1. No category

Author manuscript, published in "Statistics and Probability Letters 78, 15 (2008) 2485-2489"
DOI : 10.1016/j.spl.2008.02.037
Bahadur representation of sample quantiles for
functional of Gaussian dependent sequences under a
minimal assumption
By Jean-Fran¸
cois Coeurjolly1
SAGAG, Department of Statistics, Grenoble, FRANCE
hal-00131071, version 2 - 29 Sep 2008
September 30, 2008
We obtain a Bahadur representation for sample quantiles of nonlinear functional of Gaussian sequences with correlation function decreasing as k −α for
some α > 0. This representation is derived under a mimimal assumption.
1
Introduction
We consider the problem of obtaining a Bahadur representation of sample quantiles in a
certain dependence context. Before stating in what a Bahadur representation consists,
let us specify some general notation. Given some random variable Y , F(·) = FY (·) is
referred as the cumulative distribution function of Y , ξ(p) = ξY (p) for some 0 < p < 1 as
the quantile of order p. If F(·) is absolutely continuous with respect to Lebesgue measure,
the probability density function is denoted by f(·) = fY (·). Based on the observation
of a vector Y = (Y (1), . . . , Y (n)) of n random variables distributed as Y , the sample
cumulative distribution function and the sample quantile of order p are respectively
denoted by FbY (·; Y ) and ξbY (p; Y ) or simply by Fb (·; Y ) and ξb(p; Y ).
Let Y = (Y (1), . . . , Y (n)) a vector of n i.i.d. random variables such that F ′′ (ξ(p))
exists and is bounded in a neighborhood of ξ(p) and such that F ′ (ξ(p)) > 0, Bahadur
proved that as n → +∞,
1
p − Fb (p)
ξb(p) − ξ(p) =
+ rn ,
f(ξ(p))
Key words and phrases. Gaussian processes, Bahadur representation of sample quantiles, Hermite
expansions.
1
Bahadur representation of sample quantiles for functional of Gaussian
dependent sequences
2
with rn = Oa.s. n−3/4 log(n)3/4
where a sequence of random variables Un is said
to be Oa.s. (vn ) if Un /vn is almost surely bounded. Kiefer obtained the exact rate
n−3/4 log log(n)3/4 . Under an Assumption on F (·) which is quite similar to the one done
by Bahadur, extensions of above results to dependent random variables have been pursued in Sen and Ghosh (1972) for φ−mixing variables, in Yoshihara (1995) for strongly
mixing variables, and recently in Wu (2005) for short-range and long-range dependent
linear processes, following works of Hesse (1990) and Ho and Hsing (1996). Finally,
such a representation has been obtained by Coeurjolly (2007) for nonlinear functional of
Gaussian sequences with correlation function decreasing as k−α for some α > 0.
Ghosh (1971) proposed in the i.i.d. case a much simpler proof of Bahadur’s result
which suffices for many statistical applications. He established under a weaker assump-
hal-00131071, version 2 - 29 Sep 2008
tion on F(·) (F ′ (·) exists and is bounded in a neighborhood of ξ(p) and f(ξ(p)) > 0)
that the remainder term satisfies rn = oP (n−1/2 ), which means that n1/2 rn tends to 0
in probability. This result is sufficient for example to establish a central limit theorem
for the sample quantile. Our goal is to extend Ghosh’s result to nonlinear functional of
Gaussian sequences with correlation function decreasing as k−α . The Bahadur representation is presented in Section 2 and is applied to a central limit theorem for the sample
quantile. Proofs are deferred in Section 3.
2
Main result
Let {Y (i)}+∞
i=1 be a stationary (centered) gaussian process with variance 1, and correlation function ρ(·) such that, as i → +∞
|ρ(i)| ∼ i−α
(1)
for some α > 0.
Let us recall some background on Hermite polynomials: the Hermite polynomials
form an orthogonal system for the Gaussian measure and are in particular such that
E (Hj (Y )Hk (Y )) = j! δj,k , where Y is referred to a standard Gaussian variable. For
some measurable function g(·) defined on R such that E(g(Y )2 ) < +∞, the following
expansion holds
g(t) =
X cj
j≥τ
j!
Hj (t)
with
cj = E (g(Y )Hj (Y )) ,
Bahadur representation of sample quantiles for functional of Gaussian
dependent sequences
3
where the integer τ defined by τ = inf {j ≥ 0, cj 6= 0}, is called the Hermite rank of the
function g. Note that this integer plays an important role. For example, it is related to
the correlation of g(Y1 ) and g(Y2 ), for Y1 and Y2 two standard gaussian variables with
P
2
k)
ρk = O (ρτ ).
correlation ρ, since E(g(Y1 )g(Y2 ) = k≥τ (ck!
′
Our result is based on the assumption that Fg(Y
) (·) exists and is bounded in a neigh-
borhood of ξ(p). This is achieved if the function g(·) satisfies the following assumption
(see e.g. Dacunha-Castelle and Duflo (1982), p.33).
Assumption A(ξ(p)) : there exist Ui , i = 1, . . . , L, disjoint open sets such that
′
Ui contains a unique solution to the equation g(t) = ξg(Y ) (p), such that Fg(Y
) (ξ(p)) > 0
and such that g is a C 1 −diffeomorphism on ∪L
i=1 Ui .
hal-00131071, version 2 - 29 Sep 2008
Note that this assumption allows us to obtain
L
X
φ(gi−1 (t))
,
′ (g −1 (t))
g
i
i=1
′
Fg(Y
) (ξg(Y ) (p)) = fg(Y ) (ξg(Y ) (p)) =
where gi (·) is the restriction of g(·) on Ui and where φ(·) is referred to the probability
density function of a standard Gaussian variable.
Now, define, for some real u, the function hu (·) by:
hu (t) = 1{g(t)≤u} (t) − Fg(Y ) (u).
(2)
We denote by τ (u) the Hermite rank of hu (·). For the sake of simplicity, we set τp =
τ (ξg(Y ) (p)). For some function g(·) satisfying Assumption A(ξ(p)), we denote by
τp =
inf
γ∈∪L
i=1 g(Ui )
τ (γ),
(3)
that is the minimal Hermite rank of hu (·) for u in a neighborhood of ξg(Y ) (p). Denote
also by cj (u) the j-th Hermite coefficient of the function hu (·).
Theorem 1 Under Assumption A(ξ(p)), the following result holds as n → +∞
p − Fb ξg(Y ) (p); g(Y )
b
+ oP (rn (α, τ p )) ,
ξ (p; g(Y )) − ξg(Y ) (p) =
fg(Y ) (ξg(Y ) (p))
(4)
where g(Y ) = (g(Y (1), . . . , g(Y (n))), for i = 1, . . . , n and where the sequence (rn (α, τ p ))n≥1
is defined by
rn (α, τ p ) =


n−1/2



n−1/2 log(n)1/2



 n−ατ p /2
if ατ p > 1,
if ατ p = 1,
if ατ p < 1.
(5)
Bahadur representation of sample quantiles for functional of Gaussian
dependent sequences
4
Remark 1 The sequence rn (α, τ p ) is related to the behaviour short-range or long-range
dependent behaviour of the sequence hu (Y (1)), . . . , hu (Y (n)) for u in a neighborhood of
ξ(p). More precisely, it corresponds to the asymptotic behaviour of the sequence
1/2
X
1
ρ(i)τ p  .
n

|i|<n
Corollary 2 Under Assumption A(ξ(p)), then the following convergence in distribution hold as n → +∞
( i)
if ατ p > 1
√ d
n ξb(p; g(Y )) − ξg(Y ) (p) −→ N (0, σ 2 ),
(6)
hal-00131071, version 2 - 29 Sep 2008
p
where
σp2 =
1 X X cj (p)2
ρ(i)j with f (p) = fg(Y ) (ξg(Y ) (p)) and cj (p) = cj (ξg(Y ) (p)).
f (p)2
j!
i∈Z j≥τ p
(ii) if ατ p < 1
where
cτ p (p)
d
nατ p /2 ξb(p; g(Y )) − ξg(Y ) (p) −→
Zτ ,
τ p !f (p) p
Zτ p = K(τ p , α)
and
Z
′
Rτ p
(7)
τp
exp(i(λ1 + · · · + λτ p )) − 1 Y
e
|λj |(α−1)/2 B(dλ
j)
i(λ1 + · · · + λτ p )
K(τ p , α) =
j=1
(1 − ατ p /2)(1 − ατ p )
τ p ! (2Γ(α) sin(π(1 − α)/2))τ p
e is a Gaussian complex measure and the symbol
The measure B
of integration excludes the hyperdiagonals {λi = ±λj , i 6= j}.
!1/2
R′
.
means that the domain
The proof of this result is omitted since it is a direct application of Theorem 1
and general limit theorems adapted to nonlinear functional of Gaussian sequences, e.g.
Breuer and Major (1983) and Dehling and Taqqu (1989).
Bahadur representation of sample quantiles for functional of Gaussian
dependent sequences
5
3
3.1
Proofs
Auxiliary Lemma
Lemma 3 For every j ≥ 1 and for all positive sequence (un )n≥1 such that un → 0, as
n → +∞, we have, under Assumption A(ξ(p))
Z
Hj (t)φ(t)1{|g(t)−ξg(Y ) (p))|≤un } dt ∼ un κj ,
I=
(8)
hal-00131071, version 2 - 29 Sep 2008
R
where κj is defined, for every j ≥ 1,by

P
φ′ (gi−1 (ξ(p))

 −2 L
i=1 g ′ (g −1 (ξ(p)))
i
κj =
PL φ(j−2) (gi−1 (ξ(p))

j
 2(−1)
i=1 g ′ (g −1 (ξ(p)))
i
if j = 1,
(9)
if j > 1.
Proof. Under Assumption A(ξ(p)), there exists n0 ∈ N such that for all n ≥ n0 ,
I=
L
X
Ii
with
Ii =
Z
Ui
i=1
Hj (t)φ(t)1{ξ(p)−un ≤ g(t) ≤ ξ(p)+un } dt.
(10)
Assume without loss of generality that the restriction of g(·) on Ui (denoted by gi (·)) is
an increasing function, we have
Z
Hj (t)φ(t)1{ξ(p)−un ≤ g(t) ≤ ξ(p)+un } dt
Ii =
Ui
=
Z
gi−1 (ξ(p)+un )
gi−1 (ξ(p)−un )
Hj (t)φ(t)dt

 φ(mi,n ) − φ(Mi,n ) = (mi,n − Mi,n )
=
 (−1)j φ(j−1) (M ) − φ(j−1) (m )
i,n
i,n
if j = 1
if j > 1,
where Mi,n = gi−1 (ξ(p) + un ) and mi,n = gi−1 (ξ(p) − un ). Then, there exists ωn,i,j ∈
[mi,n , Mi,n ] for every j ≥ 1 such that

 (mi,n − Mi,n ) φ(1) (ωn,i,1 )
Ii =
 (−1)j (M − m ) φ(j−2) (ω
i,n
i,n
n,i,j )
if j = 1
,
if j > 1.
Under Assumption A(ξ(p)), we have, as n → +∞
ωn,i,j ∼ gi−1 (ξ(p))
which ends the proof.
and
Mi,n − mi,n ∼ 2un
1
g′ (gi−1 (ξ(p)))
,
Bahadur representation of sample quantiles for functional of Gaussian
dependent sequences
6
3.2
Proof of Theorem 1
For the sake of simplicity, we set ξb(p) = ξb(p; g(Y )), ξ(p) = ξg(Y ) (p), Fb (·) = Fb (·; g(Y )),
F(·) = Fg(Y ) (·) et f(·) = fg(Y ) (·) and rn = rn (α, τ p ). Define,
−1 p − F(p)
−1 b
and Wn = rn
Vn = rn ξ (p) − ξ(p)
.
f(p)
P
The result is established if Vn − Wn → 0 as n → +∞. It suffices to prove that Vn and
Wn satisfy the conditions of Lemma 1 of Ghosh (1971):
hal-00131071, version 2 - 29 Sep 2008
• condition (a) : for all δ > 0, there exists ε = ε(δ) such that P (|Wn | > ε) < δ.
• condition (b) : for all y ∈ R and for all ε > 0
lim P (Vn ≤ y, Wn ≥ k + ε)
n→+∞
and
lim P (Vn ≥ y + ε, Wn ≥ k)
n→+∞
condition (a) : from Bienaym´e-Tchebyshev’s inequality it is sufficient to prove that
−1 P
n
EWn2 = O(1). Rewrite Wn = rnn
i=ℓ+1 hξ(p) (Y (i)). Let cj (for some j ≥ 0) denote
the j-th Hermite coefficient of hξ(p) (·). Since hξ(p) (·) has at least Hermite rank τ p , then
EWn2
n
rn−2 X
= 2
E hξ(p) (Y (i1 )) hξ(p) (Y (i2 ))
n
=
=
i1 ,i2 =1
n
rn−2 X X
cj1 cj2 E (Hj1 (Y
n2
i1 ,i2 =1 j1 ,j2 ≥τp
n
−2
rn X X (cj )2
ρ(i2 − i1 )j
n2
(j)!
i1 ,i2 =1 j≥τ p

from Remark 1.
(i1 )) Hj2 (Y (i2 )))

X
1
= O rn−2 ×
ρ(i)τ p  = O (1) ,
n
|i|<n
condition (b) : let y ∈ R, we have
n
o
{Vn ≤ y} = ξb(p) ≤ y × rn + ξ(p)
n
o
= p ≤ Fb (y × rn + ξ(p)) = {Zn ≤ yn } ,
with
rn−1
Zn =
f(ξ(p))
y
b
F y × rn + ξ(p) − F √ + ξ(p)
rn
(11)
Bahadur representation of sample quantiles for functional of Gaussian
dependent sequences
7
and
yn =
rn−1 F y × rn + ξ(p) − p
f (ξ(p))
P
Under Assumption A(ξ(p)), we have yn → y, as n → +∞. Now, prove that Zn −Wn → 0.
Without loss of generality, assume y > 0. Then, we have
rn−1 b
F y × rn + ξ(p) − F y × rn + ξ(p) − Fb (ξ(p)) + F(ξ(p))
f(p)
n
X
rn−1
1
=
hξ(p),n (Y (i))
n f(ξ(p))
Wn − Zn =
i=1
hal-00131071, version 2 - 29 Sep 2008
where hξ(p),n (·) is the function defined for t ∈ R by :
hξ(p),n (t) = 1n
ξ(p)≤g(t)≤ξ(p)+y×rn
o (t) − P ξ(p) ≤ g(Y ) ≤ ξ(p) + y × rn .
For n sufficiently large, the function hξ(p),n (·) has Hermite rank τ p . Denote by cj,n the
j-th Hermite coefficient of hξ(p),n (·). From Lemma 3, there exists a sequence (κj )j≥τ p
such that, as n → +∞
cj,n ∼ κj × rn .
P
Since, for all n ≥ 1 E(hn (Y )2 ) = j≥τ p (cj,n )2 /j! < +∞, it is clear that the sequence
P
(κj )j≥τ p is such that j≥τ p (κj )2 /j! < +∞. By denoting λ a positive constant, we get,
as n → +∞
E(Wn − Zn )2 =
=
=
n
X
rn−2
1
E hξ(p),n (Y (i1 )) hξ(p),n (Y (i2 ))
2
2
n f(ξ(p))
rn−2
n2
1
f(ξ(p))2
1
rn−2
n2 f(ξ(p))2
i1 ,i2 =1
n
X
X
cj1 ,n cj2 ,n E (Hj1 (Y (i1 )) Hj2 (Y (i2 )))
i1 ,i2 =1 j1 ,j2 ≥τ p
n
X
X c2j,n
i1 ,i2 =1 j≥τ p
j!
ρ(i2 − i1 )j


X
1
rn−2 X (κj )2 2 X
≤λ
ρ(i)j = O 
ρ(i)τ p  = O(rn2 ),
rn
n
j!
n
j≥τ p
|i|<n
|i|<n
from Remark 1. Therefore, Wn − Zn converges to 0 in probability, as n → +∞. Thus,
for all ε > 0, we have, as n → +∞,
P (Vn ≤ y, Wn ≥ y + ε) = P (Zn ≤ yn , Wn ≥ y + ε) → 0.
Bahadur representation of sample quantiles for functional of Gaussian
dependent sequences
8
Following the sketch of this proof, we also have P (Vn ≥ y + ε, Wn ≤ y) → 0, ensuring
condition (b). Therefore, Wn − Zn converges to 0 in probability, as n → +∞. Thus, for
all ε > 0, we have, as n → +∞,
P (Vn ≤ y, Wn ≥ y + ε) = P (Zn ≤ yn , Wn ≥ y + ε) → 0.
Following the sketch of this proof, we also have P (Vn ≥ y + ε, Wn ≤ y) → 0, ensuring
condition (b).
Acknowledgement The author would like to thank the referee for his coments im-
hal-00131071, version 2 - 29 Sep 2008
proving the statement of Corollary 2.
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J.-F. Coeurjolly
SAGAG, Department of Statistics, Grenoble
1251 Av. Centrale BP 47
38040 GRENOBLE Cedex 09
France
E-mail: [email protected]