properties both in the iid and dependent settings. Then, we will present the asymptotic representation of the estimator derived
from the assumptions.
3.1 Assumptions
Before stating the assumptions, we introduce some notations, which will be used throughout the asymptotic analysis of our
estimator. As mentioned in the previous section, we consider a certain family of copula densities, C = {cu
, u ; θ , θ ∈ }, for
estimating the true density cu , u
. Define θ
∗
to be the unique pseudo-true copula parameter which lies in the interior of and
minimizes I
θ =
[0,1]
d+1
log c
u , u
c u
, u ; θ
dC u
, u .
Here, I θ is the classical Kullback-Leibler information cri- terion expressed in terms of copula densities instead of the
traditional densities. It is clear that when the true copula density belongs to the given family, that is, c· = c·; θ
for a certain θ , then we have θ
∗
= θ . Additionally, we let
c
X
u; θ = c
u , u
; θ du . Also define
f
θ
y|x = f y
c F
y, Fx; θ
c
X
Fx; θ
and m
τ
x; θ = arg min
y
E[ρ
τ
Y − ycF Y , Fx; θ ].
Concerning the partial derivatives of the copula density, we define
D
j
c = ∂c
∂u
j
, j = 0, . . . , d, c
′
= D
1
c, . . . , D
d
c
⊤
and
˙c =
∂c ∂θ
1
, . . . , ∂c
∂θ
p ⊤
. Here are the assumptions for our estimator.
C0 {Y
i
}
i≥1
is a strictly stationary process with β-mixing coefficient βi satisfying βi = Oi
−ν
, as i → ∞, for some ν 1.
C1 ˆ F
x − Fx = O
p
n
−12
, where
ˆ F
x =
ˆ F
1
x
1
, . . . , ˆ F
d
x
d ⊤
and ˆ
F
j
· is an estimator for F
j
·. C2 ˆθ − θ
∗
= O
p
n
−12
, where ˆθ is an estimator of θ
∗
.
C3 Let g denote either ˙c or D
j
c , j = 0, . . . , d and
x ∈ R
d
be a given point of interest such that Fx ∈
0, 1
d
. i
c 1, Fx; θ
∗
∞, c0, Fx; θ
∗
∞ and 0 c
X
Fx; θ
∗
∞;
ii u, θ → gu , u
; θ is continuous in u, θ at Fx, θ
∗
uniformly in u ∈ [0, 1];
iii u → gu
, F x; θ
∗
is continuous on [0, 1]. C4 f
is continuous at m
τ
x; θ
∗
. C5 y → f
θ
∗
y|x is continuous at m
τ
x; θ
∗
and f
θ
∗
m
τ
x; θ
∗
|x 0.
Assumption C3 is satisfied for many popular copula fami- lies. C4 and C5 are typically assumed in quantile regression.
Hence, in the following we will give some examples where C0, C1, and C2 are satisfied in the iid setting and the time series
setting. 3.1.1
IID Setting . Suppose that we have Y
i
, X
i
, i = 1, . . . , n, an independent and identically distributed iid sam-
ple of n observations generated from the distribution of Y, X. In this case, C0 is trivially satisfied. Concerning C1, it is
satisfied with the empirical distribution of X
j
and its rescaled version which is popular in copula estimation context and is
defined by ˆ
F
s j
x
j
= 1
n + 1
n i=1
I X
j,i
≤ x
j
. Additionally, we can also use kernel smoothing method for
estimating F
j
·, j = 1, . . . , d. Let k· be a function which is a symmetric probability density function and h ≡ h
n
→ 0 be a bandwidth parameter. Then, a kernel smoothing estimator ˜
F
j
is given by
˜ F
j
x
j
= 1
n
n i=1
K x
j
− X
j,i
h ,
where Kx =
x −∞
k t dt. If nh
4
→ 0 holds for the bandwidth h
, then for ˆ F
j
= ˜ F
j
, the following condition is satisfied: C1’ ˆ
F
j
x
j
= n
−1 n
i=1
I X
j,i
≤ x
j
+ o
p
n
−12
, from which C1 follows. One advantage of using ˜
F
j
is that it results in a smooth estimate ˆ
m
τ
x, whereas the empirical
distribution or its rescaled version does not. As for C2, one example of the estimator ˆ
θ that satisfies C2 in the literature
is the maximum pseudo-likelihood PL estimator ˆθ
PL
, which maximizes
L θ =
n i=1
log c ˆ
F
s
Y
i
, ˆ F
s
X
i
; θ ,
5 where ˆ
F
s
X
i
= ˆ F
s 1
X
1,i
, . . . , ˆ F
s d
X
d,i ⊤
. The estimator ˆθ
PL
was studied by several authors including Genest, Ghoudi, and Rivest
1995 , Klaassen and Wellner
1997 , Silvapulle, Kim,
and Silvapulle 2004
, Tsukahara 2005
, and Kojadinovic and Yan
2011 , etc. If the score function of cu
, u ; θ satisfies the
assumptions A.1–A.5 given in Tsukahara 2005
, the PL esti- mator satisfies C2 even when the copula family is misspecified
as checked in Noh, El Ghouch, and Bouezmarni 2013
. 3.1.2
Time Series Setting . Assumptions under which a
copula-based Markov process satisfies α-mixing or β-mixing have been studied by many authors. For example, if {Y
i
}
i≥1
is a stationary univariate first-order Markov process and the copula of Y
i
, Y
i−1
≡ Y
i
, X
i, 1
satisfies certain conditions see Proposition 2.1 of Chen and Fan
2006 , then Assumption C0
holds and hence C1 also holds with any estimator ˆ F
j
· satis- fying C1’ see Rio
2000 . Following similar ideas, R´emillard,
Papageogiou, and Soustra 2012
extended this result to the case of copula-based multivariate first-order Markov process. As for
copula-based Markov processes of higher order, unfortunately such results are rare. The only related work that we have found
is Ibragimov 2009
, who obtained a characterization of higher- order Markov processes in terms of copulas, but he did not
discuss the mixing properties of the resulting process. Concerning Assumption C2, if we consider an extension of
the maximum pseudo-likelihood estimator studied by Genest,
Downloaded by [Universitas Maritim Raja Ali Haji] at 19:25 11 January 2016
Ghoudi, and Rivest 1995
to the Markovian case, the result- ing estimator satisfies C2. For example, suppose that we have
a sample {Y
i
, X
i
: i = 1, . . . , n} of a multivariate first-order Markov process generated from F
·, F
1
·, c·, ·, ·, ·; θ
∗
, where c·, ·, ·, ·; θ
∗
is the true parametric copula density as-
sociated with Z
i
, Z
i−1
with Z
i
= Y
i
, X
i ⊤
up to the unknown value θ
∗
. If we consider the following estimator for θ
∗
, ˆθ
dep PL
= arg max
θ ∈ n
i=2
log c
ˆ U
i
, ˆ U
i−1
; θ q
ˆ U
i−1
; θ ,
6 where
ˆ F
s
· = 1
n + 1
n i=1
I Y
i
≤ ·, ˆ
F
s 1
· = 1
n + 1
n i=1
I X
i
≤ ·, ˆU
i
= ˆ
F
s
Y
i
, ˆ F
s 1
X
i ⊤
, q
u
2
, u
3
; θ =
1 1
c u
, u
1
, u
2
, u
3
; θ du
1
du ,
then C2 holds according to Theorem 1 in R´emillard, Papageo- giou, and Soustra
2012 under the assumptions A1–A4 that
they provided. For univariate copula-based first-order Markov models, a similar result can be found in Chen and Fan
2006 .
3.2 Asymptotic Representation of the Estimator ˆ
m
τ
x
To realize the theoretical analysis of our estimator, we begin by introducing a few more notations:
• ˆ F
y = n
−1 n
i=1
I Y
i
≤ y. • ǫ
i
≡ ǫ
i
x; θ
∗
= Y
i
− m
τ
x; θ
∗
.
• e
′
x = E[ψ
τ
ǫc
′
F Y , Fx; θ
∗
] and
˙ex = E[ψ
τ
ǫ˙cF Y , Fx; θ
∗
], where ψ
τ
y = τ − I y ≤ 0. Now we are ready to present the asymptotic representation of
the proposed estimator. Theorem 3.1.
Suppose that Assumptions C0–C5 hold. Then, we have
√ n
ˆ m
τ
x − m
τ
x; θ
∗
= 1
f m
τ
x; θ
∗
cF m
τ
x; θ
∗
, Fx; θ
∗
U
n
+ o
p
1, where
U
n
= √
n ˆ
F x − Fx
⊤
e
′
x +
√ n
ˆθ − θ
∗ ⊤
˙ex
− √
n ˆ
F m
τ
x; θ
∗
− F m
τ
x; θ
∗
× cF m
τ
x; θ
∗
, Fx; θ
∗
. Theorem 3.1 implies that the estimator ˆ
m
τ
x converges in
probability to m
τ
x; θ
∗
as n → ∞. Hence, when the copula family is misspecified, the estimator ˆ
m
τ
x is no more con-
sistent. In such situation, since c·; θ
∗
is just the best ap- proximation to the true copula density c·, we have a bias in
the estimation of the true conditional quantile function m
τ
x,
which is asymptotically the difference between m
τ
x and
its best approximation m
τ
x; θ
∗
among the function class
{mx; θ : θ ∈ }.
As an application of Theorem 3.1, we consider the asymptotic normality of the estimator, for which we have to make a stronger
assumption than C2: C2
′
ˆθ − θ
∗
= n
−1 n
i=1
η
i
+ o
p
n
−12
, where η
i
= ηU
0,i
, U
i
; θ
∗
is a p-dimensional random vector such that Eη = 0 and
E η
⊤
η ∞ and U
i
= U
1,i
, . . . , U
d,i ⊤
. This stronger assumption also holds for ˆθ
PL
in the iid setting and ˆθ
dep PL
in the time series setting with the same conditions for C2. For the iid case, the function η is given by
η U
, U ; θ = J
−1
θ × KU , U
; θ , 7
where J
θ =
[0,1]
d+1
− ∂
2
∂θ ∂θ
⊤
log cu
, u
; θ dC
u
, u
and KU , U
; θ is a p-dimensional vector whose kth element is
∂ ∂θ
k
log cU , U
; θ +
d j =0
[0,1]
d+1
I U
j
≤ u
j
− u
j
× ∂
2
∂θ
k
∂u
j
log cu , u
; θ dC
u , u
. Concerning the time series case, see Chen and Fan
2006 for
the univariate case and R´emillard, Papageogiou, and Soustra 2012
for the multivariate case. Replacing C2 with C2’ in the previous assumptions, we have an asymptotic linear
representation of the conditional quantile estimator ˆ m
τ
x,
which implies the asymptotic normality of ˆ m
τ
x.
Corollary 3.1. Suppose that Assumptions C0–C1, C1’,
C2’, C3–C5 hold. Then, we have √
n ˆ
m
τ
x − m
τ
x; θ
∗
= 1
√ n
n i=1
d j =1
I X
j,i
≤ x
j
− F
j
x
j
e
j
x + η
⊤ i
˙ e
x
− I
Y
i
≤ m
τ
x; θ
∗
− F m
τ
x; θ
∗
c F
m
τ
x; θ
∗
, F
x; θ
∗
[f m
τ
x; θ
∗
cF m
τ
x; θ
∗
, Fx; θ
∗
]
−1
+ o
p
1, 8
where e
′
x = e
1
x, . . . , e
d
x
⊤
. Specifically,
Corollary 3.1
implies that
√ n
ˆ m
τ
x −
m
τ
x; θ
∗
follows asymptotically a normal distribution with mean 0 and variance σ
2
= varE
1
+ 2
∞ j =1
covE
j +1
, E
1
, where E
i
denotes each summand in the summation of 8
see Rio 2000
, theorem 4.2. Especially, since σ
2
= varE
1
when the data are iid, we can estimate σ
2
by an estimator ˆ
σ
2
= n
−1 n
i=1
{ E
i
− n
−1 n
i=1
E
i
}
2
, where E
i
is an estima- tor of E
i
obtained by replacing all the unknown quantities in E
i
by their corresponding estimates, for example, θ
∗
by ˆθ and F by ˆ
F . Thanks to this estimator ˆ
σ , we can easily calculate the
confidence interval for m
τ
x; θ
∗
using Corollary 3.1. However, according to our simulation studies see Section
4 ,
the accuracy of the coverage of this confidence interval seems
Downloaded by [Universitas Maritim Raja Ali Haji] at 19:25 11 January 2016
to sensitively depend on the accuracy of the estimation of the unknown quantities involved in E
i
. Due to this, we propose to use a bootstrap method to approximate the asymptotic variance
of the estimator ˆ m
τ
x. The bootstrap that we use for the iid
