A Note on Nonparametric Estimation of Conditional
Hazard Quantile Function

El Hadj Hamel, Nadia Kadiri, Abbes Rabhi

Laboratory of Mathematics, University Djillali LIABES of Sidi Bel Abbés,
PO. BOX. 89, Sidi Bel Abbés 22000, Algeria

E-mail: rbe0222@yahoo.com, nad.kad06@yahoo.com, rabhi abbes@yahoo.fr

Abstract

In this paper, we study an kernel estimator of the conditional hazard quantile function (CHQF) of a scalar
response variable Y given a random variable (rv) X taking values in a semi-metric space and using the
proposed estimator based of the kernel smoothing method. The almost complete consistency and the
asymptotic normality of this estimate are obtained when the sample is an independante sequence.

Keywords: Asymptotic normality, conditional hazard quantile function, functional data, kernel smoothing,
nonparametric estimator.

1. Introduction

The goal of this paper is to study a nonparametric
estimator of the CHQF when the explanatory vari-
able is functional. This is motivated by the increas-
ing number of situations in which the collected data
are curves (consecutive discrete recordings are ag-
gregated and viewed as sampled values of a ran-
dom curve) where it used to be numbers and vec-
tors. Functional data analysis (see Ferraty and Vieu
(2006)) can help to analyze such data sets in a non-
parametric framework.

Recently, many authors are interested in the esti-
mation of conditional quantiles for a scalar response
and functional covariate. Ferraty et al. (2005) in-
troduced a nonparametric estimator of conditional
quantile defined as the inverse of the conditional cu-
mulative distribution function when the sample is
considered as an α -mixing sequence. They stated

its rate of almost complete consistency and used it
to forecast the well-known El Niño time series and
to build confidence prediction bands. Ezzahrioui
et al. (2008) established the asymptotic normal-
ity of the kernel conditional quantile estimator un-
der α -mixing assumption. Recently, and within the
same framework, Dabo-Niang and Laksaci (2012)
provided the consistency in Lp norm of the condi-
tional quantile estimator for functional dependent
data, Bouchentouf et al. (2015) provided the con-
sistency and asymptotic normality of the smoothing
conditional quantile density function.

In an earlier contribution of the estimator of the
CHQF (see Sankaran and Unnikrishnan (2009)) we
established the consistency and asymptotic normal-
ity of the kernel smoothing estimator for the inde-
pendent sequence and reel case. The present work
gives a generalization to the functional data, we in-
vestigate the asymptotic properties and the asymp-

Journal of Risk Analysis and Crisis Response, Vol. 7, No. 3 (October 2017) 101–107
___________________________________________________________________________________________________________

101

Received 14 March 2017

Accepted 18 April 2017

Copyright © 2017, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).



totic normality of the CHQF of a scalar response
and functional covariate. The interest comes mainly
from the fact that application fields for functional
methods need to analyze continuous-time stochastic
processes.

In what follows, The rest of the paper is orga-
nized as follows. Section 2 we present our esti-
mation procedure and recall the definition of the
The functional kernel estimates property. Section 3
formulates main results of strong consistency (with
rate) and asymptotic normality of the estimator with
gives proofs of the main results. Section 4 is devoted
provides a brief conclusion of the study.

2. The functional kernel estimates

We consider a random pair (X,Y ) where Y is valued
in R and X is valued in some infinite dimensional
semi-metric vector space (F,d(·,·)). Let (Xi,Yi),i =
1,...,n be the statistical sample of pairs which are
identically distributed like (X,Y ),but not necessarily
independent. From now on, X is called functional
random variable f.r.v. Let x be fixed in F and let
FY |X (y,x) be the conditional cumulative distribution
function (cond-cdf) of Y given X = x, is defined by:

∀y ∈ R,FY |X (x,y) = P(Y = y|X = x).

Let QY |X (γ) be the γ -order quantile of the dis-
tribution of Y given X = x. From the cond-cdf
FY |X (·,x), it is easy to give the general definition of
the γ -order quantile:

Q(γ|X = x) ≡ QY |X (γ) = in f {t : FY |X (t,x) = γ},0 6 γ 6 1.

Then, the definition of conditional quantile im-
plies that

FY |X (QY |X (γ)) = γ.

On differentiating partially w.r.t. γ we get

fY |X (QY |X (γ)) =
1

∂
∂ γ (QY |X (γ))

.

Thus, the condition quantile density function can
be written as follows

qY |X (γ) =
1

fY |X (QY |X (γ))
.

Let us now, define the kernel estimator F̂Y |X (·,x)
of FY |X (·,x)

F̂Y |X (x,y) =

n

∑
i=1

K(h−1K d(x,Xi))H(h
−1
H (y −Yi))

n

∑
i=1

K(h−1K d(x,Xi))
.

(1)
where K is a kernel function, H a cumulative dis-
tribution function and hK = hK,n(resp.hH = hH,n) a
sequence of positive real numbers. Roussas (1969)
introduced some related estimate but in the special
case when X is real, while Samanta (1980) produced
previous asymptotic study. As a by-using of Nair
and Sankaran (2009) and Xiang (1995), it is easy to
derive an estimator Q̂Y |X of QY |X :

QY |X (γ) = inf{t : F̂Y |X (t,x) = γ} = F−1Y |X (QY |X (γ)).

Let now defined the conditional density function
is the derivative of conditional distribution function.

f̂Y |X (x,y) =

h−1H
n

∑
i=1

K(h−1K d(x,Xi))H(h
−1
H (y −Yi))

n

∑
i=1

K(h−1K d(x,Xi))
.

(2)
Parzen (1979) and Jones (1992) defined the

quantile density function as the derivative of Q(γ),
that is, q(γ) = Q′(γ). Note that the sum of two
quantile density functions is again a quantile density
function.

Nair and Sankaran (2009) have defined the haz-
ard quantile function as follows:

r(γ) = r(Q(γ)) =
f (Q(γ))

1 − F(Q(γ))
= ((1 − γ)q(γ))−1.

(3)
Thus hazard rate of two populations would be

equal if and only if their corresponding quantile den-
sity functions are equal. This has been used to con-
struct tests for testing equality of failure rates of two
independent samples. Now, from this definition, let

Journal of Risk Analysis and Crisis Response, Vol. 7, No. 3 (October 2017) 101–107
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102



us introduce the γ -order conditional quantile of the
conditional hazard function

r(γ) = rY |X (QY |X (γ)) =
fY |X (QY |X (γ))

1 − FY |X (QY |X (γ))
(4)

=((1 − γ)qY |X (γ))−1. (5)

Consequently, the conditional quantiles of condi-
tional hazard function operator is defined in a natural
way and can be estimated by using kernel smoothing
methods by

rn(γ) = r̂Y /X (Q̂Y /X (γ)) =
f̂Y /X (Q̂Y /X (γ))

1 − F̂Y /X (Q̂Y /X (γ))
. (6)

Now we proposed the other estimator of rn(γ)
using the kernel smoothing method, define by:

rn(γ) =
1

hH

∫ 1
0

1

[1 − F̂Y |X (Q̂Y |X (t))].q̂Y |X (t)
H
(

t − γ
hH

)
dt.

(7)
In the next section derive the asymptotic proper-

ties of our conditional quantile hazard function

3. Assumptions and main results

3.1. General Assumptions

Our results are stated under some assumptions we
gather hereafter for easy reference.

(H1) For all h > 0, P(X ∈ B(x,h)) =: ϕx(h) > 0.
Moreover, ϕx(h) > 0 −→ 0 as h −→ 0.

(H2) For all i ̸= j ,0 < supi ̸= j P[(Xi,X j) ∈ B(x,h)×
B(x,h)] = P(Wi 6 h,Wj 6 h) 6 ψx(h), where
ψx(h) −→ 0 as h −→ 0.

Furthermore, we assume that ψx(h) =
O(ϕ x2 (h)).

(H3) H is such that, for all (y1,y2) ∈ R2, |H(y1)−
H(y2)| = C|y1 − y2|
and its first derivative H(1) verifies∫
|t|b2 H(1)(t)dt < ∞.

(H4) K is a nonnegative bounded kernel of class C1
over its support [0,1] such that K(1) > 0.

The derivative K′ exists on [0,1] and satisfy
the condition K′(t) < 0 , for all t ∈ [0,1] and∫ 1

0
(K) j(t)dt < ∞ for j = 1,2.

(H5) lim
n−→∞

hK = 0 with lim
n−→∞

logn
nϕx(hK)

= 0.

Remark 1. Hypothesis (H1) is the classical con-
centration assumption.(H3) allows to get the conver-
gence rate in the independent case.

Assumption H3 is classical in nonparametric es-
timation and is satisfied by usual kernels such as
Epanechnikov, Biweight, whereas the Gaussian den-
sity K is also possible, it suffices to replace the com-
pact support assumption by:

∫
Rd |t|

b2 H(t)dt < ∞ .

Assumption H3 ensures the existence and
uniqueness of the quantile estimate qγ (x), see Fer-
raty et al. (2005).

A mild regularity hypothesis (H4) is assumed for
the distribution function. Hypothesis (H3) is tech-
nical and is imposed only for the brevity of proofs.
Finally The choice of bandwidth is given by (H5).

3.2. Asymptotic properties

In this section, we prove strong consistency and
asymptotic normality of the estimator 7.

Theorem 1. Let FY |X be continuous.Assume that
K(·) satisfies the conditions (H1)-(H5) in Secestima-
tor rn(γ) is uniformly strong consistent.

Proof.

We can write Equation (7) as

Journal of Risk Analysis and Crisis Response, Vol. 7, No. 3 (October 2017) 101–107
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103



rn(γ)

=
1

hH

∫ 1
0

H
(

t − γ
hH

)
dt

[1 − F̂Y |X (Q̂Y |X (t))].q̂Y |X (t)

−
1

hH

∫ 1
0

H
(

t − γ
hH

)
dt

[1 − FY |X (QY |X (t))].q̂Y |X (t)

+
1

hH

∫ 1
0

H
(

t − γ
hH

)
dt

[1 − FY |X (QY |X (t))].q̂Y |X (t)

=
1

hH

∫ 1
0

H
(

t − γ
hH

)
1

q̂Y |X (t)

[ 1
[1 − F̂Y |X (Q̂Y |X (t))]

−
1

[1 − FY |X (QY |X (t))]

]
dt

+
1

hH

∫ 1
0

H
(

t − γ
hH

)
dt

[1 − FY |X (QY |X (t))].q̂Y |X (t)

=
1

hH

∫ 1
0

H
(

t − γ
hH

)
F̂Y |X (Q̂Y |X (t)− FY |X (QY |X (t))dt

q̂Y |X (t)[1 − FY |X (QY |X (t))][1 − F̂Y |X (Q̂Y |X (t))]

+
1

hH

∫ 1
0

H
(

t − γ
hH

)
dt

[1 − FY |X (QY |X (t))]q̂Y |X (t)

(8)

Since

sup
t
|F̂Y |X (t)− FY |X (t)| −→ 0

almost surely, equation (8) is asymptotically equal
to

rn(γ) =
1

hH

∫ 1
0

H
(

t − γ
hH

)
dt

(1 − FY |X (QY |X (t))q̂Y |X (t)
.

Thus,

rn(γ)− r(γ)

=
1

hH

∫ 1
0

H
(

t − γ
hH

)
1

(1 − FY |X (QY |X (t)))[
1

q̂Y |X (t)
−

1
qY |X (t)

]
dt+

1
hH

∫ 1
0

H
(

t − γ
hH

)
1

qY |X (t)

dt
(1 − FY |X (QY |X (t)))

−
1

(1 − γ)qY |X (γ)

=
1

hH

∫ 1
0

1
(1 −t)

H
(

t − γ
hH

)
[
qY |X (t)− q̂Y |X (t)

]
dt

q̂Y |X (t)qY |X (t)
dt +

1
hH

∫ 1
0

H
(

t − γ
hH

)
dt

(1 − FY |X (QY |X (t)))qY |X (t)
−

1
(1 − γ)qY |X (γ)

(9)

Denoting

K∗(t,γ) = (H((t − γ)/hH))/(1 − t)q̂Y |X (t)qY |X (t),

on using integration by parts, equation (9) reduces
to

rn(γ)− r(γ)

=
1

hH

∫ 1
0
(QY |X (t)− Q̂Y |X (t))dK∗(t,γ)+

1
hH

∫ 1
0

H
(

t − γ
hH

)
dt

(1 −t)qY |X (t)

−
1

(1 − γ)qY |X (γ)
.

Since supt |Q̂Y |X (t) − QY |X (t)| −→ 0 almost
surely, equation (10) is asymptotically equal to

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104



rn(γ)− r(γ)

=
1

hH

∫ 1
0

H
(

t − γ
hH

)
dt

(1 − t)qY |X (t)

−
1

(1 − γ)qY |X (γ)
.

(10)

Setting (t − γ)/hH = v, in equation (10),

rn(γ)− r(γ)

=
1

hH

∫ (1−γ)/hH
−γ/hH

H(v)
fY |X QY |X (γ + vhH)

1 −(γ + vhH)
dv

−
1

(1 − γ)qY |X (γ)

=
1

hH

∫ (1−γ)/hH
−γ/hH

H(v)
1

1 − γ

[
1 −

vhH
1 − γ

]−1
fY |X QY |X (γ + vhH)dv −

1
(1 − γ)qY |X (γ)

.

(11)

By Taylor’s series expansion of QY |X (γ + vhH)
around γ , equation (11) becomes

rn(γ)− r(γ)

=
1

hH(1 − γ)

∫ (1−γ)/hH
−u/hH

(
H(v)

[
1 +

vhH
1 − γ

+······
]

× fY |X
[
QY |X (γ)+ vhH dQY |X (γ)+ ······

])
dv

−
1

(1 − γ)qY |X (γ)
.

(12)

As n −→ ∞,we have hn −→ 0 and
∫ +∞
−∞ H(v)dv =

1, so that equation (12) reduces to

|rn(γ)− r(γ)| =
∣∣∣∣ fY |X (QY |X )(γ)1 − γ − 1(1 − γ)qY |X (γ)

∣∣∣∣,
which tends to zero as n −→ ∞. This completes the
proof.

3.3. Asymptotic normality

In this section we give the asymptotic normality of
rn(γ).

Theorem 2. Under assumptions (H1)(H5) and
suppose that FY |X is continuous, for 0 < γ < 1,√

n(rn(γ) − r(γ)) is asymptotically normal with
mean zero and variance σ 2(γ) as given in Equation
(13).

σ 2(γ)

= n
1

h2H
E
[∫ 1

0
QY |X (t)dM

′(t,γ)

+
∫ 1

0
F̂Y |X (Q̂Y |X )(t)

M(t,γ)
(1 − t)

dQY |X (t)
]2
.

(13)

M(t,γ) = H((t − γ)/hH)/qY |X (t)

and M′(t,γ) is the derivative of M(t,γ) with respect
to t.

Proof.

√
n(rn(γ)− r(γ))

=
√

n
1

hH

∫ 1
0

H
(

t − γ
hH

)
[

1

1 − F̂Y |X (Q̂Y |X )(t)
.

1
q̂Y |X (t)

−
1

(1 − F̂Y |X (Q̂Y |X )(t))qY |X (t)

]
dt

+
√

n
1

hH

∫ 1
0

H
(

t − γ
hH

)
dt

(1 − F̂Y |X (Q̂Y |X )(t))qY |X (t)
−

√
n

(1 − γ)qY |X (t)
,

Journal of Risk Analysis and Crisis Response, Vol. 7, No. 3 (October 2017) 101–107
___________________________________________________________________________________________________________

105



which can be written as

√
n(rn(γ)− r(γ))

=

√
n

hH

∫ 1
0

H
(

t − γ
hH

)
1

(1 − F̂Y |X (Q̂Y |X )(t))[
qY |X (t)− q̂Y |X (t)

qY |X (t)q̂Y |X (t)

]
dt

+

√
n

hH

∫ 1
0

H
(

t − γ
hH

)
1

qY |X (t)


(
F̂Y |X (Q̂Y |X )(t)− FY |X (QY |X )(t)

)
dt

(1 − F̂Y |X (Q̂Y |X )(t))(1 − FY |X (QY |X )(t))




+

√
n

hH

∫ 1
0

H
(

t − γ
hH

)
dt

(1 − FY |X (QY |X )(t))qY |X (t)
−

√
n

(1 − γ)qY |X (γ)
.

(14)

Since supt |F̂Y |X (Q̂Y |X )(t)− FY |X (QY |X )(t)| −→ 0
and FY |X (QY |X )(t) = t, equation (14) is asymptoti-
cally equal to

=
√

n
1

hH

∫ 1
0

H
(

t − γ
hH

)
(qY −X (t)− q̂Y −X (t))dt

(1 − t)(qY |X (t))2

+

√
n

hH

∫ 1
0

H
(

t − γ
hH

)
1

qY |X (t)
1

(1 − t)2[
F̂Y |X (Q̂Y |X )(t)− FY |X (QY |X )(t)

]
dt

+
√

n
1

hH

∫ 1
0

H
(

t − γ
hH

)
dt

(1 − FY |X (QY |X )(t))qY |X (t)

−
√

n
(1 − γ)qY |X (γ)

. (15)

Setting M(t,γ) = H((t − γ)/hH)/qY |X (t) in
equation (15) and applying integration by parts, we

obtain
√

n(rn(γ)− r(γ))

=

√
n

hH

∫ 1
0

M(t,γ)qY |X (t)
(1 − t)[

F̂Y |X (Q̂Y |X )(t)− FY |X (QY |X )(t)
]

dt

+

√
n

hH

∫ 1
0

H
(

t − γ
hH

)
dt

(1 − FY |X (QY |X )(t))qY |X (t)

−
√

n
(1 − γ)qY |X (γ)

+

√
n

hH

∫ 1
0
(Q̂Y |X (t)− QY |X (t))dM′(t,γ),

(16)

where M′(t,γ) is the derivative of M(t,γ) with re-
spect to t. From equation (12), we can obtain equa-
tion (16) as

√
n(rn(γ)− r(γ)) =

√
n

hH

∫ 1
0

M(t,γ)qY |X (t)
(1 − t)[

F̂Y |X (Q̂Y |X )(t)− FY |X (QY |X )(t)
]

+

√
n

hH

∫ 1
0
(Q̂Y |X (t)− QY |X (t))dM′(t,γ). (17)

Note that from Ezzahrioui and Ould-Saı̈d (2008),
Laksaci et al. (2011) and Chaouch and Khardani
(2015) for 0 6 γ 6 1,

√
n(Q̂Y |X (γ) − QY |X (γ)) is

asymptotically normal with mean zero and variance

σ 2(γ) =
1√

ϕx(h)
β2
β 21

γ(1 − γ)
( fY |X (QY |X (γ)))2

,

β j = K j(1)−
∫ 1

0
(K j)′(s)τ0(s)ds,

and τ0 is a nondecreasing bounded function such
that, uniformly in s ∈ [0,1],

ϕ(hs)
ϕ(h)

= τ0(s)+ o(1) as h ↓ 0

and for j > 1,∫ 1
0
((K) j(t))′τ0(t)dt < ∞.

Thus
√

n(F̂Y |X (Q̂Y |X )(γ)−FY |X (QY |X )(γ)) is also
asymptotically normal with mean zero and variance
σ 21 (γ), since d/dγ FY |X (QY |X )(γ) = 1.

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106



Now from equation (17), we can show that√
n(rn(γ) − r(γ)) is asymptotically normal with

mean zero. The expression of variance can be ob-
tained from equation (17), which is given by

σ 21 (γ) =
n

h2H
E
[∫ 1

0
Q̂Y |X (t)dM

′(t,γ)

+
∫ 1

0
F̂Y |X (Q̂Y |X )(t)

M(t,γ)
(1 − t)

dQY |X (t)
]
.(18)

This completes the proof.

Remark 2.
• The function τ0(·) defined in by there ex-

ists a function τ0(·) s.t. for all s ∈ [0,1],
lim

r−→0
ϕx(sr)/ϕx(r) = ϕx(s), permits to get the vari-

ance term explicitly.
This condition is classical and related to a non
vanishing conditional density. The second one
means that a small amount a concentration is
needed in order to ensure asymptotic normality.

• The present study provided a nonparametric es-
timator, for the conditional hazard quantile func-
tion, its based and using the kernel smoothing
method, and the asymptotic properties of the ker-
nel estimator were studied. The kernel based es-
timator seems to perform satisfactorily except for
large values of γ .

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