arXiv:1604.00039v1 [] 31 Mar 2016

Pointwise Adaptive Estimation of the Marginal
Density of a Weakly Dependent Process
Karine Bertin∗

Nicolas Klutchnikoff†

April 4, 2016

Abstract
This paper is devoted to the estimation of the common marginal
density function of weakly dependent processes. The accuracy of
estimation is measured using pointwise risks. We propose a datadriven procedure using kernel rules. The bandwidth is selected using
the approach of Goldenshluger and Lepski and we prove that the
resulting estimator satisfies an oracle type inequality. The procedure
is also proved to be adaptive (in a minimax framework) over a scale of
Hölder balls for several types of dependence: stong mixing processes, λdependent processes or i.i.d. sequences can be considered using a single
procedure of estimation. Some simulations illustrate the performance
of the proposed method.
Keywords. Adaptive minimax rates, Density estimation, Hölder spaces,
Kernel estimation, Oracle inequality, Weakly dependent processes

1

Introduction

Let X = (Xi )i∈Z be a real-valued weakly dependent process admitting a
common marginal density f : R → R. We consider the problem of estimating
f at a fixed point x0 based on observation of X1 , . . . , Xn with n ∈ N∗ . The
accuracy of an estimator f˜n is evaluated using the pointwise risk defined, for
fixed x0 ∈ R and q > 0, by


Rq (f˜n , f ) = E|f˜n (x0 ) − f (x0 )|q
∗

1/q

,

CIMFAV, Universidad de Valparaíso, General Cruz 222, Valparaíso, Chile, tel/fax:
0056322303623
†
Crest-Ensai and Université de Strasbourg

1

where E denotes the expectation with respect to the distribution of the process
X. The main interest in considering such risks is to obtain estimators that
adapt to the local behavior of the density function to be estimated.
The aim of this paper is to obtain adaptive estimators of f on Hölder
classes of regularity s > 0 for this risk and different types of weakly dependent
processes.
In the independent and identically distributed (i.i.d.) case, the minimax
rate of convergence is n−s/(2s+1) (see Tsybakov (2009), and references therein).
Adaptive procedures based on the classical Lepki procedure (see Lepski (1990))


s/(2s+1)

have been obtained (see Butucea (2000)) with rates of the form logn n
.
In the context of dependent data, Ragache and Wintenberger (2006) as
well as Rio (2000) studied kernel density estimators from a minimax point
of view for pointwise risks. They obtained the same rate of convergence
as in the independent case when the coefficients of dependence decay at
a geometric rate. Several papers deal with the adaptive estimation of the
common marginal density of a weakly dependent process. Tribouley and
Viennet (1998) and Comte and Merlevède (2002) proposed Lp –adaptive
estimators under α–mixing or β–mixing conditions that converge at the
previously mentioned rates. Gannaz and Wintenberger (2010) extend these
results to a wide variety of weakly dependent processes including λ–dependent
processes. Note that, in these papers, the proposed procedures are based
on nonlinear wavelet estimators and only integrated risks are considered.
Moreover, the thresholds are not explicitly defined since they depend on an
unknown multiplicative constant. As a consequence, such methods can not
be used directly for practical purposes.
Our main purpose is to prove similar results for pointwise risks. We propose
here a kernel density estimator with a data-driven selection of the bandwidth
where the selection rule is performed using the so-called Goldenshluger-Lepski
method (see Goldenshluger and Lepski, 2008, 2011, 2014). This method
was successfully used in different contexts such as in Comte and GenonCatalot (2012), Doumic, Hoffmann, Reynaud-Bouret, and Rivoirard (2012),
Bertin, Lacour, and Rivoirard (2014), Rebelles (2015), but only with i.i.d.
observations. However there are at least two practical motivations to consider
dependent data. Firstly, obtaining estimators that are robust with respect to
slight perturbations from the i.i.d. ideal model can be useful. Secondly, many
econometric models (such as ARCH or GARCH) deal with dependent data
that admit a common marginal density. These two motivations suggest to
consider a class of dependent data as large as possible and to find a single
procedure of estimation that adapts to each situation of dependence.
Our contribution is the following. We obtain the adaptive rate of con2

vergence for pointwise risks over a large scale of Hölder spaces in several
situations of dependence, such as α–mixing introduced by Rosenblatt (1956)
and the λ–dependence defined by Doukhan and Wintenberger (2007). This
partially generalizes previous results obtained in i.i.d. case by Butucea (2000)
and Rebelles (2015). To the best of our knowledge, this is the first adaptive
result for pointwise density estimation in the context of dependent data.
To establish it, we prove an oracle type inequality: the selected estimator
performs almost as well as the best estimator in a given large finite family
of kernel estimators. Our data-driven procedure depends only on explicit
quantities. This implies that this procedure can be directly implemented in
practice. As a direct consequence, we get a new method to choose an accurate
local bandwidth for kernel estimators.
The rest of this paper is organized as follows. Section 2 is devoted to the
presentation of our model and of the assumptions on the process X. The
construction of our procedure of estimation is developed in Section 3. The
main results of the paper are stated in Section 4 whereas their proofs are
postponed to Section 6. A simulation study is performed in Section 5 to
illustrate the performance of our method in comparison to other classical
estimation procedures. The proofs of the technical results are presented in
the appendix.

2

Model

In what follows X = (Xi )i∈Z is a real-valued discrete time process and the
observation consists of the vector (X1 , . . . , Xn ). We assume that the Xi ’s
are identically distributed and we aim at estimating the common marginal
density f at a fixed point x0 ∈ R. In this section, basic assumptions on the
distribution of X are stated. Examples of processes are given, which illustrate
the variety of models covered in this work.

2.1

Assumptions

We first assume that f is bounded in a neigborhood of x0 .
Assumption 1. The marginal density f satisfies
sup f (x) ≤ B,
x∈Vn (x0 )

where Vn (x0 ) = [x0 −



1
log n

1/q

, x0 +



1
log n

3

1/q

] and B is a positive real constant.

Before giving the two last assumptions, we need some notation. For
i ∈ Zu , we consider the random variable Xi = (Xi1 , . . . , Xiu ) with values in
Ru . If i ∈ Zu and j ∈ Zv , we define the gap between j and i by γ(j, i) =
min(j) − max(i) ∈ Z. For any positive integer u, the functional class Gu
consists of real-valued functions g defined on Ru such that the support of g is
included in (Vn (x0 ))u ,
kgk∞,u =
and

sup |g(x)| < ∞,
x∈[−1,1]u

|g(x) − g(y)|
< +∞.
Lipu (g) = sup Pu
x6=y
i=1 |xi − yi |

We now define the sequence ρ(X) = (ρr (X))r∈N∗ by


ρr (X) = sup

sup

sup sup

u,v∈N∗ (i,j)∈Zu ×Zv g∈Gu g̃∈Gv
γ(j,i)≥r

Cov g(Xi ), g̃(Xj )
Ψ(u, v, g, g̃)



,

where Ψ(u, v, g, g̃) = max(Ψ1 (u, v, g, g̃), Ψ2 (u, v, g, g̃)) where for g ∈ Gu and
g̃ ∈ Gv
Ψ1 (u, v, g, g̃) = 4kgk∞,u kg̃k∞,v
and
Ψ2 (u, v, g, g̃) = ukg̃k∞,v Lipu (g) + vkgk∞,u Lipv (g̃) + uv Lipu (g) Lipv (g̃).
Both Assumptions 2 and 3 given below, allow us to control the covariance
of functionals of the process. Assumption 2 deals with the type of dependence,
whereas Assumption 3 is technical (see Lemma 1, Section 6).
Assumption 2. For some positive constants a, b, and c, the sequence ρ(X)
is such that
ρr (X) ≤ c exp(−arb ), ∀r ∈ N.
Assumption 3. The exists a positive constant C such that
µ(X) =

sup

sup

g,g̃∈G1 \{0} i6=j

where kgk1 =

R
R

|E (g(Xi )g̃(Xj ))|
≤ C,
kgk1 kg̃k1

|g(t)|dt.

4

2.2

Comments

On Assumption 1. We are assuming that the marginal density f is
bounded on a neighborhood of x0 . Such an assumption is classical in density estimation (see Goldenshluger and Lepski, 2011, and references therein).
Note also that stationarity of X is not assumed. Thus, re-sampled processes
of stationary processes can be considered in this study as in Ragache and
Wintenberger (2006).
On Assumption 2. Recall that a process X is called weakly dependent
if, roughly speaking, the covariance between functionals of the past and
the future of the process decreases as the gap from the past to the future
increases (for a broader picture of weakly-dependent processes, as well as
examples and applications, we refer the reader to Dedecker, Doukhan, Lang,
León R., Louhichi, and Prieur (2007) and references therein). Assumption 2
ensures that the decay occurs at a geometric rate. Under similar assumptions,
Doukhan and Neumann (2007), Merlevède, Peligrad, and Rio (2009) proved
Bernstein-type inequalities that are used in the proof of Theorem 1.
Note also that the type of weak-dependence considered in this paper
includes usual types of dependence such as strong mixing as well as classical
weak-dependence assumptions used in econometrics as illustrated below.
Strongly mixing processes. The process X is called α-mixing if the
sequence α(X) = (αr (X))r∈N defined by
αr (X) = sup sup

sup |P(A ∩ B) − P(A)P(B)|

n
+∞
n∈Z A∈F−∞
B∈Fn+r

tends to 0 as r goes to infinity, where Fk` is defined for any k, ` ∈ Z as the
σ-algebra generated by (Xi )k≤i≤` . Recall that for any u, v ∈ N∗ , g : Ru → R
and g̃ : Rv → R such that kgk∞,u < +∞ and kg̃k∞,v < +∞ we have
sup
(i,j)∈Zu ×Zv
γ(j,i)≥r

|Cov(g(Xi ), g̃(Xj ))| ≤ 4kgk∞,u kg̃k∞,v αr (X).

This readily implies that, for any r ∈ N, we have ρr (X) ≤ αr (X).
λ–dependent processes. The process X is called λ–dependent if the
sequence (λr (X))r defined by


λr (X) = sup

sup

sup sup

u,v∈N∗ (i,j)∈Zu ×Zv g∈Gu g̃∈Gv
γ(j,i)≥r

5



Cov g(Xi ), g̃(Xt )
Ψ2 (u, v, g, g̃)

,

tends to 0 as r tends to infinity. Then ρr (X) ≤ λr (X) for any r ∈ N.
Example. Bernoulli shifts are, under some conditions, λ–dependent. Indeed, let us consider the process X defined by:
Xi = H((ξi−j )j∈Z ),

i∈Z

(1)

where H : RZ → [0, 1] is a measurable function and the variables ξi are i.i.d.
and real-valued. In addition, assume that there exists a sequence (θr )r∈N∗
such that
E|H((ξi )i∈Z ) − H((ξi0 )i∈Z )| ≤ θr
(2)
where, for any r ∈ N∗ , (ξi0 )i∈Z is an i.i.d. sequence such that ξi0 = ξi if |i| ≤ r and
ξi0 is an independent copy of ξi otherwise. It can be proved (see Doukhan and
Louhichi, 1999) that such processes are strongly stationary and λ–dependent
with rate ρr ≤ 2θ[r/2] . Remark also that θr can be evaluated under both
regularity conditions on the function H and integrability conditions on the ξi ,
i ∈ Z. Indeed, if we assume that there exist b ∈ (0, 1] and positive constants
P
(ai )i∈Z such that |H((xi )i ) − H((yi )i )| ≤ i∈Z ai |xi − yi |b with ξi ∈ Lb (R) for
all i ∈ Z, then
X
ai E|ξi |b .
θr =
|i|≥r

Moreover, under the weaker condition that (ξi )i∈Z is λ–dependent and
stronger assumptions on H (see Doukhan and Wintenberger, 2007), the
process X inherits the same properties. Finally, we point out that classical
econometrics models such as AR, ARCH or GARCH can be viewed as causal
Bernoulli shifts (that is, they obey (1) with j ∈ N).
On Assumption 3 This technical assumption is satisfied in several situations. In what follows, we offer examples of sufficient conditions such that
Assumption 3 holds.
Situation 1. We assume that for any i, j ∈ Z the pair (Xi , Xj ) admits a
density function fi,j with respect to the Lebesgue measure in R2 . Moreover
we assume that there exists a constant F such that
sup

fi,j (x) ≤ F.

sup

i,j∈Z x,y∈Vn (x0 )

Under this assumption and using Fubini’s theorem, we readily obtain that
Assumption 3 holds with C = F. Note that this assumption is used in Gannaz
and Wintenberger (2010).

6

Situation 2. We consider the infinite moving average process, with i.i.d.
innovations (ξj )j∈Z , given by :
Xi =

X

t ∈ Z,

aj ξi−j ,

j∈Z

where (aj )j∈N and (a−j )j∈N are decreasing sequences of deterministic positive
real numbers. We assume that ξ1 admits a density function pξ (·) bounded
above by a positive constant P. Set i, j ∈ Z and denote by Ξ the σ-algebra
generated by the process (ξi )i∈Z\{i,j} . For g, g̃ in G1 we have
E (g(Xi )g̃(Xj )) = E [E (g(Xi )g̃(Xj )|Ξ)]
= E [E (g(a0 ξi + aj−i ξj + A)g̃(a0 ξj + ai−j ξi + B)|Ξ)] ,
where A and B are Ξ-mesurable random variables. A simple change of
variables gives:
|E (g(Xi )g̃(Xj )|Ξ)| ≤

P2
kgk1 kg̃k1 .
a20 − aj−i ai−j

Since a20 − aj−i ai−j ≥ a20 − a1 a−1 , Assumption 3 is fulfilled with C = P2 (a20 −
a1 a−1 )−1 .
Situation 3. We consider a GARCH(1, 1) model. Let α, β and γ be
positive real numbers. Let (ξi )i∈Z be i.i.d. innovations with marginal density
pξ (·), bounded above by a positive constant B, and denote by (Fi )i∈Z the
natural filtration associated with this proccess. Assume that the process X is
such that, for any i ∈ Z:
Xi = σi ξi

with

2
2
σi2 = γ + αXi−1
+ βσi−1
.

(3)

Consider i, j ∈ Z such that i < j. For g, g̃ in G1 we have
E (g(Xi )g̃(Xj )) = EE [g(Xi )g̃(Xj )|Fi ]
= E (g(Xi )E [g̃(Xj )|Fi ]) .
Now remark that, since σj ∈ Fj−1 and ξj is independent of Fj−1 , we have
E [g̃(Xj )|Fi ] = E (E [g̃(ξj σj )|Fj−1 ] |Fi )
1
x
= g̃(x)E
pξ
σj
σj
R
Z

Since σj ≥

√

!

!

|Fi dx.

γ we obtain:
|E (g(Xi )g̃(Xj ))| ≤

B
kgk1 kg̃k1 .
γ

Assumption 3 is thus fulfilled with C = B/γ.
7

3

Estimation procedure

In this section, we describe the construction of our procedure which is based
on the so-called Goldenshluger-Lepski method (GLM for short). It consists
in selecting, in a data driven way, an estimator in a given family of linear
kernel density estimators. Consequently, our method offers a new approach
to select an optimal bandwidth for kernel estimators in order to estimate the
marginal density of a process in several situations of weak dependence. This
leads to a procedure of estimation which is well adapted to inhomogeneous
smoothness of the underlying marginal density. Notice also that our procedure
is completely data-driven: it depends only on explicit constants that do not
need to be calibrated by simulations or using the so-called rule of thumb.

3.1

Kernel density estimators

We consider kernels K : R → R that satisfy the following assumptions.
Assumption 4. The kernel KRis compactly supported on [−1, 1]. Its Lipschitz
constant Lip(K) is finite and R K(x) dx = 1.
Assumption 5. There exists m ∈ N such that the kernel K is of order m.
That is, for any 1 ≤ ` ≤ m, we have
Z

K(x)x` dx = 0.

R

Let h∗ = n−1 exp
define Hn = {2−k
defined by,

√



log n and h∗ = (log n)−1/q be two bandwidths and
: k ∈ N} ∩ [h∗ , h∗ ]. We consider the family of estimators fˆh

n
1X
ˆ
fh (x0 ) =
Kh (x0 − Xi ),
n i=1

h ∈ Hn

where Kh (·) = h−1 K(h−1 ·).

3.2

Bandwidth selection

Following Goldenshluger and Lepski (2011), we first define for h ∈ Hn the
two following quantities
n

o

c (h, h)
A(h, x0 ) = max |fˆh∨h (x0 ) − fˆh (x0 )| − M
n
h∈Hn

and
n (h)

c
M

=

v
u
u
t2q|log h|

8

+

!

δn
,
n (h) +
nh

Jb

(4)

where δn = (log n)−1/2 , {y}+ = max(0, y) for any y ∈ R, and h ∨ h =
max(h, h) for any h, h ∈ Hn . We also consider
c (h, h) = M
c (h) + M
c (h ∨ h),
M
n
n
n

and
Jbn (h) =

n
1 X
K 2 (x0 − Xi ).
n2 i=1 h

Then our procedure consists in selecting the bandwidth ĥ(x0 ) such that




c (h) .
ĥ(x0 ) = arg min A(h, x0 ) + M
n

(5)

h∈Hn

The final estimator of f (x0 ) is defined by
fˆ(x0 ) = fˆĥ(x0 ) (x0 ).
Remark. The Goldenshluger-Lepski method consists in selecting a data-driven
bandwidth that makes a trade-off between the two quantities A(h, x0 ) and
c (h). Hereafter we explain how the minimization in (5) can be viewed
M
n
as an empirical version of the classical trade-off between a bias term and a
penalized standard deviation term.
c (h) can be viewed as a penalized upper bound of the
1. The quantity M
n
standard deviation of the estimator fˆh . Indeed, Lemma 2 implies that

Var(fˆh (x0 )) ≤ Jn (h) +
where
Jn (h) =

δn
6nh

1Z 2
Kh (x0 − x)f (x)dx
n

(6)

would be the variance of fˆh (x0 ) if the data were i.i.d. Moreover (see
the proof of Theorem 1 in Section 6), for n large enough and with high
probability
δn
δn
Jn (h) +
≤ Jˆn (h) +
.
6nh
nh
2. The quantity A(h, x0 ) is a rough estimator of the bias term of fˆh . Indeed
(see proof of Theorem 1), we have
A(h, x0 ) ≤ max |Efˆh∨h (x0 ) − Efˆh (x0 )| + 2T
h∈Hn

≤2 max|Kh ? f (x0 ) − f (x0 )| + 2T,
h≤h

9

(7)

where
o

n

c (h)
T = max |fˆh (x0 ) − Efˆh (x0 )| − M
.
n
+
h∈Hn
√
The quantity T is negligible with respect to 1/ nh with high probability
and maxh≤h |Kh ? f (x0 ) − f (x0 )| is of the same order of the bias of fˆh
over Hölder balls.

4

Results

We prove two results. Theorem 1 is an oracle-type inequality: under appropriate assumptions, our estimator performs almost as well as the best linear
kernel estimator in the considered family. Theorem 2 proves that our procedure achieves classical minimax rates of convergence (up to a multiplicative
logarithmic factor) over a wide scale of Hölder spaces.
Theorem 1. Under Assumptions 1, 2, 3 and 4 we have:
Rqq (fˆ, f ) ≤ C1∗ min

h∈Hn





max |Kh ? f (x0 ) − f (x0 )|q +


 h≤h

h∈Hn


!q/2 

|log h|

nh

(8)




where C1∗ is a positive constant that depends only on a, b, c, B, C and K.
Proof of Theorem 1 is postponed to Section 6.
Remark. The right hand side term of (8) can be viewed as a tight upper
bound for minh∈Hn E|fˆh (x0 ) − f (x0 )|q since it is the sum of an approximation
of the bias term and the standard deviation term (up to a multiplicative
logarithmic term) of fˆh . That means that our procedure performs almost as
well as the best kernel density estimator in the considered family.
Now using Theorem 1, we obtain in Theorem 2 the adaptive rate of
convergence on Hölder classes. Let s, L and B be positive real numbers. The
Hölder class C s (L, B) is defined as the set of functions f : R → R such that
Assumption 1 is fulfilled with the constant B, f is ms = sup{k ∈ N : k < s}
times differentiable and
|f (ms ) (x) − f (ms ) (y)| ≤ L|x − y|s−ms ,

∀x, y ∈ R.

Theorem 2. Let a, b, c, B, C, m and L be positive constants. Let K be
a kernel such that Assumptions 4 and 5 are fulfilled (in particular, K is a

10

kernel of order m) and set s such that ms ≤ m. There exists a constant C2∗
that depends only on a, b, c, B, C, s, L, K and q such that:
sup

sup

sup



E|fˆ(x0 ) − f (x0 )|q

ρ(X)∈R(a,b,c) µ(X)≤C f ∈C s (L,B)

1/q

≤

C2∗

log n
n

!

s
2s+1

,

where
n

o

R(a, b, c) = (ρr )r∈N : ρr ≤ c exp(−arb ) .
This result is a direct consequence of Theorem 1, since it can be easily
proved that
sup

max |Kh ? f (x0 ) − f (x0 )|q ≤ C3∗ hsq ,

f ∈C s (L,B) h≤h
h∈Hn

for any bandwidth h > 0, where C3∗ depends only on s, L, K and q. This
implies that, for n large enough, there exists hn (s, L, K, q) ∈ Hn such that
the right hand side of (8) is bounded, up to a multiplicative constant, by the
expected rate.
Remark.
1. Recall that the expectation E is taken with respect to the
distribution of the process X. Note also that the sequence ρ(X), µ(X)
and f depend only on this distribution. As a consequence our procedure
of estimation is minimax (up to a multiplicative log n term) with respect
to any distribution of X that satisfies the conditions:
ρ(X) ∈ R(a, b, c),

µ(X) ≤ C and f ∈ C s (L, B).

Indeed, in the i.i.d. case (which is included in our framework since
ρ(X) ≡ 0 and µ(X) ≤ B2 ), the minimax rate of convergence over the
Hölder class C s (L, B) is of order n−s/(2s+1) and can be obtained from
the results of Hasminskii and Ibragimov (1990) or Tsybakov (2009).
Moreover, note that fˆ does not depend on the constants a, b, c, B, C, L
and s that appear in these conditions. Thus, our procedure is adaptive,
up to the log n term, to both the regularity of f and the “structure” of
dependence.
2. It can be deduced from our proofs that the minimax rate of convergence
over Hölder classes, under Assumptions 1, 2 and 3, is upper bounded,
up to a multiplicative constant, by n−s/(2s+1) . This result was previously
obtained in a similar setting by Ragache and Wintenberger (2006) and
Rio (2000). Given that this rate is minimax optimal in the i.i.d. case, it
is also the minimax rate of convergence under our assumptions.
11

3. The extra log n term in the rate of convergence obtained in Theorem 2
is unavoidable. Indeed, for pointwise estimation, even in the i.i.d. case
(see Lepski, 1990, Klutchnikoff, 2014, Rebelles, 2015, among others)
the adaptive rate of convergence is of this form. This ensures that our
procedure attains the adaptive rate of convergence over Hölder classes.
4. Note that δn , that appears in (4), allows us to control the covariance
terms of the development of Var(fˆh (x0 )) under Assumption 2. If we
only consider the i.i.d. case, the covariance terms vanish, and the choice
δn = 0 can be considered. The resulting procedure still satisfies an
oracle inequality and remains adaptive in this case.
5. As far as we know, this result is the first theoretical pointwise adaptive
result for the estimation of the marginal density in a context of weak
dependence. Moreover, integrating the pointwise risk on a bounded
domain, we obtain that our procedure converges adaptively at the rate
s/(2s+1)
(n−1 log n)
in Lp –norm (p =
6 ∞) over Hölder balls. This extends
the results of Gannaz and Wintenberger (2010).

5

Simulation study

In this section, we study the performance of our procedure using simulated
data. More precisely, we aim at estimating three density functions, for three
types of dependent processes. In each situation, we study the accuracy of
our procedure as well as classical competitive methods, calculating empirical
risks using p = 10000 Monte-Carlo replications. In the following, we detail
our simulation scheme and comment the obtained results.
Simulation scheme
Density functions. We consider three density functions to be estimated. The
first one is:




f1 (x) = 1.28 sin((3π/2 − 1)x)I[0,0.65] (x) + I(0.65,1] (x) + cI[0,1] (x) ,
where c is a positive constant such that f1 is a density. The second one is the
density of a mixture of three normal distributions restricted to the support
[0, 1]


f2 (x) =

1
1
1
φ0.5,0.1 (x) + φ0.6,0.01 (x) + φ0.65,0.95 (x) + c I[0,1] (x),
2
4
4


12

where φµ,σ stands for the density of a normal distribution with mean µ and
standard deviation σ and c is a positive constant such that f2 is a density. Note
that very similar densities were also considered in Gannaz and Wintenberger
(2010). The third one is:

f3 (x) =

5
X
k=1

1
k
+
2 − 40 x −
10 20

!

I( k−1 , k ] (x) + 0.5I(0.5,1] (x).
10

10

The function f1 is very smooth except in the discontinuity point x = 0.65.
The function f2 is a classical example where rule-of-thumb bandwidths do
not work. The third function has several spikes in [0, 0.5] and is constant
on [0.5, 1]. As a consequence, a global choice of bandwidth can fail to catch
the two different behaviors of the function. The three densities are bounded
from above (Assumption 1 is then satisfied) and their inverse cumulative
distribution functions are Lipschitz.
Types of dependence: We simulate data (X1 , . . . , Xn ) with density f ∈
{f1 , f2 , f3 } in three cases of dependence. Denote by F the cumulative distribution function of f .
Case 1. The Xi are independent variables given by F −1 (Ui ) where the
Ui are i.i.d. uniform variables on [0, 1]. Assumptions 2 and 3 are clearly
satisfied.
Case 2. The Xi are λ–dependent given by F −1 (G(Yi )) where the Yi
satisfy of the non-causal equation:
Yi = 2(Yi−1 + Yi+1 )/5 + 5ξi /21,

i ∈ Z.

Here (ξi )i∈Z is an i.i.d. sequence of Bernoulli variables with parameter
1/2. The function G is the marginal distribution function of the Yi and of
0
the variable U +U3 +ξ0 , where U and U 0 are independent uniform variables
on [0, 1] (see Gannaz and Wintenberger, 2010, for more details).
Case 3. The Xi are given by F −1 (G(Yi )) where Y = (Yi )i∈Z is an
ARCH(1) process given by (3) where the (ξ)i∈Z are i.i.d. standard
normal variables, α = 0.5, β = 0 and γ = 0.5. In this case the function
G is estimated using the empirical distribution function on a simulated
process Ỹ independent of Y with the same distribution.
It remains to verify that Assumptions 2 and 3 hold for the processes X in the
last two cases. Firstly, note that the process Y is λ–dependent with
λr (Y) ≤ c exp(−ar),
13

(9)

for some positive constants a and c. Indeed, in the second case, Yi is of the
form (1) since it satisfies Yi =

P

j∈Z aj ξi−j with aj =

 r
1

1
3

 |j|
1
2

and the sequence

θr (see (2)) such that θr ∝ 2 . In the third case, since α < 1, Yi is β–mixing
at a geometric rate and then it is α–mixing and λ–dependent at a geometric
rate.
Now, in both cases, since F −1 and G are Lipschitz and the process (Yi )i∈Z
is λ–dependent, using Proposition 2.1 of Dedecker et al. (2007), we have that
the process (Xi )i∈Z is also λ–dependent with λr (X) satisfying (9) with some
positive constants a and c. As a consequence, Assumption 2 is fulfilled in the
last two cases.
Secondly, Assumption 3 holds for the process (Yi )i∈Z (see Situation 2 and
Situation 3 in Subsection 2.2). Then, for g, g̃ ∈ G1 and i, j ∈ Z with i =
6 j,
we have
|E (g(Xi )g̃(Xj ))| ≤Ckg ◦ F −1 ◦ Gk1 kg̃ ◦ F −1 ◦ Gk1
CB2
≤ 2 kgk1 kg̃k1
D
where D = min{G0 (x) : x ∈ G−1 ◦ F (Vn (x0 ))} is bounded from below as soon
as x0 ∈ (0, 1) and n is large enough. As a consequence, Assumption 3 is also
fulfilled for the procces X.
Estimation procedures
We propose to compare in this simulation study the following procedures.
• Our procedure (GL) fˆ performed with, in (4), q = 2 and δn =
(log n)−1/2 .
• The leave-one-out cross validation (CV) performed on the family of
kernel estimators given in Subsection 3.1.
• The kernel procedure with bandwidth given by the rule-of-thumb (RT).
• The procedure developed by Gannaz and Wintenberger (2010).
In the first three procedures, we use the uniform kernel.
Quality criteria
For each density function f ∈ {f1 , f2 , f3 } and each case of dependence,
we simulate p = 10000 sequences of observations (X1 , . . . , Xn ) with n = 1000.
14

f1

f2

f3

GL
CV
RT
GL
CV
RT
GL
CV
RT

Case 1

Case 2

Case 3

0.036
0.027
0.036
0.181
0.079
0.965
0.090
0.172
0.263

0.033
0.034
0.040
0.203
0.116
0.975
0.098
0.180
0.266

0.044
0.049
0.054
0.222
0.162
0.971
0.118
0.190
0.286

Table 1: Mean of ISE for the two densities f1 and f2 , the three cases of
dependence and the three procedures GL, CV and RT.

Given an estimation procedure, we calculate p estimators fˆ(1) , . . . , fˆ(p) . We
consider the empirical integrated square error:
d =
ISE

p Z

2
1X
f (x) − f (j) (x) dx.
p j=1 [0,1]

Results
Our results are summarized in Table 1.
For the estimation of the function f1 , our procedure gives better results
than the CV or RT methods in cases of dependence (2 and 3). We also
outperform the results of Gannaz and Wintenberger (2010) for case 1 and 2
where the ISE was around 0.09 (case 3 was not considered in their paper).
For the estimation of f2 which is quite smooth, the cross validation method
is about two times better than GL method and as expected, the RT method
does not work. For the estimation of f3 that contains several peaks, the GL
procedure is about two times better than the CV method.
To conclude, in the considered examples, our procedure has similar or
better performances than already existing methods used for dependent data.
Moreover, it gives better results when the density function to be estimated
presents irregularities. This illustrates the fact that our method adapts locally
to the irregularities of the function thanks to the use of local bandwidths. An
other important point is that the choice of the bandwidth depends on explicit
constants that can be used directly in practice and do not need previous
calibration. Additionally, our GL procedure is about 25 times faster than
cross-validation.
15

6
6.1

Proofs
Basic notation

For the sake of readability, we introduce in this section some conventions and
notations that are used throughout the proofs. Moreover, here and later, we
assume that Assumptions 1, 2 and 3 hold.
Firstly, let us consider, for any h ∈ Hn , the functions gh and ḡh defined,
for any y ∈ R, by gh (y) = Kh (x0 − y) and
ḡh (y) =

gh (y) − Egh (X1 )
.
n

Note that we have:
fˆh (x0 ) − Efˆh (x0 ) =

n
X

ḡh (Xi ),

h ∈ Hn .

i=1

Next, we introduce some constants. Let us consider:
C1 = kKk21 (B2 + C),

C2 = BkKk22 ,

and

C3 = 2kKk∞ .

Moreover we define L = Lip(K) and
C4 = 2C3 L + L2

6.2

and

1/4

C5 = (2C1 )3/4 C4 .

Auxiliary results

In the three following lemmas, assume that Assumptions 1, 2 and 3 are
satisfied. The first lemma provides bounds on covariance terms for functionals
of the past and the future of the observations. The considered functionals
depend on the kernel K.
Lemma 1. For any h ∈ Hn , we define
D1 (h) = D1 (n, h) =

C3
nh

and

D2 (h) = D2 (n, h) =

C5
.
n2 h

Then for any u, v and r in N, if (i1 , . . . , iu , j1 , . . . jv ) ∈ Zu+v is such that
i1 ≤ . . . , iu ≤ iu + r ≤ j1 ≤ . . . ≤ jv , we have
Cov

u
Y
k=1

ḡh (Xik ),

v
Y

!

≤ Φ(u, v)D1u+v−2 (h)D2 (h)ρr1/4 ,

ḡh (Xjm )

m=1

where Φ(u, v) = u + v + uv.
16

The following lemma provides a moment inequality for the classical kernel
estimator.
Lemma 2. There exists a positive integer N0 that depends only on a, b, c,
B, C and K such that, for any n ≥ N0 , we have


E

2

n
X



C2
δn
δn
≤
+
,
ḡh (Xi )  ≤ Jn (h) +
6nh
nh 6nh
i=1

where Jn is defined by (6). Moreover for q > 0, we have
n
X

E

q!

≤ Cq (nh)−q/2 (1 + o(1)),

ḡh (Xi )

i=1

where Cq is a positive constant. Here the o(·)–terms depend only on a, b, c,
B, C and K.
The following result is an adaptation of the Bernstein-type inequality
obtained by Doukhan and Neumann (2007).
Lemma 3 (Bernstein’s inequality). We have:
P

n
X

!

ḡh (Xi ) ≥ λ(t) ≤ C7 exp(−t/2)

i=1

where,

√
λ(t) = σn (h) 2t + Bn (h) (2t)2+1/b ,
σn (h) = Jn (h) +

t≥0

δn
6nh

(10)
(11)

and

C6
(12)
nh
with C6 and C7 positive constants that depend only on a, b, c, B, C and K.
Bn (h) =

6.3

Proof of Theorem 1

Let us denote γ = q(1 + δn /(12 max(C2 , 1/6))). For convenience, we split the
proof into several steps.
Step 1. Let us consider the random event
(

A =

\
h∈Hn

δn
n (h) − Jn (h) ≤
2nh

Jb

17

)

and the quantities Γ1 and Γ2 defined by :
q
Γ1 = E fˆ(x0 ) − f (x0 ) IA

and

1/2



2q ˆ
Γ2 = max R2q
(fh , f )P(A c )
h∈Hn

where IA is the characteristic function of the set A . Using Cauchy-Schwarz
inequality, it follows that:
Rqq (fˆ, f ) ≤ Γ1 + Γ2 .
We define
Mn (h, a) =

v
u
u
t2q|log h|

!

aδn
Jn (h) +
.
nh

Now note that if the event A holds, we have:


Mn

3
1
c (h) ≤ M
≤M
.
h,
n
n h,
2
2






Steps 2–5 are devoted to control the term Γ1 whereas Γ2 is upper bounded in
Step 6.
Step 2. Let h ∈ Hn be an arbitrary bandwidth. Using triangular inequality
we have:
|fˆ(x0 )−f (x0 )| ≤ |fˆĥ (x0 )− fˆh∨ĥ (x0 )|+|fˆh∨ĥ (x0 )− fˆh (x0 )|+|fˆh (x0 )−f (x0 )|.
If h ∨ ĥ = ĥ, the first term of the right hand side of this inequality is equal to
0, and if h ∨ ĥ = h, it satisfies
n

o

c (h, ĥ)
|fˆĥ (x0 ) − fˆh∨ĥ (x0 )| ≤ |fˆĥ (x0 ) − fˆh∨ĥ (x0 )| − M
n
n

+

c (h, ĥ)
+M
n
o

c (h, h)
≤ max |fˆh (x0 ) − fˆh∨h (x0 )| − M
n
h∈Hn

+

c (h, ĥ)
+M
n

c (ĥ) + M
c (h).
≤ A(h, x0 ) + M
n
n

Applying the same reasoning to the term |fˆh∨ĥ (x0 ) − fˆh (x0 )| and using (5),
this leads to




c (h) + |fˆ (x ) − f (x )|.
|fˆ(x0 ) − f (x0 )| ≤ 2 A(h, x0 ) + M
n
h 0
0

Using this equation, we obtain that, for some positive constant cq ,
 





Γ1 ≤ cq E Aq (h, x0 )IA + Mqn h,
18

3
+ Rqq (fˆh , f )) .
2




(13)

Step 3. Now, we upper bound A(h, x0 ). Using basic inequalities we have:
A(h, x0 ) ≤ max

n

h∈Hn

+ max
≤ max

h∈Hn

+

+ max

o

n

c (h)
fˆh (x0 ) − Efˆh (x0 ) − M
n

h∈Hn

o

n

c (h ∨ h)
fˆh∨h (x0 ) − Efˆh∨h (x0 ) − M
n

h∈Hn

n

o

Efˆh∨h (x0 ) − Efˆh (x0 )

o

Efˆh∨h (x0 ) − Efˆh (x0 )

+

+

+ 2T,

where
T = max

n

h∈Hn

o

c (h)
fˆh (x0 ) − Efˆh (x0 ) − M
n

+

( n
X

= max

h∈Hn

)
c (h)
ḡh (Xi ) − M
n

.

i=1

+

Using (7), we obtain
A(h, x0 ) ≤ 2 max {|Kh ? f (x0 ) − f (x0 )|}+ + 2T.

(14)

h≤h
h∈Hn

Denoting by Eh (x0 ) the first term of the right hand side of (14), we obtain






E (Aq (h, x0 )IA ) ≤ cq Ehq (x0 ) + E T q IA

(15)

,

for some positive constant cq .
Step 4. It remains to upper bound E(T q IA ). To this aim, notice that,
E(T q IA ) ≤ ET̃ q ,
where
T̃ = max

h∈Hn

( n
X

ḡh (Xi ) − Mn

i=1

(16)


1
h,
2

)

.
+

Now, we define r(·) by
r(u) =

q

2σn2 (h)u + 2d−1 Bn (h) (2u)d ,

u≥0

where Bn (h) and σ√n (h) are given by (11) and (12) and d = 2 + b−1 . Since
h ≥ h∗ = n−1 exp( log n), we have, for n large enough:
d−1

2

Bn (h)(2γ|log h|) ≤

s

δn

d

q

12 2 max(C2 , 1/3)

2q|log h|
.
nh

(17)

Moreover, we have
q

2σn2 (h)γ|log h| ≤

v
u
u
t2q|log h|

≤

v
u
u
t2q|log h|

δn
Jn (h) +
6nh

!

δn
1+
12 max(C2 , 1/6)

!

!

δn
Jn (h) +
.
3nh
19

(18)

+

Last inequality comes from the fact that Jn (h) is upper bounded by C2 /(nh).
Now, using (17) and (18), we obtain:


r(γ| log h|) ≤ Mn h,

1
.
2


(19)

Thus, doing the change of variables t = (r(u))q and thanks to (19), we
obtain:
q

ET̃ ≤

X Z ∞

0

q−1

r (u)r(u)

n
X

P

0



!

1
h,
+ t1/q dt
2


!

ḡh (Xi ) ≥ r(γ| log h|) + r(u) du,

i=1

X Z ∞
h∈Hn

ḡh (Xi ) ≥ Mn

i=1

X Z ∞
h∈Hn

≤C

P

0

h∈Hn

≤C

n
X

−1

q

u λ(u) P

0

n
X

!

ḡh (Xi ) ≥ λ(γ| log h| + u) du,

i=1

where λ(·) is defined by (10). Using Lemma 3, we obtain
q

ET̃ ≤ C

X Z ∞
h∈Hn

≤C

X

u

−1

q

σn2 (h)u

0

+ Bn (h)u

3

q

(

)

u γ| log h|
exp − −
du
2
2

σnq (h)hγ/2 .

h∈Hn

Using the definitions of the quantities that appear in this equation and using
(16), we readily obtain:
q

−q/2

ET IA ≤ Cn

X

(2

(γ−q)/2 −k

)

k∈N

≤C

n−q/2


δn log 2
1 − exp − 24 max(C
2 ,1/6)



q

≤ Cn−q/2 log n
≤ C(nh)−q/2 .

(20)

Step 5. Lemma 2 implies that:


E|fˆh (x0 ) − f (x0 )|q ≤ cq |Eh (x0 )|q + (nh)−q/2



(21)

for some positive constant cq .
Combining (13), (15), (20) and (21), we have:




Γ1 ≤ C ∗ min max |Kh ? f (x0 ) − f (x0 )|q +
h∈Hn  h≤h
h∈Hn

20


!q/2 

|log h|

nh




(22)

where C ∗ is a positive constant that depends only on a, b, c, B, C and K.
Step 6. Using Lemma 3 where in g h , K is replaced by K 2 , we obtain that
P

Z
n
hX
δn x
2
Kh (x0 − Xi ) − h Kh2 (x0 − x)f (x)dx >
n i=1
2

!

≤ exp(−C1 n2 h2 ),

where C1 is a constant that depends only on a, b, c, B, C and δ. Then this
implies that
q


log n
P (A c ) ≤
exp − C1 exp(2 log n)
log 2
and then


1
Γ2 = o q/2 .
(23)
n
Now, using (22) and (23), Theorem 1 follows.

A

Proof of Lemma 1

In order to prove this lemma, we derive two different bounds for the term
Υh (u, v) = Cov

u
Y

ḡh (Xik ),

v
Y

!

ḡh (Xjm )

.

m=1

k=1

The first bound is obtained by a direct calculation whereas the second one
is obtained thanks to the dependence structure of the observations. For the
sake of readability, we denote ` = u + v throughout this proof.
Direct bound. The proof of this bound is composed of two steps. First, we
assume that ` = 2, then the general case ` ≥ 3 is considered.
Assume that ` = 2. If Assumptions 1 and 3 are fulfilled, we have
n2 Υh (u, v) ≤ |E (gh (Xi )gh (Xj ))| + (Egh (X1 ))2
≤ (C + B2 )kgh k21 ≤ C1 .
Then, we have
|Cov (ḡh (Xi ), ḡh (Xj ))| ≤ C1 n−2 .

(24)

Let us now assume that ` ≥ 3. Without loss of generality, we can assume
that u ≥ 2 and v ≥ 1. We have:
Υh (u, v) ≤ A + B
where

A

= E ( uk=1 ḡh (Xik ) vm=1 ḡh (Xjm ))
B = E (Qu ḡh (Xi )) E (Qv
k=1
m=1 ḡh (Xjm )) .
k
Q

Q

21

Remark that both A and B can be bounded, using (24), by
kḡh k(u−2)+v
Cov(ḡh (Xi1 ), ḡh (Xi2 )) ≤
∞



C3
nh

`−2

C1
.
n2

This implies our first bound, for all ` ≥ 2:
Υh (u, v) ≤

2C1
n2



C3
nh

`−2

(25)

.

Structural bound. Using Assumption 2, we obtain that




Υh (u, v) ≤ Ψ u, v, ḡh⊗u , ḡh⊗v ρr .
Now using that
nhḡh
C3
and
nhḡh
Lip
C3

!⊗u

≤1
∞

nhḡh
≤ Lip
C3

!

≤

L
,
C3 h

we obtain, since h ≤ h∗ , that
C3
Υh (u, v) ≤
nh


`

Φ(u, v)

C4
ρr .
C32 h2

This implies that
Υh (u, v) ≤

1
n2



C3
nh

`−2

C4
Φ(u, v)ρr .
h4

(26)

Conclusion. Now combining (25) and (26) we obtain:
1 C3 `−2
C4
Υh (u, v) ≤ 2
(2C1 )3/4
Φ(u, v)ρr
n nh
h4


C5 C3 `−2
≤ 2
Φ(u, v)ρr1/4 .
n h nh






This proves Lemma 1.

22

1/4

B

Proof of Lemma 2

Proof of this result can be readily adapted from the proof of Theorem 1 in
Doukhan and Louhichi (2001) (using similar arguments that ones used in the
proof of Lemma 1). The only thing to do is to bound explicitely the term
A2 (ḡh ) = E

n
X

!2

ḡh (Xi )

.

i=1

Set R = h−1/4 . Remark that
A2 (ḡh ) = nEḡh (X1 )2 +

X

Eḡh (Xi )ḡh (Xj )

i6=j

= Jn (h) + 2

n−1
X n−i
X

Eḡh (Xi )ḡh (Xi+r ).

i=1 r=1

Using Lemma 1 and (24), we obtain:
A2 (ḡh ) ≤ Jn (h) + 2n

n−1
X
C1
(n − r)Φ(1, 1)ρr1/4
+
2D
(h)
2
2
n
r=1
r=R+1
R
X





∞
X
1 
≤ Jn (h) +
ρr1/4 
(2C1 )h3/4 + (6C5 )
nh
r=R+1





∞
X
1  2C1
≤ Jn (h) +
+ (6C5 )
ρr1/4  .
3/4
nh (log n)
r=1+(log n)1/4

Last inequality holds sine h ≤ h∗ ≤ (log n)−1 . Using Assumption 2, there
exists N0 = N0 (K, B, C, a, b, c) such that, for any n ≥ N0 we have:
A2 (ḡh ) ≤ Jn (h) +

δn
.
6nh

This equation, combined with the fact that Jn (h) ≤ C2 (nh)−1 , completes the
proof.

C

Proof of Lemma 3

First, let us remark that Lemma 6.2 in Gannaz and Wintenberger (2010)
and Assumption 2 imply that there exist positive constants L1 and L2 (that
depend on a, b and c) such that, for any k ∈ N we have,
X

(1 + r)k ρ1/4
≤ L1 Lk2 (k!)1/b .
r

r∈N

23

This implies that, using Lemma 2, one can apply the Bernstein-type inequality
obtained by Doukhan and Neumann (2007, see Theorem 1). First, remark
that, using Lemma 2, for n large enough, we have
E

n
X

!2

ḡh (Xi )

≤ σn (h) and Bn (h) =

i=1

2L2 C3
.
nh

where the theoretical expression of Bn (h) given in Doukhan and Neumann
(2007). Let us now denote d = 2 + b−1 . We obtain:
P

n
X



!

u /2

ḡh (Xi ) ≥ u ≤ exp −
σn2 (h)

i=1



2

1
d

+ Bn (h)u

2d−1
d

.

√
Now, let us remark that, on the one hand λ(t) ≥ σn (h) 2t and thus
On the other hand, λ2+

λ2 (t) ≥ 2σn2 (h)t.
1

λ2 (t) ≥ (2Bnd (h)t)λ
and thus, finally:

2d−1
ad

1−2d
d

1

(t) ≥ 2Bnd (h)t and thus
1

(t). This implies that λ2 (t) ≥ t(σn2 (h)+Bnd (h)λ



σn2 (h)

1
d

(t))



λ2 (t)

exp −

2d−1
d

+ Bn (h)λ

2d−1
d



≤ exp(−t/2).

(t)

This implies the results.

D

Acknowledgments

The authors have been supported by Fondecyt project 1141258. Karine
Bertin has been supported by the grant Anillo ACT-1112 CONICYT-PIA,
Mathamsud project 16-MATH-03 SIDRE and ECOS project C15E05.

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