degenerate V -statistics of dependent data
has only been investigated by Leucht [22]. However, her findings cannot be applied
directly in the present context, cf. proof of Proposition 2.2.
Proposition 2.2. Suppose that assumptions (S1) and (S3) hold and that Sbn∗ is
generated via the afore-mentioned algorithm. Then, under H0 ,
d
Sbn∗ −→ Z1
in probability,
as n → ∞, where Z1 is defined as in Theorem 2.1. Moreover, if var(Z1 ) > 0,
P
sup P ∗ Sbn∗ ≤ x − P Sbn ≤ x −→ 0.
−∞<x<∞
The latter assertion implies that, under the null hypothesis, the bootstrap-based
test has asymptotically the correct size. Moreover, consistency and asymptotic
unbiasedness under Pitman alternatives of the bootstrap-aided test can be verified.
Although Proposition 2.1 merely yields unbiasedness, we conjecture that our test
has non-trivial power against some alternatives of the structure H1,n . In the i.i.d.
case for instance, one can verify non-trivial power for certain local alternatives based
on the asymptotic results of U -statistics under contiguous alternatives of Gregory
[17]. However, technical conditions were imposed that depend on the underlying
distribution in a complicated way and that can hardly be checked here. Since things
become even more involved in the case of dependent observations, we do not pursue
these investigations here.
Corollary 2.1. Suppose that assumptions (S1) and (S3) are fulfilled and let α ∈
(0, 1). The proposed bootstrap test based on the algorithm above satisfies
(
α
if H0 is true and var(Z1 ) > 0,
∗
b
lim P Sn > tα =
n→∞
1
if H1 is true.
Under H1,n and the additional assumptions of Proposition 2.1,
lim inf PH1,n Sbn > t∗α ≥ α.
n→∞
3. A goodness-of-fit test for the marginal distribution
of a time series
3.1. The test statistics and its asymptotics. We extend the test originally
proposed by Fan [13] for i.i.d. random variables to weakly dependent observations.
10
Suppose X1 , . . . , Xn are Rd -valued observations with distribution PX and consider
the test problem
H0 : PX ∈ {P θ | θ ∈ Θ ⊆ Rp }
H1 : PX ∈
/ {P θ | θ ∈ Θ ⊆ Rp }
vs.
which is equivalent to
H0 : cX = c(·, θ0 ) for some θ0 ∈ Θ
vs.
H1 : cX 6= c(·, θ) ∀ θ ∈ Θ.
Here, cX and c(·, θ) denote the characteristic functions associated with PX and
P θ , θ ∈ Θ, respectively. Fan [13] suggested a test statistic of the form
Z
bn = n
cn (t) − c(t, θbn )2 w(t) dt,
G
Rd
√
where θbn is a n-consistent estimator of θ0 and w is an appropriate weight function.
b n , we make the following assumptions:
In order to derive the limit distribution of G
P∞
2
(G1)(i) (Xn )n is a stationary, Rd -valued, τ -dependent process with r=1 r(τr )δ <
(2−δ)/(1−δ)
∞ and EkX1 kl1
< ∞ for some δ > 0.
(ii) The sequence of estimators θbn admits the expansion
n
1X
l(Xk , θ0 ) + oP (n−1/2 ),
θbn − θ0 =
n
(3.1)
k=1
where El(X1 , θ0 ) = 0 and Ekl(X1 , θ0 )k2ν
l1 < ∞ for some ν > (2 − δ)/(1 − δ).
Moreover, kl(x, θ0 ) − l(x̄, θ0 )kl1 ≤ fl (x, x̄, θ0 ) kx − x̄kl1 where, fl (·, θ0 ) is
e1 of X1 and some
continuous and such that for any independent copy X
A > 0,
sup E
j,k∈N
E
max
|fl (Xj + a, Xk , θ0 )|2(2−δ)/(1−δ) < ∞,
max
e1 , θ0 )|2(2−δ)/(1−δ) < ∞.
|fl (X1 + a, X
a∈[−A,A]d
a∈[−A,A]d
(iii) The weight function w : Rd → R is measurable, positive
almost everywhere
R
w.r.t. the Lebesgue measure on Rd and satisfies Rd (1 + ktk2l1 ) w(t) dt < ∞.
(iv) The
function c(t, ·) is twice continuously differentiable ∀ t ∈ Rd with
R
kc(1) (t, θ0 )k2ν
l1 w(t)dt < ∞. Additionally, there exists a neighbourhood
Rd
U (θ0 ) ⊆ Θ such that for all θ ∈ U (θ0 ) every
of c(2) (t, θ) can be
R element
bounded by a symmetric function M with Rd M 2 (t) w(t) dt < ∞.
The expansion (3.1) of θbn − θ0 is a standard assumption in the field of hypothesis
testing. It is often satisfied when maximum-likelihood estimators or the method of
moments are applied. Even though the smoothness assumption on the associated
function l has a rather technical structure, it is satisfied e.g. by polynomial functions
as long as the sample variables have sufficiently many finite moments. Under the
b n can be approximated by a degenerate V -statistic and thus the
null hypothesis, G
limit distribution can be derived similarly to the previous section.
Theorem 3.1. Suppose that assumption (G1) holds. Then, under H0 ,
11
b n − Gn = oP (1), where
(i) G
n
2
1X
(t)
−
c(t,
θ
)
−
[l(Xj , θ0 )]0 c(1) (t, θ0 ) w(t) dt.
cn
0
n
d
R
j=1
Z
Gn = n
(c)
(ii) Moreover, there are families of constants (λj;k1 ,k2 (θ0 ))c,j,k1 ,k2 , (Cj;k1 ,k2 )j,k1 ,k2
and centered, jointly normal random variables (Zj;k )j,k such that
∞
X
d
b n −→
G
Z2 := lim
c→∞
X
(c)
λj;k1 ,k2 (θ0 ) Zj;k1 Zj;k2 + Cj;k1 ,k2
j=1 k1 ,k2 ∈Zd
Z
+E
Rd
2
0
it X1
− c(t, θ0 ) − [l(X1 , θ0 )]0 c(1) (t, θ0 ) w(t) dt.
e
Here, the r.h.s. converges in the L2 -sense.
As in the previous section, the test statistic is asymptotically unbounded under
fixed alternatives.
Lemma 3.1. Suppose that (G1)(i) and (iii) hold. Moreover, assume that there
P
exists a constant θ0 ∈ Θ such that θbn −→ θ0 . Then, under H1 ,
b n > K −→ 1 ∀ K < ∞.
P G
n→∞
We conclude with a result concerning the behaviour of the test statistic under local
alternatives
H1,n :
dPXn,1
gn
= 1 + √ , where kgn − gk∞ −→ 0 for some measurable, bounded
n→∞
dPX1
n
Z
0
function g with
eit x g(x)PX1 (dx) 6= 0 for some t ∈ Rd .
Rd
Proposition 3.1. Suppose that the conditions (G1) and (S2) are satisfied. AddiPn
tionally, assume that θbn,n = θbn (Xn,1 , . . . , Xn,n ) = n−1 k=1 l(Xn,k , θ0 )+oP (n−1/2 )
with
sup
sup E
n∈N 1≤j,k≤n
max
a∈[−A,A]d
|fl (Xn,j + a, Xn,k )|2(2−δ)/(1−δ) < ∞.
Then, under H1,n ,
d
b n −→
G
Z2,loc := lim
∞
X
c→∞
X
(c)
λj;k1 ,k2 (θ0 ) {Zj;k1 + Dj;k1 }{Zj;k2 + Dj;k2 } + Cj;k1 ,k2
j=1 k1 ,k2 ∈Zd
Z
+E
Rd
0
2
it X1
− c(t, θ0 ) − [l(X1 , θ0 )]0 c(1) (t, θ0 ) w(t) dt
e
with a family of constants (Dj;k )j,k . If additionally var(Z2 ) > 0, then
h
i
b n > x − PH G
b n > x ≥ 0 ∀ x ∈ R.
lim inf PH1,n G
0
n→∞
12
3.2. Bootstrap validity. Again we propose to invoke a parametric bootstrap procedure to determine critical values of the test. For this purpose we assume:
(G2)(i) (S3)(i) and (S3)(ii) are satisfied.
(2−δ)/(1−δ) P
(2−δ)/(1−δ)
(ii) E∗ kX1∗ kl1
−→ EkX1 kl1
.
(iii) The sequence of estimators θbn∗ admits the expansion
n
1X
l(Xk∗ , θbn ) + oP ∗ (n−1/2 ),
θbn∗ − θbn =
n
k=1
∗
l(X1∗ , θbn )
where E
= 0 and E∗ kl(X1∗ , θbn )k2ν
l1 = OP (1) for some ν > (2 −
δ)/(1−δ). Additionally, kl(x, θbn )−l(x̄, θbn )kl1 ≤ fl (x, x̄, θbn )kx− x̄kl1 , where
e ∗ of X ∗ , some A > 0 and a K < ∞,
for any independent copy X
1
1
∗
∗
max |fl (Xj + a, Xk∗ , θbn )|2(2−δ)/(1−δ)
P sup E
j,k∈N
+ E∗
a∈[−A,A]d
max
a∈[−A,A]d
e1∗ , θbn )|2(2−δ)/(1−δ) ≤ K
|fl (X1∗ + a, X
(iv) The first derivative of c w.r.t. θ satisfies
U (θ0 ).
R
Rd
−→ 1.
n→∞
kc(1) (t, θ)k2ν
l1 w(t)dt < ∞, ∀ θ ∈
In the i.i.d. case, the validity of parametric bootstrap was derived by Fan [13]. She
employed the bootstrap test statistic
Z
∗
∗
e
cn (t) − c(t, θbn∗ )2 w(t) dt,
Gn = n
Rd
Pn
0
∗
where c∗n is the empirical counterpart of cn , i.e. c∗n (t) = n−1 k=1 eit Xk . When
b n under weak dependence, similar problems as in
bootstrapping the test statistic G
the previous section occur. Again the bootstrap counterpart G∗n of the approximating V -statistic Gn is not degenerate in general. In order to establish a consistent
bootstrap method to determine critical values of the test statistic, we do therefore
e ∗n directly but include the additional term c(t, θbn ) − c∗ (t) in its integrand.
not use G
∗
Here, c is the characteristic function of X1∗ , conditionally on X1 , . . . , Xn . We
suggest the application of the following bootstrap algorithm:
(1) Determine θbn such that (G1)(ii) is satisfied.
(2) Generate X1∗ , . . . , Xn∗ such that (G2)(i) and (ii) hold.
(3) Determine θbn∗ such that (G2)(iii) is satisfied.
(4) Compute the bootstrap test statistic
Z
∗
∗
b
cn (t) − c∗ (t) + c(t, θbn ) − c(t, θbn∗ )2 w(t) dt.
Gn := n
Rd
(5) Define the critical value t∗α as the (1 − α)-quantile of the (conditional)
b ∗n . Reject H0 if P (G
b n > t∗α ) > α.
distribution of G
Remark 3.1. The bootstrap characteristic function c∗ is often unknown and has
to be approximated by simulation. However, note that there are cases where the
bootstrap variables can be generated such that c∗ (t) = c(t, θbn ); see Taufer and Leob ∗n coincides with the exact bootstrap counterpart
nenko [30]. In these situations G
∗
e n of G
bn .
G
13
Proposition 3.2. Suppose that the assumptions (G1) and (G2) are satisfied. Then,
under H0 ,
d
b ∗ −→
G
Z2 in probability,
n
as n → ∞, where Z2 is defined as in Theorem 3.1. Moreover, if var(Z2 ) > 0,
P
b∗ ≤ x − P G
b n ≤ x −→
0.
sup P ∗ G
n
−∞<x<∞
The proposition assures that the bootstrap-based test is asymptotically of correct
size under H0 . Additionally, it is consistent w.r.t. fixed alternatives and asymptotically unbiased against Pitman local alternatives. The asymptotic behaviour of our
bootstrap-aided test can be summarized as follows.
Corollary 3.1. Suppose that the assumptions (G1) and (G2) are fulfilled and let
α ∈ (0, 1). The proposed bootstrap test based on the algorithm above satisfies
(
α
if H0 is true and var(Z2 ) > 0,
∗
b
lim P Gn > tα =
n→∞
1
if H1 is true.
Moreover, under the additional assumptions of Proposition 3.1,
b n > t∗α ≥ α.
lim inf PH1,n G
n→∞
4. Proofs
Throughout this section, k · k := k · kl1 and C denotes a generic finite constant
that may change its value even within a single calculation.
4.1. Proofs of the main results.
Proof of Lemma 2.1. Define
Z n
2
1 X
e
√
sin(t0 (Xk − µ0 )) − (b
µn − µ0 )0 t cos(t0 (Xk − µ0 )) w(t) dt.
Sn :=
n
Rd
k=1
Step 1: Sbn − Sen = oP (1).
The application of an addition formula for trigonometric functions yields
Z
n
1 X
b
e
sin(t0 (Xj − µ0 )) sin(t0 (Xk − µ0 ))[cos2 (t0 (µ0 − µ
bn )) − 1] w(t) dt
Sn − Sn =
Rd n j,k=1
Z
n
2 X +
sin(t0 (Xj − µ0 )) cos(t0 (Xk − µ0 ))
Rd n j,k=1
× [sin(t0 (µ0 − µ
bn )) cos(t0 (µ0 − µ
bn )) + t0 (b
µn − µ0 )] w(t) dt
Z
n
1 X +
cos(t0 (Xj − µ0 )) cos(t0 (Xk − µ0 ))
n
d
R
j,k=1
× [sin2 (t0 (µ0 − µ
bn )) − (t0 (b
µn − µ0 ))2 ] w(t) dt
=: T1 + T2 + T3 .
√
Pn
√
Since n (b
µn − µ0 ) = n−1/2 k=1 l(Xk , µ0 ) + oP (1), Lemma 4.1 implies n (b
µn −
µ0 ) = Op (1). This leads to | cos2 (t0 (µ0 − µ
bn )) − 1| ≤ ktk oP (1), which in turn
14
R
Pn
implies that T1 ≤ oP (1) Rd [n−1/2 k=1 sin(t0 (Xk − µ0 ))]2 ktk w(t) dt. In order to
prove that the integral is of order OP (1), it remains to show that for some C < ∞
the inequality
"
#2
Z
n
X
1
E p
sin(t0 (Xk − µ0 )) ktk (1 + ktk) w(t) dt < C, ∀ n ∈ N,
n(1 + ktk) k=1
Rd
holds. Applying Lemma 4.1 with gn (x) = sin(t0 (x − µ0 ))/(1 + ktk), we obtain
Pn
supt E[(n(1 + ktk))−1/2 k=1 sin(t0 (Xk − µ0 ))]2 < ∞. The integrability constraint
on the function w finally leads to T1 = oP (1).
Furthermore, the equality 2 sin(t0 (µ0 − µ
bn )) cos(t0 (µ0 − µ
bn )) = sin(2t0 (µ0 − µ
bn ))
0
and a Taylor expansion of sin(2t (µ0 − µ
bn )) in the origin yield
Z X
n
bn )) − 2t0 (µ0 − µ
bn )| w(t) dt
sin(t0 (Xk − µ0 )) |sin(2t0 (µ0 − µ
|T2 | ≤
d
R k=1
Z X
n
0
bn )]2 w(t) dt.
sin(t (Xk − µ0 )) [t0 (µ0 − µ
≤2
Rd k=1
In the same manner as before, we get |T2 | = oP (1).
Finally, the quantity T3 can be approximated by applying the identity 2 sin2 (t0 (µ0 −
µ̂n )) = 1 − cos(2t0 (µ0 − µ
bn )) and by Taylor expansion of cos(2t0 (µ0 − µ
bn )) in the
origin:
Z 1
0
0
2
|T3 | ≤ n
bn ))] − [t (µ0 − µ
bn )] w(t) dt ≤ OP n−1/2 .
2 [1 − cos(2t (µ0 − µ
d
R
Step 2: Sen − Sn = oP (1).
We split up
Z
n
2 X
Sen − Sn =
sin(t0 (Xj − µ0 ))
Rd n j,k=1
× cX−µ0 (t) l(Xk , µ0 )0 t − (b
µn − µ0 )0 t cos(t(Xk − µ0 )) w(t) dt
Z
n
1 X h
+
cos(t0 (Xj − µ0 )) cos(t0 (Xk − µ0 )) [t0 (b
µn − µ0 )]2
Rd n j,k=1
i
− c2X−µ0 (t) l(Xj , µ0 )0 t l(Xk , µ0 )0 t w(t) dt
= R1 + R 2 .
Concerning the first summand one gets
Z n
n
1 X
√
2 X
|R1 | ≤
sin(t0 (Xj − µ0 ))|cX−µ0 (t)| √
l(Xk , µ0 )0 t − n(b
µn − µ0 )0 t
√
n
n
d
R
j=1
k=1
Z n
n
X
1X
2
× w(t) dt +
sin(t0 (Xj − µ0 ))cX−µ0 (t) −
cos(t0 (Xk − µ0 ))
√
n
n
d
R
j=1
k=1
√
0
× | n(b
µn − µ0 ) t| w(t) dt
=oP (1)
15
by (S1)(iii) and Lemma 4.1. The latter result additionally implies
n
Z 1 X
|R2 | = oP (1) + l(Xi , µ0 )0 t l(Xj , µ0 )0 t
Rd n i,j=1
n
h1 X
i
cos(t0 (Xk − µ0 )) cos(t0 (Xm − µ0 )) − c2X−µ0 (t) w(t) dt
2
n
k,m=1
Z X
n
1
= oP (1) + OP (1)
cos(t0 (Xk − µ0 )) − cX−µ0 (t) ktk2 w(t) ,
Rd n
×
k=1
which is asymptotically negligible.
d
Proof of Theorem 2.1. In view of Lemma 2.1 it suffices to show that Sn −→ Z1 .
The statistic Sn is of degenerate V -type with symmetric kernel h = hµ0 given by
Z
hµ0 (x, y) =
[sin(t0 (x − µ0 )) − cX−µ0 (t) t0 l(x, µ0 )]
Rd
×[sin(t0 (y − µ0 )) − cX−µ0 (t) t0 l(y, µ0 )] w(t) dt.
We obtain its limit from Theorem 2.2 in Leucht [22] if her assumptions (A1), (A2)
and (A4) are satisfied. The first condition follows from (S1)(i). Obviously, hµ0 is
degenerate under the null hypothesis, i.e. Ehµ0 (x, X1 ) = 0 ∀x ∈ Rd , and satisfies
e1 )|ν < ∞. This implies (A2), where X
e1 is
supk∈N E|hµ0 (X1 , X1+k )|ν + E|hµ0 (X1 , X
an i.i.d. copy of X1 . Moreover, hµ0 has the following continuity property:
|hµ0 (x, y) − hµ0 (x̄, ȳ)|
≤ C (1 + kl(x, µ0 )k + kl(y, µ0 )k + kl(x̄, µ0 )k + kl(ȳ, µ0 )k) (kx − x̄k + ky − ȳk)
=: f (x, x̄, y, ȳ) (kx − x̄k + ky − ȳk) .
Due to stationarity, Lipschitz continuity of l and the moment constraints on l and
X1 , we obtain by Hölder’s inequality
sup E
max
f (Yk1 , Yk2 + c1 , Yk3 , Yk4 + c2 )η kYk5 k
k1 ,...,k5 ∈N
c1 ,c2 ∈[−a,a]d
h
i
≤ C 1 + (Ekl(X1 , µ0 )k2ν )η/(2ν) (EkX1 k2ν/(2ν−η) )(2ν−η)/(2ν) < ∞
for η := 1/(1 − δ), some a > 0, and any (Yk01 , . . . , Yk05 )0 consisting of independent subvectors (Yk0j
d
1 (m)
, . . . , Yk0j (m) )0 = (Xk0 j
l
1 (m)
, . . . , Xk0 j (m) )0 , l, m = 1, . . . , 5.
l
To this end, also note that 2ν/(2ν − η) < 1 + (3 − 2δ)−1 which actually implies
EkX1 k2ν/(2ν−η) < ∞ by virtue of (S1)(ii). Consequently, (A4) of Leucht [22] holds
and her Theorem 2.2 can be applied in order to determine the asymptotic distribution of Sn : Let φ and ψ denote R-valued, Lipschitz continuous, compactly
supported scale and wavelet functions
associated with
R∞
R ∞a one-dimensional multiresolution analysis and such that −∞ φ(x) dx = 1 and −∞ ψ(x) dx = 0. Set
(
φ for i = 0,
(i)
ϕ :=
ψ for i = 1
16
(e)
and Φj;k (x) = 2jd/2
(c)
Qd
ϕ(ei ) (2j xi − ki ) for x ∈ Rd , k ∈ Zd , e ∈ {0, 1}d . Let hµ0
(c)
denote the truncated version of hµ0 defined as the degenerate counterpart of e
hµ0 ,


hµ0 (x, y), for |hµ0 (x, y)| ≤ ch ,
e
h(c)
:=
−ch ,
for hµ0 (x, y) < −ch ,
µ0


ch ,
for hµ0 (x, y) > ch
i=1
with ch := maxx,y∈[−c,c]d |hµ0 (x, y)|. The limit distributions of Leucht [22] and the
present theorem coincide when we define
RR
(c)
(e )
(e )
h (x, y)Φj;k11 /2 (x)Φj;k22 /2 (y)dx dy

Rd ×Rd µ0



for k21 , k22 ∈ Zd , (e01 , e02 )0 ∈ {0, 1}2d \{02d }
(c)
(4.1)
γj;k1 +e1 ,k2 +e2 (µ0 ) :=
k1 k 2
d
0
0 0

or
,
∈
Z
,
(e
,
e
)
=
0
,
j
=
0,

2d
1
2
2
2


0
else,
(e )
(e )
Cj;k1 +e1 ,k2 +e2 := cov(Φj;k11 /2 (X1 ), Φj;k22 /2 (X1 )) and if the family (Zj;k )j,k of jointly
normal random variables is chosen such that
(e )
(e )
cov(Zj1 ;k1 +e1 , Zj2 ;k2 +e2 ) = cov Φj11;k1 /2 (X1 ), Φj22;k2 /2 (X1 )
+
∞ h
X
i
(e )
(e )
(e )
(e )
cov Φj11;k1 /2 (X1 ), Φj22;k2 /2 (Xt ) + cov Φj11;k1 /2 (Xt ), Φj22;k2 /2 (X1 ) .
t=2
R
Proof of Lemma 2.2. Under H1 and the assumptions concerning w, Rd [=(cX−µ0 (t))]2
R
w(t) dt > 0 holds true. Thus, it is sufficient to show that P (|n−1 Sbn − Rd [=(cX−µ0 (t))]2
w(t) dt| ≥ ε) −→n→∞ 0 ∀ ε > 0 in order to verify the claim. According to the integrability assumption on w, it remains to prove that
!
Z
it0 µ
bn
P
cn (t)) − =(cX−µ0 (t)) w(t) dt > ε −→ 0 ∀ ε > 0. (4.2)
=(e
[−T,T ]d
n→∞
Clearly, [−T,T ]d (·)(t) w(t) dt is a continuous mapping from C([−T, T ]d ) to R+ , both
endowed with the corresponding uniform metrics. The continuous mapping theoPn
P
bn )])t∈[−T,T ]d −→ (=[cX−µ0 (t)])t∈[−T,T ]d
rem implies (4.2) if (n−1 j=1 sin[t0 (Xj − µ
Pn
holds true. To this end, we first show pointwise convergence n−1 j=1 sin[t0 (Xj −
R
P
µ
bn )] −→ =[cX−µ0 (t)] for any fixed t ∈ Rd . Afterwards it remains to prove that
n
1 X
sin[s0 (Xj − µ
bn )] − =(cX−µ0 (s))
lim lim sup P
sup δ→0 n→∞
n
ks−tk<δ
j=1
(4.3)
0
− sin[t (Xj − µ
bn )] + =(cX−µ0 (t)) ≥ ε = 0
for all ε > 0. In order to derive pointwise convergence, note that
n
1 X
sin[t0 (Xj − µ
bn )] − =[cX−µ0 (t)]
n j=1
n
1 X
≤
sin[t0 (Xj − µ0 )] − =[cX−µ0 (t)] + oP (1).
n j=1
17
The remaining sum converges to zero in probability due to the law of large numbers;
see Lemma 5.1 of Leucht [22]. For proving the relation (4.3), it suffices to verify
that
)
(
!
n
1 X
ε
0
0
lim lim sup P
sup sin[s (Xj − µ
bn )] − sin[t (Xj − µ
bn )] ≥
2
δ→0 n→∞
ks−tk<δ n j=1
vanishes according to the uniform continuity of the characteristic function. This
term can be bounded by
!
n
ε
δX
ε
+ lim lim sup P
lim lim sup P δ kb
µn − µ0 k ≥
kXj − µ0 k ≥
= 0.
δ→0 n→∞
δ→0 n→∞
4
n j=1
4
The latter probability vanishes asymptotically since n−1
µ0 k under (S1)(i).
Pn
j=1
P
kXj −µ0 k −→ EkX1 −
Proof of Proposition 2.1. This is an immediate consequence of Proposition 4.1 and
Lemma 4.3.
Proof of Proposition 2.2. Proceeding as in the proof of Lemma 2.1, we obtain
!2
Z
n
1 X
0
∗
∗
∗
0
0
∗
∗
b
√
sin(t (Xk − µ
bn )) − =[c (t)] − (b
µn − µ
bn ) t cos(t (Xk − µ
bn )) w(t) dt
Sn =
n
Rd
k=1
+ oP ∗ (1)
!2
Z
n
1 X
0
∗
∗
0
∗
√
=
sin(t (Xk − µ
bn )) − =[c (t)] − cX−µ0 (t) t l(Xk , µ
bn ) w(t) dt
n
Rd
k=1
+ oP ∗ (1).
d
Therefore, it suffices to show that Sn∗ −→ Z1 in probability, where
!2
Z
n
1 X
0
∗
∗
0
∗
∗
√
sin(t (Xk − µ
bn )) − =[c (t)] − cX−µ0 (t) t l(Xk , µ
bn ) w(t) dt.
Sn :=
n
Rd
k=1
The direct application of Theorem 3.1 of Leucht [22] is not possible since Sn∗ is not
the bootstrap counterpart of Sn , i.e. the kernel functions of both statistics do not
d
coincide. However, her proof of this result implies that nV̄n∗ −→ Z1 , in probability,
where
Z
n
1 Xh
∗
∗
∗
V̄n := 2
h(Xj , Xk , µ
bn ) −
h(x, Xk∗ , µ
bn )PX1∗ |X1 ,...,Xn (dx)
n
d
R
j,k=1
Z
−
h(Xj∗ , y, µ
bn ) PX1∗ |X1 ,...,Xn (dy)
d
R
ZZ
i
+
h(x, y, µ
bn ) PX1∗ |X1 ,...,Xn (dx) PX1∗ |X1 ,...,Xn (dy)
Rd ×Rd
and
Z
h(x, y, µ
bn ) =
Rd
[sin(t0 (x − µ
bn )) − cX−µ0 (t) t0 l(x, µ
bn )]
× [sin(t0 (y − µ
bn )) − cX−µ0 (t) t0 l(y, µ
bn )] w(t) dt.
18
Actually, n V̄n∗ and Sn∗ coincide which finally yields the first assertion of the proposition. The second one is a consequence of the previous part if Z1 has a continuous
distribution function. This in turn follows from var(Z1 ) > 0 by Lemma 3.2 of
Leucht [22].
Proof of Corollary 2.1. Step 1: limn→∞ P (Sbn > t∗α ) = α.
This is a consequence of Proposition 2.2.
Step 2: limn→∞ P (Sbn > t∗α ) = 1.
Note that Sbn∗ − S̄n∗ = oP ∗ (1) holds under H1 . Here, the statistic S̄n∗ is the bootstrap
counterpart of a degenerate V -statistic S̄n with kernel
Z h̄µ0 (x, y) :=
sin(t0 (x − µ0 )) − =[cX−µ0 (t)] − <[cX−µ0 (t)] t0 l(x, µ0 )
Rd
× sin(t0 (y − µ0 )) − =[cX−µ0 (t)] − <[cX−µ0 (t)] t0 l(y, µ0 ) w(t) dt.
Thus, the asymptotic (1 − α)-quantiles, α ∈ (0, 1), of S̄n are bounded. One obtains
equivalence of the limits of S̄n and S̄n∗ in analogy to Proposition 2.2. Consequently,
the bootstrap quantiles are bounded in probability and the claim follows immediately from Lemma 2.2.
Step 3: lim inf n→∞ PH1,n (Sbn > t∗α ) ≥ α.
First note that the bootstrap algorithm imitates the null situation. In conjunction
with Proposition 2.2 and the continuity of the distribution function of Z1 , we obtain
P (t∗α ∈ [tα+δ , tα−δ ]) −→ 1 ∀ δ > 0 with α ± δ ∈ (0, 1),
n→∞
(4.4)
where tx denotes the (1 − x)-quantile of the distribution of Z1 . Let ε > 0 arbitrary
but fixed, then we have to show lim inf n→∞ PH1,n (Sbn > t∗α ) − α ≥ −ε. According to
step 1 of the proof it suffices to show that lim inf n→∞ [PH1,n (Sbn > t∗α ) − PH0 (Sbn >
t∗α )] > −ε. The application of (4.4) and Proposition 2.1 lead to
lim inf PH1,n (Sbn > t∗α ) − PH0 (Sbn > t∗α )
n→∞
≥ lim PH0 (Sbn > tα−δ ) − PH0 (Sbn > tα+δ )
n→∞
= P (Z1 > tα−δ ) − P (Z1 > tα+δ )
which is greater than −ε for any sufficiently small δ > 0 as the cumulative distribution function of Z1 is continuous.
Proof of Theorem 3.1.
(i) Define
Z
e n := n
cn (t) − c(t, θ0 ) − (θbn − θ0 )0 c(1) (t, θ0 )2 w(t) dt.
G
Rd
bn = G
e n + oP (1).
Step 1: G
Recall that for any characteristic function c the equality c(t) = c(−t) holds
19
for all t ∈ Rd , where the bar denotes the complex conjugate. Taylor expansion gives
Z
0
e θbn − θ0 w(t) dt
bn − G
en = − n
G
[cn (t) − c(t, θ0 )] θbn − θ0 c(2) (−t, θ)
2 Rd
Z
0
n
e θbn − θ0 w(t) dt
−
[cn (−t) − c(−t, θ0 )] θbn − θ0 c(2) (t, θ)
2 Rd
Z 0
0
n
e θbn − θ0 w(t) dt
θbn − θ0 c(1) (t, θ0 ) θbn − θ0 c(2) (−t, θ)
+
2 Rd
Z 0
0
n
e θbn − θ0 w(t) dt
θbn − θ0 c(1) (−t, θ0 ) θbn − θ0 c(2) (t, θ)
+
2 Rd
Z 0
2
n
e θbn − θ0 w(t) dt
θbn − θ0 c(2) (t, θ)
+
4 d
R
√
with some random θe between θbn and θ0 . Since θbn is a n-consistent estimator, the latter three summands vanish asymptotically due to assumption (G1). The analyses of the remaining expressions are equal to each
other. The absolute value of the first term can be bounded from above by
Z
1/2 Z
1/2
2
2
oP (1) + OP (1)
|cn (t) − c(t, θ0 )| w(t) dt
M (−t) w(t) dt
.
Rd
Rd
Similar to the consideration of the quantity T1 in the proof of Lemma 2.1,
asymptotic negligibility of the latter expression is a consequence of
E|cn (t) − c(t, θ0 )|2
2
2 X ek )
≤ + 2
E sin(t0 Xj ) − =c(t, θ0 ) sin(t0 Xk ) − sin(t0 X
n n
j<k
2 X ek )
+ 2
E cos(t0 Xj ) − <c(t, θ0 ) cos(t0 Xk ) − cos(t0 X
n
(4.5)
j<k
∞
≤
Cktk X
2
+
τr .
n
n r=1
ek denotes a suitable copy of Xk that is independent of Xj .
Here, X
e n = Gn + oP (1).
Step 2: G
This follows easily from (4.5) in conjunction with (G1)(ii) and (iv).
(ii) Under the assumption (G1) this assertion is an immediate consequence of
part (i) in conjunction with Theorem 2.2 of Leucht [22]. We just remark
that Gn can be formulated as a degenerate V -statistic with kernel
Z
hθ0 (x, y) :=
(<[g(x, t, θ0 )] <[g(y, t, θ0 )] + =[g(x, t, θ0 )] =[g(y, t, θ0 )]) w(t) dt
Rd
(4.6)
0
and g(x, t, θ0 ) = eit x − c(t, θ0 ) − [l(x, θ0 )]0 c(1) (t, θ0 ). The function hθ0 satisfies the moment constraint
e1 )|ν < ∞,
sup E|hθ0 (X1 , X1+k )|ν + E|hθ0 (X1 , X
k∈N
20
e1 is an independent copy of X1 . Furthermore, note that h exhibits
where X
the continuity property
|hθ0 (x, y) − hθ0 (x̄, ȳ)| ≤ C(θ0 ) (1 + |fl (x, x̄, θ0 )|)(1 + kl(y, θ0 )k + kl(ȳ, θ0 )k)
+ (1 + |fl (y, ȳ, θ0 )|)(1 + kl(x, θ0 )k + kl(x̄, θ0 )k)
× (kx − x̄k + ky − ȳk)
=:fθ0 (x, x̄, y, ȳ) (kx − x̄k + ky − ȳk) .
Thus the prerequisites of Theorem 2.2 of Leucht [22] are satisfied and the
rest of the proof can be carried out in complete analogy to the proof of
Theorem 2.1 above.
Proof of Lemma 3.1. It is sufficient to verify the existence of some η > 0 with
b n > η) −→n→∞ 1. According to the continuity of characteristic functions,
P (n−1 G
there exist −∞ < T1,i < T2,i < ∞, i = 1, . . . , d, such that
|cX (t) − c(t, θ0 )| > c0 ,
∀ t ∈ ΩT := [T1,1 , T2,1 ] × · · · × [T1,d , T2,d ],
for some c0 > 0. This implies
bn > η
P n−1 G
Z cn (t) − cX (t) + cX (t) − c(t, θ0 ) − (θbn − θ0 )0 c(1) (t, θ0 )
≥P
ΩT
1
e (θbn − θ0 )2 w(t) dt > η
− (θbn − θ0 )0 c(2) (t, θ)
2
Z n
Z cX (t) − c(t, θ0 )2 w(t) dt > 2η, 4
cn (t) − cX (t)
≥P
ΩT
ΩT
o
1
e (θbn − θ0 ) w(t) dt ≤ η
+ (θbn − θ0 )0 c(1) (t, θ0 ) + (θbn − θ0 )0 c(2) (t, θ)
2
for some random θe between θbn and θ0 . For sufficiently small η > 0 we obtain
b n > η)
P (n−1 G
Z n
cn (t) − cX (t)
≥P
ΩT
o
1
e (θbn − θ0 ) w(t) dt ≤ η/4
+ (θbn − θ0 )0 c(1) (t, θ0 ) + (θbn − θ0 )0 c(2) (t, θ)
2
Z 1
b
e (θbn − θ0 )w(t) dt ≤ η/8
≥ P
(θn − θ0 )0 c(1) (t, θ0 ) + (θbn − θ0 )0 c(2) (t, θ)
2
Ω
ZT
+P
cn (t) − cX (t) w(t) dt ≤ η/8 − 1.
ΩT
The first probability on the r.h.s. tends to one. To show that
Z
P
|cn (t) − cX (t)| w(t) dt < η/8 −→ 1,
(4.7)
n→∞
ΩT
one can proceed as in the proof of Lemma 2.2. On the one hand,
!
lim lim sup P
δ→0 n→∞
sup |cn (s) − cX (s) − cn (t) + cX (t)| ≥ ε
ks−tk≤δ
21
tends to zero according to uniform Lipschitz continuity of (cn )n∈N and uniform
P
continuity of c. On the other hand, pointwise convergence cn (t) −→ cX (t), t ∈ Rd ,
follows from inequality (4.5) in the proof of Theorem 3.1.
Proof of Proposition 3.1. This is an immediate consequence of Proposition 4.1 and
Lemma 4.3.
Proof of Proposition 3.2. Similarly to the previous proof, the bootstrap statistic
can be approximated as follows:
Z 2
∗
∗
b
Gn = n
cn (t) − c∗ (t) − (θbn∗ − θbn )0 c(1) (t, θbn ) w(t) dt + oP ∗ (1)
Rd
n
2
1 X
0
∗
l(Xj∗ , θbn ) c(1) (t, θbn ) w(t) dt + oP ∗ (1).
cn (t) − c∗ (t) −
n j=1
Rd
Z
=n
d
Thus, it suffices to show that G∗n −→ Z2 in probability, where
G∗n = n
n
2
0
1 X
∗
l(Xj∗ , θbn ) c(1) (t, θbn ) w(t) dt.
cn (t) − c∗ (t) −
n j=1
Rd
Z
The direct application of the result of Leucht [22] is again not possible since G∗n is
not the bootstrap counterpart of Gn , but
ZZ
n
1 X h
Ven∗ =
hθbn (Xj∗ , Xk∗ ) +
hθbn (x, y)PX1∗ |X1 ,...,Xn (dx)PX1∗ |X1 ,...,Xn (dy)
n
Rd ×Rd
j,k=1
Z
Z
i
hθbn (Xj∗ , y)PX1∗ |X1 ,...,Xn (dy)
−
hθbn (x, Xk∗ )PX1∗ |X1 ,...,Xn (dx) −
Rd
d
−→ Z2
Rd
in probability,
where hθbn is defined by a substitution of θ0 through θbn in (4.6). Moreover, straightforward calculations yield that G∗n = Ven∗ . That is, with probability tending to
one, G∗n is the artificially degenerated bootstrap counterpart of Gn . Therefore,
d
G∗n −→ Z2 in probability, which also implies the second assertion of the proposition due to the continuity of the distribution function of Z2 in case of a positive
variance.
b n > t∗α ) = α.
Proof of Corollary 3.1. Step 1: limn→∞ P (G
This assertion follows from Proposition 3.2.
b n > t∗α ) = 1.
Step 2: limn→∞ P (G
∗
∗
b − G = oP ∗ (1) holds under H1 as well and that G∗ is still degenerate.
Note that G
n
n
n
Thus, the bootstrap quantiles are bounded in probability. Now, the claim follows
immediately from Lemma 3.1.
b n > t∗α ) ≥ α.
Step 3: lim inf n→∞ PH1,n (G
With the same arguments as in the proof of Corollary 2.1, this inequality can be
deduced from Proposition 3.1.
22
4.2. Auxiliary results on V -statistics. First we state an auxiliary result concerning variances of degree-1 V -statistics of τ -dependent observations which is extensively applied within the proofs of our main results.
Lemma 4.1. Let (Xn,k )nk=1 , n ∈ N, be a triangular scheme of Rd -valued random
variables such that (S2)(i) holds for some δ ∈ (0, 1). Suppose that the sequence of
functions gn : Rd → R satisfies supn∈N E|gn (Xn,1 )|(2−δ)/(1−δ) < ∞ and
|gn (x) − gn (y)| ≤ fn (x, y) kx − ykl1 ,
∀ x, y ∈ Rd ,
where the functions fn : R2d → R fulfil supn∈N E{[fn (X, Y )]1/(1−δ) (kXkl1 +kY kl1 )} <
∞ for i.i.d. random variables X, Y with X ∼ PXn,1 . Then,
"
#
n
1 X
sup var √
gn (Xn,k ) < ∞.
n
n∈N
k=1
Proof. We have
#
"
n
n
1X
2
1 X
gn (Xn,k ) =
var(gn (Xn,k ))+
var √
n
n
n
k=1
k=1
X
cov(gn (Xn,j ), gn (Xn,k )).
1≤j<k≤n
en,k a copy of Xn,k that is independent of Xn,j and that satisfies
Denote by X
e
EkXn,k − Xn,k k ≤ τ̄k−j . According to (1.1) such a random variable exists, at
least after enlarging the underlying probability space. Based on this relation we
can bound the covariance terms by an iterative application of Hölder’s inequality,
|cov(gn (Xn,j ), gn (Xn,k ))|
en,k )]|
= |Egn (Xn,j ) [gn (Xn,k ) − gn (X
1−δ
en,k )|)δ
en,k )|
(E|gn (Xn,k ) − gn (X
≤ E|gn (Xn,j )|1/(1−δ) |gn (Xn,k ) − gn (X
δ(1−δ)
2
en,k )]1/(1−δ) (kXn,k k + kX
en,k k)}
≤ C E{[fn (Xn,k , X
(τ̄k−j )δ .
Eventually, this leads to
#
"
n
1 X
C
gn (Xn,k ) ≤ C +
var √
n
n
k=1
which in turn implies the assertion.
"
X
1≤j<k≤n
2
(τ̄k−j )δ ≤ C 1 +
∞
X
#
(τ̄r )δ
2
< ∞,
r=1
We devise a general result on V -statistics for triangular schemes of random
variables. Besides (S2) we assume
(A1) (i) The kernel h : Rd × Rd → R is a symmetric, measurable function and
degenerate under PX1 , i.e. Eh(x; X1 ) = 0, ∀ x ∈ Rd .
(ii) For a δ satisfying (S2), the following moment constraints hold true
e1 of X1 :
with some ν > (2 − δ)/(1 − δ) and an independent copy X
sup
1≤k<n,n∈N
e1 )|ν + E|h(X1 , X1 )| < ∞.
E|h(Xn,1 , Xn,1+k )|ν < ∞ and E|h(X1 , X
23
(iii) With a continuous function f : R4d → R , the kernel satisfies
|h(x, y) − h(x̄, ȳ)| ≤ f (x, x̄, y, ȳ) [kx − x̄kl1 + ky − ȳkl1 ] , ∀ x, x̄, y, ȳ ∈ Rd .
sup
1≤k1 ,...,k5 ≤n,
n∈N
Moreover, let δ ∈ (0, 1) be such that (A1)(ii) holds. Then
[f (Yn,k1 , Yn,k2 + a1 , Yn,k3 , Yn,k4 + a2 )]1/(1−δ) kYn,k5 kl1
E
max
a1 ,a2 ∈[−A,A]d
0
0
is finite for some A > 0, and any (Yn,k
, . . . , Yn,k
)0 consisting of in1
5
0
0
0
dependent subvectors (Yn,kj (m) , . . . , Yn,kj (m) ) , l, m = 1, . . . , 5, where
1
0
either (Yn,k
j
l
d
1
0
0
, . . . , Yn,k
)0 = (Xn,k
(m)
j (m)
j
l
1 (m)
0
, . . . , Xn,k
)0 or
j (m)
l
d
0
0
(Yn,k
, . . . , Yn,k
)0 = (Xk0 j (m) , . . . , Xk0 j (m) )0 . Here, (Xn )n is dej1 (m)
jl (m)
1
l
fined as in (S2).
This assumption can be seen as the counterpart of (A2) and (A4) of Leucht [22],
who considered stationary sequences of random variables, for triangular schemes of
random variables. In particular, note that any kernel H that is degenerate under
PX1 is asymptotically degenerate under PXn,1 in the sense of
Z
e1 ) −→ 0.
≤ 1 kgn k2∞ EH(X1 , X
H(x,
y)P
(dx)P
(dy)
Xn,1
Xn,1
n
d
n→∞
R
This turns out to be crucial for proving the asymptotic result below.
Proposition 4.1. Suppose that the assumptions (S2) and (A1) are fulfilled. Then,
n Vn,n :=
n
1 X
d
h(Xn,j , Xn,k ) −→ Zloc ,
n
j,k=1
Zloc := lim
c→∞
∞
X
(c)
γj;k1 ,k2 (Zj;k1 + Dj;k1 ) (Zj;k2 + Dj;k2 ) − Cj;k1 ,k2
X
j=0 k1 ,k2 ∈Zd
(4.8)
+ Eh(X1 , X1 ),
(c)
where the r.h.s. converges in the L2 -sense for certain families of constants (γj;k1 ,k2 )c,j,k1 ,k2 ,
(Cj;k1 ,k2 )j,k1 ,k2 and (Dj;k )j,k and centered, jointly normal random variables (Zj;k )j,k .
Proof. According to the law of large numbers (Lemma 5.1 of Leucht [22]), we obtain
n
1X
P
h(Xn,k , Xn,k ) −→ Eh(X1 , X1 ).
n
k=1
Thus it remains to investigate the limit of the U -statistic
1 X
n Un,n :=
h(Xn,j , Xn,k )
n
1≤j6=k≤n
The main ideas of the proof are already contained in the proof of Theorem 2.2
in Leucht [22]. Therefore we only point out the additional calculations that are
required for the extension of her results on sequences of random variables to triangular schemes. In a first step one reduces the problem to statistics with bounded
24
kernels. To this end, we introduce degenerate kernels
Z
Z
e
e
hc (x, y) = e
hc (x, y) −
hc (x, y)PX1 (dx) −
hc (x, y)PX1 (dy)
Rd
Rd
ZZ
e
+
hc (x, y)PX1 (dx)PX1 (dy)
Rd ×Rd
such that
(1) e
hc (x, y) = h(x, y), ∀ x, y ∈ [−c, c],
(2) e
hc is bounded and (hc )c satisfies (A1)(iii) uniformly,
e1 )|ν + E|hc (X1 , X1 )| <
(3) supc∈R+ supk∈N E|hc (Xn,1 , Xn,1+k )|ν + E|hc (X1 , X
∞,
e
e.g. hc = (h ∧ maxx,y∈[−c,c]d |h(x, y)|) ∨ (− maxx,y∈[−c,c]d |h(x, y)|). Denote the
corresponding U -statistic by Un,n,c . In order to show that lim supn→∞ n2 E(Un,n −
Un,n,c )2 −→ 0 as c → ∞, first note that Un,n − Un,n,c is a U -statistic with kernel
H = h − hc . The second moment of this statistic can be bounded using the
decomposition
n2 E(Un,n − Un,n,c )2 ≤ 8 sup E|H(Xn,1 , Xn,1+k )|2 +
1≤k<n
n−1 4
8 X X (t)
Z
n r=1 t=1 n,r
with
n
e (r) )H(X
e (r) , X
e (r) )
EH(Xn,i , X
n,j
n,k
n,l
X
(1)
Zn,r
:=
1≤i<j;k<l;j≤l≤n
r:=min{j,k}−i≥l−max{j,k}
(2)
Zn,r
o
e (r) )H(X
e (r) , X
e (r) ) ,
+ EH(Xn,i , Xn,j )H(Xn,k , Xn,l ) − EH(Xn,i , X
n,j
n,k
n,l
n
X
(r)
e )
:=
EH(Xn,i , Xn,j )H(Xn,k , X
n,l
1≤i<j;i≤k;k<l≤n
r:=l−max{j,k}>min{j,k}−i
(3)
Zn,r
o
e (r) ) ,
+ EH(Xn,i , Xn,j )H(Xn,k , Xn,l ) − EH(Xn,i , Xn,j )H(Xn,k , X
n,l
n
X
(r)
(r)
(r)
e )H(X
e ,X
e )
:=
EH(Xn,i , X
n,j
n,k
n,l
1≤i≤k<l<j≤n
r:=k−i≥j−l
(4)
Zn,r
o
e (r) )H(X
e (r) , X
e (r) ) ,
+ EH(Xn,i , Xn,j )H(Xn,k , Xn,l ) − EH(Xn,i , X
n,j
n,k
n,l
n
X
e (r) )H(Xn,k , Xn,l )
:=
EH(Xn,i , X
n,j
1≤i≤k<l<j≤n
r:=j−l>k−i
o
e (r) )H(Xn,k , Xn,l ) .
+ EH(Xn,i , Xn,j )H(Xn,k , Xn,l ) − EH(Xn,i , X
n,j
(1)
(3)
e (r)0 , X
e (r)0 , X
e (r)0 )0 is chosen such
In every summand of Zn,r and Zn,r the vector (X
n,j
n,k
n,l
d
e (r)0 , X
e (r)0 , X
e (r)0 )0 =
that it is independent of the random variable Xn,i , (X
n,j
(2)
n,k
n,l
(4)
0
0
0
(Xn,j
, Xn,k
, Xn,l
)0 , and (1.1) holds. Within Zn,r (respectively Zn,r ) the random
e (r) (respectively X
e (r) ) is chosen to be independent of the vector
variable X
n,l
n,j
25
d
0
0
0
0
0
0
e (r) =
(Xn,i
, Xn,j
, Xn,k
)0 (respectively (Xn,i
, Xn,k
, Xn,l
)0 ) such that X
Xn,l (respecn,l
d
(r)
e = Xn,j ) and (1.1) holds. This may possibly require an enlargement of
tively X
n,j
the underlying probability space. Moreover, note that the number of summands
(t)
of Zn,r , t = 1, . . . , 4, is bounded by (r + 1)n2 . Asymptotic negligibility of the sec(t)
ond summands of Zn,r can be verified along the lines of the proof of Lemma 3.1
of Leucht [22]. The remaining terms, that are equal to zero her setting, can be
bounded with the help of L1 -coupling techniques, the asymptotic degeneracy of H
and the properties of the truncated kernels hc . Exemplarily we consider the first
summands of Zn,2 which can be estimated from above by
n−1
1 X
n2 r=1
+
+
+
1
n2
1
n2
1
n2
e (r) )
EH(Xn,i , Xn,j )H(Xn,k , X
n,l
X
1≤i<j≤k<l≤n; r:=l−k>j−i; k−j≥j−i
n−1
X
X
e (r) )
EH(Xn,i , Xn,j )H(Xn,k , X
n,l
r=1 1≤i<j≤k<l≤n; r:=l−k>j−i>k−j
n−1
X
e (r) )
EH(Xn,i , Xn,j )H(Xn,k , X
n,l
X
r=1 1≤i≤k<j<l≤n; r:=l−j>k−i; j−k≥k−i
n−1
X
X
e (r) ) .
EH(Xn,i , Xn,j )H(Xn,k , X
n,l
r=1 1≤i≤k<j<l≤n; r:=l−j>k−i>j−k
We investigate the first summand only since the remaining terms can be treated in
en,k ∼ PX be independent of the vector (X 0 , X 0 )0
an analogous manner. Let X
n,k
n,i
n,j
en,k − Xn,k k ≤ τ̄k−j . (This may require an enlargement of the
and such that EkX
underlying probability space.) Now an iterative application of Hölder’s inequality
yields
e (r) )
EH(Xn,i , Xn,j )H(Xn,k , X
n,l
Z
en,k , y) PX (dy) ≤ EH(Xn,i , Xn,j )
H(Xn,k , y) − H(X
n,1
Rd
e (r) )
+ EH(Xn,i , Xn,j ) EH(Xn,k , X
n,l
(1−δ)/(2−δ)
2
δ
≤ C τ̄k−j
E|H(Xn,i , Xn,j )|(2−δ)/(1−δ)
ZZ
kgn k∞
+ |EH(Xn,i , Xn,j )|
|H(x, y)| PX1 (dx) PX1 (dy)
n
d
d
R ×R
h
(1−δ)/(2−δ) δ2
kgn k∞ δ i
≤ C E|H(Xn,i , Xn,j )|(2−δ)/(1−δ)
τ̄k−j +
τ̄j−i
n
Z Z
2
kgn k∞
+C
|H(x, y)| PX1 (dx) PX1 (dy)
.
n
d
d
R ×R
26
Within these calculations, the inequality
Z en,k , y) PX (dy)
E
H(Xn,k , y) − H(X
n,1
Rd
≤
δ
τ̄k−j
≤C
1−δ
[f1 (x, z, y, y)] (kxk + kzk)PXn,1 (dx)PXn,1 (dy)PXn,1 (dz)
Z Z Z
η
Rd ×Rd ×Rd
δ
τ̄k−j
is applied which is a consequence of (A1)(iii), where the function f1 is given by
Z
Z
f1 (x, x̄, y, ȳ) := 2f (x, x̄, y, ȳ) +
f (x, x̄, y, y)PX1 (dy) +
f (x, x, y, ȳ)PX1 (dx).
Rd
Rd
Similarly to the investigation of the term E1 in the proof of Lemma 2.1 in Leucht
[22], it can be shown that there exists a family (εc )c∈R+ with εc −→c→∞ 0 such
that
(1−δ)/(2−δ)
e1 )| ≤ εc .
sup
E|H(Xn,1 , Xn,1+k )|(2−δ)/(1−δ)
+ E|H(X1 , X
1≤k<n; n∈N
e1 denotes an independent copy of X1 . Hence, the relation
Here, X
n−1
X
1 X
e (r) ) ≤ C εc −→ 0
lim sup 2
EH(Xn,i , Xn,j )H(Xn,k , X
n,l
c→∞
n→∞ n
r=1
1≤i<j≤k<l≤n
r:=l−k>j−i; k−j≥j−i
holds. After similar estimations for the remaining terms one obtains asymptotic
negligibility of the error n2 E(Un,n − Un,n,c )2 . In a next step, the statistics with
bounded kernels are approximated by their (finite) multi-scale wavelet series expansion. Similar arguments as in the respective estimation steps in the proof of
Theorem 2.2 in Leucht [22] lead to
lim
(K,L) 2
lim lim sup n2 E(Un,n,c − Un,n,c
) = 0.
K→∞ L→∞ n→∞
Here,
J(K)
(K,L)
Un,n,c
=
X
X
(c)
γj;k1 ,k2
n
j=0 k1 ,k2 ∈{−L,...,L}d
n
1 X Ψj;k1 (Xn,s ) − EΨj;k1 (X1 ) Ψj;k2 (Xn,t ) − EΨj;k2 (X1 )
n s,t=1
(4.9)
n
o
1 X
−
Ψj;k1 (Xn,t ) − EΨj;k1 (X1 ) Ψj;k2 (Xn,t ) − EΨj;k2 (X1 )
n t=1
(e)
(e)
with Ψj;k := Φj;l/2 , l + e = k, l/2 ∈ Zd and e ∈ {0, 1}d . The functions Φj;k and
(c)
the constants γj;k1 ,k2 are defined as in the proof of Theorem 2.1 (with hc instead of
(c)
hµ0 ). The subtrahend of the r.h.s. of (4.9) converges in probability to Cj;k1 ,k2 due
to the weak law of large numbers, where (Cj;k1 ,k2 )j,k1 ,k2 is defined as in the proof
of Theorem 2.1. Moreover, we get
Z
P
√
n EΨj;k (Xn,1 )−EΨj;k (X1 ) −→ Dj;k :=
[Ψj;k (x)−EΨj;k (X1 )] g(x) PX1 (dx).
Rd
27
In conjunction with a CLT, Slutsky’s Lemma, and the continuous mapping, this
yields
d
(K,L)
Un,n,c
−→ Zc(K,L) .
J(K)
Zc(K,L) :=
X
h
i
(c)
γj;k1 ,k2 (Zj;k1 + Dj;k1 ) (Zj;k2 + Dj;k2 ) − Cj;k1 ,k2 ,
X
j=0 k1 ,k2 ∈{−L,...,L}d
where (Zj;k )j,k is defined as in the proof of Theorem 2.1. Here, a slight modification
of the CLT of Neumann and Paparoditis [25] can be employed; see the proof of
Theorem 3.1 in Leucht [22] for details. Since additionally
(K,L) 2
lim lim sup n2 E(Un,n − Un,n,c
) ,
lim lim
c→∞ K→∞ L→∞ n→∞
Theorem 3.2 of Billingsley [5] implies that it remains to show
2
lim lim lim E Zc(K,L) − Zloc + Eh(X1 , X1 ) = 0.
c→∞ K→∞ L→∞
This in turn follows from the corresponding relation of the U -statistics by an iterative application of Theorem 3.4 of Billingsley [5].
According to Leucht [22], the conditions (S2), (A1) and supk∈N E|h(X1 , Xk+1 )|ν <
∞ are sufficient for
n
1 X
d
h(Xj , Xk ) −→ Z
n Vn =
n
j,k=1
with
Z := lim
c→∞
∞
X
X
(c)
γj;k1 ,k2 [Zj;k1 Zj;k2 − Cj;k1 ,k2 ] + Eh(X1 , X1 ).
(4.10)
j=0 k1 ,k2 ∈Zd
Below we compare the asymptotic behaviour of n Vn and n Vn,n when the kernels
are of the form
Z
h(x, y) =
[g(x, t)]0 g(y, t) w(t) dt
(4.11)
Rd
with some vector-valued function g.
Lemma 4.2. Suppose that the conditionsR (S2) and (A1) are satisfied. Moreover, let
the kernel h be of the form (4.11) with E Rd kg(X1 , t)k2ν
2 w(t) dt < ∞ and var(Z) >
0. Then,
P (Zloc > x) ≥ P (Z > x), ∀ x ∈ R,
where Z and Zloc are defined as in (4.10) and Proposition 4.1, respectively.
Proof. First note that the limits of Vn and Vn,n can alternatively expressed using
a uni-scale wavelet decomposition, i.e.
X
d
(c)
Z = lim lim
αJ;k1 ,k2 [ZJ,k1 ZJ,k2 − AJ;k1 ,k2 ]
c→∞ J→∞
k1 ,k2 ∈Zd
and
d
Zloc = lim lim
c→∞ J→∞
X
k1 ,k2 ∈Zd
(c)
αJ;k1 ,k2 [(ZJ,k1 + CJ;k1 )(ZJ,k2 + CJ;k2 ) − AJ;k1 ,k2 ],
28
(0 )
where the r.h.s. converge in the L2 -sense. Based on ΦJ;kd as in the proof of Theorem 2.1, (ZJ,k )J,k are centered and jointly normal random variables with covariance
structure
(0 )
(0 )
cov(ZJ,s , ZJ,t ) = cov(ΦJ;sd (X1 ), ΦJ;td (X1 ))
∞ h
i
X
(0 )
(0 )
(0 )
(0 )
+
cov(ΦJ;sd (X1 ), ΦJ;td (Xk )) + cov(ΦJ;sd (Xk ), ΦJ;td (X1 )) ,
k=2
CJ;k :=
(0 )
(c)
R
Rd
[ΦJ;k (x)−EΦJ;k (X1 )]g(x)PX1 (dx), αJ;k1 ,k2 :=
(0 )
(0 )
RR
(0 )
Rd ×Rd
hc (x, y) ΦJ;kd 1 (x)
ΦJ;kd 2 (y) dx dy, and AJ;k1 ,k2 := cov(ΦJ,kd1 (X1 ), ΦJ;kd 2 (X1 )). To define hc , we first introduce Gc (t) := maxx∈[−c,c]d |g(x, t)| and truncated versions of g,


g(x, t) for |g(x, t)| ≤ Gc (t),
g (c) (x, t) := −Gc (t) for g(x, t) < −Gc (t),


Gc (t)
for g(x, t) > Gc (t).
The truncated versions (hc )c of the kernel h itself are then defined through
Z
g (c) (x, t) −
g (c) (z, t)PX1 (dz)
Rd
Rd
Z
× g (c) (y, t) −
g (c) (z, t)PX1 (dz) w(t) dt.
Z
hc (x, y) :=
Rd
We show that for all ε > 0 and x ∈ R the inequality P (Zloc > x) − P (Z > x) ≥ −ε
holds true. W.l.o.g. x can assumed to be a continuity point of the cumulative
distribution function of Zloc − Eh(X1 , X1 ). It suffices to verify
e(J,L) > x − P Z
ec(J,L) > x ≥ 0,
P Z
loc,c
∀ c ≥ c0 , L ≥ L0 (c), J ≥ J0 (c, L),
where
e(J,L) :=
Z
loc,c
(c)
X
αJ;k1 ,k2 [(ZJ,k1 + CJ;k1 )(ZJ,k2 + CJ;k2 ) − AJ;k1 ,k2 ]
k1 ,k2 ∈{−L,...,L}d
and
(c)
X
ec(J,L) :=
Z
αJ;k1 ,k2 [ZJ,k1 ZJ,k2 − AJ;k1 ,k2 ].
k1 ,k2 ∈{−L,...,L}d
(J,L)
Let the constants J, L and c be fixed. Then, with y = x + Bc
P
(c)
k1 ,k2 ∈{−L,...,L}d αJ;k1 ,k2 AJ;k1 ,k2 , we have to prove
P
h
≥P
(J,L)
Z
h
Z
+C
(J,L)
(J,L)
i0
i0
∆c(J,L)
∆(J,L)
Z (J,L)
c
h
Z
(J,L)
>y .
+C
(J,L)
i
(J,L)
and Bc
:=
>y
(4.12)
29
(J,L)
Here, Z (J,L) := (ZJ,−L1d , . . . , ZJ,L1d )0 , C (J,L) := (CJ;−L1d , . . . , CJ;L1d )0 and ∆c
(c)
is a symmetric matrix of the corresponding coefficients αJ;k1 ,k2 . The latter inequality is equivalent to
h
i0
i
h
1/2
1/2
0
0
0
(J,L)
P
S(J,L) Y + O(J,L)
C (J,L) O(J,L)
∆(J,L)
C
>
y
O
S
Y
+
O
(J,L)
c
(J,L)
(J,L)
h
i0
h
i
1/2
1/2
0
≥P
S(J,L) Y O(J,L)
∆(J,L)
O(J,L) S(J,L) Y > y
c
1/2
for some a diagonal matrix S(J,L) with decreasing diagonal elements σ1 ≥ · · · ≥
σN > σN +1 = · · · = σ(2L+1)d = 0 for an N ≤ (2L + 1)d , an orthogonal matrix
O(J,L) , and a standard normal vector Y . Note that N ≥ 1 if the indices c, J, L are
chosen sufficiently large. Moreover, we have
z
0
∆(J,L)
z
c
(2L+1)d
Z
X
=
Rd
!2
Z
zk
ΦJ,k (x)[g
(c)
(x, t) − Eg
(c)
d
(J,L)
for arbitrary z = (z1 , . . . , z(2L+1)d )0 ∈ R(2L+1) , i.e. ∆c
This in turn implies positive semi-definiteness of
P
h
w(t) dt ≥ 0
0
O(J,L)
is positive semi-definite.
(J,L)
∆c
O(J,L) Therefore,
i0
h
i
1/2
1/2
0
0
(J,L)
0
S(J,L) Y + O(J,L)
C (J,L) O(J,L)
S
Y
+
O
C
>
y
∆(J,L)
O
(J,L)
c
(J,L)
(J,L)
(2L+1)d
=P
(X1 , t)] dx
Rd
k=1
X
(2L+1)d
2
λk (Yk + δk ) +
k=1
X
!
ηk > y
k=1
(4.13)
with certain nonnegative constants λk , real-valued constants δk , ηk , k = 1, . . . , (2L+
P(2L+1)d
1)d , and k=1
ηk ≥ 0 while
(2L+1)d
P [Z (J,L) ]0 ∆(J,L)
Z (J,L) > y = P
c
X
!
λk Yk2 > y .
(4.14)
k=1
Note that for a standard normal random variable Y , the variable (Y +a)2 is stochastically larger than Y 2 for any a 6= 0. Finally, since Y1 , . . . , Y(2L+1)d are independent
standard normal variables, the comparison of (4.13) and (4.14) implies (4.12). Combining Proposition 4.1 and Lemma 4.2 yields the following assertion.
Lemma 4.3. Under the assumptions of Lemma 4.2, the following inequality holds
true:
lim inf P [(n Vn,n > x) − P (n Vn > x)] ≥ 0, ∀ x ∈ R.
n→∞
Proof. We show that for each ε > 0
lim inf [P (n Vn,n > x) − P (n Vn > x)] ≥ −ε,
n→∞
∀ x ∈ R.
30
To this end, let the cumulative distribution function of Zloc be continuous at x + δ
for some δ > 0 that is specified below. We obtain
lim inf [P (n Vn,n > x) − P (n Vn > x)] ≥ lim [P (n Vn,n > x + δ) − P (n Vn > x)]
n→∞
n→∞
= [P (Zloc > x + δ) − P (Z > x)]
≥ [P (Z > x + δ) − P (Z > x)] ,
where the latter difference is greater than −ε for sufficiently small δ.
Acknowledgment . The author is grateful to Michael H. Neumann for many fruitful discussions on this work. This research was funded by the German Research
Foundation DFG (projects: NE 606/2-1 and NE 606/2-2).
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