Parameter estimators of nonlinear quantile regression
We have obtained the asymptotic normality of parameter estimators of a nonlinear quantile regression with nonsymmetric random noise.
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irk-123456789-44282009-11-10T12:00:38Z Parameter estimators of nonlinear quantile regression Ivanov, A.V. Orlovsky, I.V. We have obtained the asymptotic normality of parameter estimators of a nonlinear quantile regression with nonsymmetric random noise. 2005 Article Parameter estimators of nonlinear quantile regression / A.V. Ivanov, I.V. Orlovsky // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 82–91. — Бібліогр.: 6 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4428 519.21 en Інститут математики НАН України |
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We have obtained the asymptotic normality of parameter estimators of a nonlinear
quantile regression with nonsymmetric random noise. |
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Article |
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Ivanov, A.V. Orlovsky, I.V. |
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Ivanov, A.V. Orlovsky, I.V. Parameter estimators of nonlinear quantile regression |
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Ivanov, A.V. Orlovsky, I.V. |
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Ivanov, A.V. |
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Parameter estimators of nonlinear quantile regression |
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Parameter estimators of nonlinear quantile regression |
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Parameter estimators of nonlinear quantile regression |
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Parameter estimators of nonlinear quantile regression |
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Parameter estimators of nonlinear quantile regression |
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parameter estimators of nonlinear quantile regression |
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Інститут математики НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/4428 |
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Parameter estimators of nonlinear quantile regression / A.V. Ivanov, I.V. Orlovsky // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 82–91. — Бібліогр.: 6 назв.— англ. |
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AT ivanovav parameterestimatorsofnonlinearquantileregression AT orlovskyiv parameterestimatorsofnonlinearquantileregression |
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2025-07-02T07:40:35Z |
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2025-07-02T07:40:35Z |
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1836520083800719360 |
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Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 82–91
UDC 519.21
A. V. IVANOV AND I. V. ORLOVSKY
PARAMETER ESTIMATORS OF
NONLINEAR QUANTILE REGRESSION
We have obtained the asymptotic normality of parameter estimators of a nonlinear
quantile regression with nonsymmetric random noise.
Introduction
Here, we examine the asymptotic normality of Koenker and Basset estimators [1] or the
generalized least moduli estimators (GLME) of nonlinear regression model parameters
that generalize least moduli estimators for non-symmetric observation errors.
The consistency property of GLME has been considered in [2].
1. Assumptions and the main result
Suppose that an observation Xj is a r.v. with values in (R1,B1) (R1 is a real line, B1
- σ-algebra of its Borel subsets) and distribution Pj . We also assume that the unknown
distribution Pj belongs to a certain parametric family {Piθ, θ ∈ Θ}. We call the triple
Ej = {R1,B1, Pjθ , θ ∈ Θ} a statistical experiment generated by the observation Xj .
We say that a statistical experiment En = {Rn,Bn, Pn
θ , θ ∈ Θ} is the product of
the statistical experiments Ei, i = 1, ..., n, if Pn
θ = P1θ × ... × Pnθ (Rn - n-dimensional
Euclidean space and Bn - σ-algebra of its Borel subsets). We say that the experiment
En is generated by n independent observations X = (X1, .., Xn).
Let the observations have the form
(1.1) Xj = g(j, θ) + εj , j = 1, ..., n ,
where g(j, θ) is a non-random sequence of functions defined on Θc, Θc is the closure of
an open convex set Θ ⊂ R
q in R
q, and
A1. εj are independent identically distributed random variables (r.v.) with zero
mean, distribution function P , and
(1.2) P(0) = β, β ∈ (0, 1).
It is not supposed that the functions g(j, θ) are the linear forms of coordinates of the
vector θ.
Definition. GLME of the parameter θ ∈ Θ obtained by the observations Xj, j = 1, ..., n
of the form (1.1) is said to be any random vector θ̂n = θ̂n(Xj , j = 1, ..., n) ∈ Θc having
the property
(1.3) Sβ(θ̂n) = inf
τ∈Θc
Sβ(τ), Sβ(τ) =
∑
ρβ(Xj − g(j, τ)),
2000 AMS Mathematics Subject Classification. Primary 62J02; Secondary 62J99.
Key words and phrases. Nonlinear quantile regression, parameter estimator.
82
PARAMETER ESTIMATORS OF NONLINEAR QUANTILE REGRESSION 83
where
∑
=
∑n
j=1 and
(1.4) ρβ(x) =
{
βx, x ≥ 0
(β − 1)x, x < 0
, β ∈ (0, 1).
Since Pn
θ {Xj < g(j, θ)} = Pn
θ {εj < 0} = F (0) = β, the observation model (1.1) can
be interpreted as a nonlinear quantile regression [1]. Indeed, θ̂n estimates the β-quantile
g(j, θ) of observations Xj , j = 1, ..., n.
Let us impose some restrictions on r.v. εj :
A2. μs = E|εj |s <∞ for some natural s.
A3. R.v. εj has a bounded density p(x) = P ′(x) with the property
|p(x)− p(0)| ≤ H |x|, p(0) > 0,
where H < ∞ is a certain constant.
Example. A r.v. ξ = χ2
2m − 2m, where χ2
2m has chi-squared distribution with even
degrees of freedom, satisfies conditions A1-A3.
Denote, by Cq ⊂ Bq, the class of all convex Borel subsets of R
q and, by T ⊂ Θ, some
compact.
Let us introduce the notation
gi(j, τ) =
∂
∂τ i
g(j, τ), gil(j, τ) =
∂2
∂τ i∂τ l
g(j, τ),
d2
in(θ) =
∑
g2
i (j, θ), d2
il,n(τ) =
∑
g2
il(j, τ), τ ∈ Θc, i, l = 1, ..., q.
Here, d2
n(θ) is a diagonal matrix with elements d2
in(θ), i = 1, ..., q on the diagonal.
Consider the change of variables u = n−1/2dn(θ)(τ − θ), i.e.
g(j, τ) = g(j, θ + n1/2d−1
n (θ)u) = f(j, u),
assuming that θ is a true value of the parameter. Under this change of variables, the
set Θ turns to the set Ũn(θ) = n−1/2Un(θ), where Un(θ) = dn(θ)(Θ− θ), and GLME θ̂n
turns to a normed random vector ûn = n−1/2dn(θ)(θ̂n − θ).
We will denote positive constants by the letter k. Suppose that
B1. Functions g(j, θ), j ≥ 1 are continuous on Θc together with all the first par-
tial derivatives, and gi(j, θ), i = 1, ..., q, j ≥ 1, are continuously differentiable in Θ.
Moreover, for any R ≥ 0,
(i) sup
θ∈T
sup
u∈v(R)∩Uc
n(θ)
max
1≤j≤n
|fi(j, u)|
din(θ)
≤ ki(R)n−1/2, i = 1, ..., q,(1.5)
(ii) sup
θ∈T
sup
u∈v(R)∩Uc
n(θ)
dil,n(θ + n1/2d
−1/2
n (θ)u)
din(θ)dln(θ)
≤ kil(R)n−1/2, i, l = 1, ..., q.(1.6)
It follows from (1.5) that
(1.7) sup
θ∈T
sup
u1, u2∈vc(R)∩Uc
n(θ)
n−1 Φn(u1, u2)
|u1 − u2|2 ≤ k(R),
where Φn(u1, u2) =
∑
(f(j, u1)− f(j, u2))
2.
Similarly, relation (1.6) yields the inequality
(1.8) sup
θ∈T
sup
u1, u2∈vc(R)∩Uc
n(θ)
Φ(i)
n (u1, u2)
d2
in(θ)|u1 − u2|2 ≤ k̃(i)(R),
84 A. V. IVANOV AND I. V. ORLOVSKY
with Φ(i)
n (u1, u2) =
∑
((fi(j, u1)− fi(j, u2))2, i = 1, ..., q.
Suppose that GLME is consistent, namely:
C. For any r > 0
sup
θ∈T
Pn
θ {|n−1/2dn(θ)(θ̂n − θ)| ≥ r} =
{
O(n−s+1), s ≥ 2,
o(1), s = 1.
.
The sufficient conditions for C to be fulfilled are stated in [2].
Let us denote
I(θ) =
(
d−1
in (θ)d−1
ln (θ)
∑
gi(j, θ)gl(j, θ)
)q
i,l=1
, θ ∈ Θ.
The matrix I(θ) is symmetric and non-negative definite. Let λmin(I(θ)) be the smallest
eigenvalue of I(θ). Assume that
B2. For n > n0, infθ∈T λmin(I(θ)) ≥ λ0 > 0.
Let l be an arbitrary direction in R
q, and τ ∈ Θ. Then
∂
∂l
Sβ(τ) =
∑
〈∇g(j, τ), l〉 (χ{Xj ∗ g(j, τ) − β}),
where ”∗” denotes ”≤” if 〈∇g(j, τ), l〉 ≥ 0 and ”<” if 〈∇g(j, τ), l〉 < 0. Let r0 be a
distance between T and R
q\Θ. If an event {|θ̂n − θ| < r} occurs for θ ∈ T and r < r0,
then, for any direction l,
∂
∂l
Sβ(θ̂n) ≥ 0.
This remark will be used in the proof of the main result.
Theorem. If conditions A1 - A3, B1, B2, and C are fulfilled, then
(1.9) sup
θ∈T
sup
C∈Cq
∣∣∣∣∣Pn
θ
{
p(0)√
β(1− β)
I1/2(θ)dn(θ)(θ̂n − θ) ∈ C
}
− Φ(C)
∣∣∣∣∣ −→n→∞ 0,
where Φ(C) =
∫
C
1
(2π)q/2
e−
‖x‖2
2 dx.
In other words, the normal distribution N
(
0, β(1−β)
p2(0) I−1(θ)
)
is the accompanying law
for the distribution of the normed estimator dn(θ)(θ̂n − θ).
2. Auxiliary assertions
We carry out the proof by the scheme of the theorem on asymptotic normality of the
least moduli estimators [3], by using the method of partitioning a parametric set [4,5].
Let l1, ..., lq be the positive directions of the coordinate axes. Let us consider the
vectors S±
β (τ) with coordinates
S±
iβ(τ) = d−1
in (θ)
(
∂
∂(±li)
)
Sβ(τ), i = 1, ..., q,
and the vectors En
θ S±
β (θ) with coordinates
En
θ S±
iβ(τ) = ±d−1
in (θ)
∑
gi(j, τ)[P(g(j, τ) − g(j, θ))− β], i = 1, ..., q.
Clearly,
En
θ S±
β (θ) = 0,
PARAMETER ESTIMATORS OF NONLINEAR QUANTILE REGRESSION 85
due to assumption A1. Let us denote S∗±
β (u) = S±
β (θ + n1/2d−1
n (θ)u) and
z±n (θ, u) =
∣∣∣S∗±
β (u)− S∗±
β (0)− En
θ S∗±
β (u)
∣∣∣
1 +
∣∣∣En
θ S∗±
β (u)
∣∣∣ .
Lemma 1. Under the conditions of the theorem, for any ε > 0 and sufficiently small
r > 0,
(2.1) sup
θ∈T
Pn
θ
{
sup
u∈vc(r)∩Ũc
n(θ)
z±n (θ, u) > ε
}
−→
n→∞ 0.
Proof. We will proof the statement for z+
n (θ, u). Assume, for simplicity, that r = 1 and
the inner supremum in (2.1) is defined in a cube
C0 =
{
u : |u|0 = max
1≤i≤q
|ui| ≤ 1
}
⊃ v(1).
Let us cover the cube C0 with N0 = O(ln n) cubes C(1), ..., C(N0) in the following way.
For the number t ∈ (0, 1), we consider a concentric system of sets
C(m) ={u : |u|0 ∈ [(1− t)m+1, (1 − t)m]}, m = 0, . . . , m0 − 1,
C(m0) ={u : |u|0 ≤ (1− t)m0}.
We cover each of the sets C(m) by identical cubes with sides
am = (1− t)m − (1− t)m+1 = t(1− t)m
and enumerate these cubes. They form the required covering
C(1), . . . , C(N0−1), C(N0) =def C(m0).
Let us choose m0 = m0(n) from the condition (1− t)m̃0 = n−γ , m0 = [m̃0], γ ∈ (1
2 , 1).
We denote, by | · |0, the distance from C(j) to 0 which is equal to
r(j) = (1 − t)n−γm/m̃0 ,
and, by | · |0, the diameter of C(j) which is equal to
a(j) = tn−γm/m̃0
for some m = m(j), j = 1, ..., N0 − 1. Moreover, if the cube C(j) is an element of the
covering of the sets C(m), then
a(j) = am, r(j) = t(1 − t)m+1 + ... + t(1− t)m0−1 + (1− t)m0 .
The number of cubes C(j) covering each set C(m) can be made not depending on m and,
consequently, on n. In order to verify this, let us consider any octant in R
q. The volume
occurring in its part of the set C(m) is (1 − t)mq − (1− t)(m+1)q, and the volume of the
sets C(j) is equal to aq(j) = tq(1− t)mq. In this way, the maximum number of cubes C(j)
that can be ”placed” in the part of C(m) that belongs to the given octant is equal to
(1 − t)mq − (1− t)(m+1)q
tq(1− t)mq
=
1− (1− t)q
tq
cubes. Since m0 = O(ln n), N0 = O(ln n) as well. Let us fix θ ∈ T . Then
(2.2) Pn
θ
{
sup
u∈C0
z+
n (θ, u) > ε
}
≤
N0∑
j=1
Pn
θ
{
sup
u∈C(j)
z+
n (θ, u) > ε
}
.
86 A. V. IVANOV AND I. V. ORLOVSKY
Let us estimate each term in (2.2). The general element of the derivative matrix Dn(u)
of the mapping
u −→ En
θ S∗+
β (u)
has the form
Dil
n (u) =
∂
∂ul
En
θ S∗+
iβ (u)
= n1/2d−1
in (θ)d−1
ln (θ)
∑
fil(j, u)[P(g(j, τ) − g(j, θ))− β]
+ n1/2d−1
in (θ)d−1
ln (θ)
∑
fi(j, u)fl(j, u) p(g(j, τ) − g(j, θ))
= 1D
il
n (u) +2 Dil
n (u).
Taking into account (1.6), (1.7), and the inequality
sup
x∈R1
p(x) = p0 <∞,
we obtain, for |u| < r,
n−1/2|1Dil
n (u)| ≤ n1/2d−1
in (θ)d−1
ln (θ)dil,n(θ + n1/2d−1
n (θ)u)×
×
(
n−1
∑
(P(f(j, u)− f(j, 0))− P(0))2
)1/2
≤ k(il)(r)k1/2(r)p0|u|.(2.3)
On the other hand,∣∣∣n−1/2
2D
il
n (u)− p(0)Iil(θ)
∣∣∣ ≤
≤ p0
[
d−1
in (θ) din(θ + n1/2d−1
n (θ)u) d−1
ln (θ)
(
Φ(l)
n (u, 0)
)1/2
+ d−1
in (θ)
(
Φ(i)
n (u, 0)
)1/2
]
(2.4) +d−1
in (θ)d−1
ln (θ)
∣∣∣∑ gi(j, θ)gl(j, θ)(p(f(j, u) − f(j, 0))− p(0))
∣∣∣ .
It follows from (1.5) and (1.8) that the terms in square brackets are bounded by the
quantity
p0
(
(k̃(i))1/2 + k(i)(r)(k̃(l))1/2
)
|u|.
For another term on the right-hand side of (2.4), we can find, by using condition A3 and
(1.5), the upper bound
n1/2d−1
in (θ) max
1≤j≤n
|gi(j, θ)|
(
n−1
∑
(p(f(j, u)− f(j, 0))− p(0))2
)1/2
(2.5) ≤ k(i)(r)Hk1/2(r)|u|.
Since the matrix n−1/2Dn(0) = p(0)I(θ) is positive definite by condition B2, it follows
from the above-presented considerations that, for sufficiently small u (for simplicity we
assume that u ∈ C0) and some k0 > 0,
(2.6) inf
θ∈T
∣∣∣En
θ S+
β (θ + n1/2d−1
n (θ)u)
∣∣∣ ≥ k0n
1/2|u|0.
Let l
= N0, and let v ∈ C(l) be an arbitrary point. Then, in view of (2.6), we can write
sup
u∈C(l)
z+
n (θ, u) ≤
(
sup
u∈C(l)
M (l)
n (θ, u, v) + L(l)
n (θ, v)
)
(1 + k0n
1/2r(l))−1,
M (l)
n (θ, u, v) =
4∑
λ=1
M
(l)
λn(θ, u, v) ( mod Pn
θ )
PARAMETER ESTIMATORS OF NONLINEAR QUANTILE REGRESSION 87
M
(l)
1n (θ, u, v) =
∣∣∣d−1
n (θ)
∑
∇f(j, u) (χ{Xj ∗ f(j, u)} − χ{Xj < f(j, v)})
∣∣∣
M
(l)
2n (θ, u, v) =
∣∣∣d−1
n (θ)
∑
(∇f(j, u)−∇f(j, v))(χ{Xj < f(j, v)} − β)
∣∣∣
M
(l)
3n (θ, u, v) =
∣∣∣d−1
n (θ)
∑
∇f(j, u) (P(f(j, u)− f(j, 0))− P(f(j, v)− f(j, 0)))
∣∣∣
M
(l)
4n (θ, u, v) =
∣∣∣d−1
n (θ)
∑
(∇f(j, u)−∇f(j, v))(P(f(j, v) − f(j, 0))− β)
∣∣∣
L(l)
n (θ, v) =
∣∣∣d−1
n (θ)
∑
(∇f(j, v)(χ{Xj < f(j, v)} − β) −∇f(j, 0)(χ{εj ∗ 0} − β)
−∇f(j, v)(P(f(j, v) − f(j, 0))− β)| ( mod P n
θ ).
By (1.8) and for u, v ∈ C(l), we obtain
(2.7) n−1/2M
(l)
2n (θ, u, v) ≤ β′
(
q∑
i=1
d−2
in (θ)Φ(i)
n (u, v)
)1/2
≤ k1a(l).
Furthermore, in accordance with (1.5), (1.7), and A3, we get
(2.8) n−1/2M
(l)
3n (θ, u, v) ≤ p0n
−1/2Φ1/2
2n (u, v)
(
q∑
i=1
d2
in(θ + n1/2d−1
n (θ)u)
d2
in(θ)
)1/2
≤ k2a(l).
Analogously,
(2.9) n−1/2M
(l)
4n (θ, u, v) ≤ p0n
−1/2Φ1/2
2n (v, 0)
(
q∑
i=1
d−2
in (θ)Φ(i)
n (u, v)
)1/2
≤ k3a(l).
Let us estimate M
(l)
1n (θ, u, v). For any u, v ∈ C(l),
|χ{Xj ∗ f(j, u)} − χ{Xj < f(j, v)}|
≤ χ
{
inf
u∈C(l)
f(j, u)− f(j, 0) ≤ εj ≤ sup
u∈C(l)
f(j, u)− f(j, 0)
}
= χj ( mod Pn
θ ).
Consequently, by (1.5),
n−1/2M
(l)
1n (θ, u, v) ≤ n−1/2
(
q∑
i=1
(
d−1
in (θ) max
1≤j≤n
|fi(j, u)|
)2
)1/2∑
χj
≤ k4n
−1
∑
χj .(2.10)
Using the formula for finite increments, we find
n−1
∑
En
θ χj = n−1
∑(
P
(
sup
u∈C(l)
f(j, u)− f(j, 0)
)
− P
(
inf
u∈C(l)
f(j, u)− f(j, 0)
))
≤ p0n
−1
∑
sup
u1,u2∈C(l)
|f(j, u1)− f(j, u2)|
(2.11) ≤ p0q
1/2
⎛⎝ q∑
i=1
(
n1/2d−1
in (θ) sup
u∈C(l)
max
1≤j≤n
|fi(j, u)|
)2
⎞⎠1/2
a(l) ≤ k5a(l).
Estimates (2.7)-(2.11) show that there exist constants k6 and k7 such that
Pn
θ
{
sup
u∈C(l)
M (k)
n (θ, u, v)(1 + k0n
1/2r(l))−1 >
ε
2
}
88 A. V. IVANOV AND I. V. ORLOVSKY
(2.12) ≤ Pn
θ
{
k6n
−1
∑
(χj − En
θ χj) >
ε
2
r(l)− k7a(l)
}
.
Note that
ε
2
r(l)−k7a(l) =
( ε
2
(1− t)− k7t
)
n−γm/m̃0 > 0, if t is chosen sufficiently small.
Therefore, probability (2.12) can be estimated, with the help of the Chebyshev inequality
and (2.11), by the quantity
(2.13)
4k2
6
(ε(1− t)− 2k7t)2
n−2+2γm/m̃0
∑
En
θ χj ≤ k8n
−1+γm/m̃0.
Using the notation
L1i(j) = (fi(j, v) − fi(j, 0))(χ{Xj < f(j, v)} − β),
L2i(j) = fi(j, 0)(χ{Xj < f(j, v)} − χ{εj ∗ 0}), i = 1, ..., q,
we obtain
P1 = Pn
θ
{
L(k)
n (θ, v)(1 + k0n
1/2r(l))−1 >
ε
2
}
(2.14) ≤ 4
n(k0ε)2r2(l)
q∑
i=1
d−2
in (θ)
2∑
λ=1
En
θ
(∑
(Lλi(j)− En
θ Lλi(j))
)2
,
(2.15) Dn
θ (
∑
L1i(j)) ≤ Φ(i)
2n(v, 0),
Dn
θ (
∑
L2i(j)) ≤
∑
f2
i (j, 0)|P(f(j, v)− f(j, 0))− P(0)|
(2.16) ≤ p0 max
1≤j≤n
|gi(j, θ)|din(θ)Φ1/2
2n (v, 0).
It follows from relations (2.14)-(2.16) and the conditions of the theorem that
P1 ≤ 4n−1
(k0ε)2
[
(r(l) + a(l))2
r2(l)
q∑
i=1
k̃(i)(1) +
r(l) + a(l)
r2(l)
p0k
1/2(1)
q∑
i=1
k(i)(1)
]
(2.17) ≤ k9n
−1
[
(1 − t)−2 + (1 − t)−2nγm/m̃0
]
= O
(
n−1+γm/m̃0
)
.
Inequalities (2.13) and (2.17) show that, for l = 1, ..., N0 − 1 and some m = m(l) < m0,
(2.18) sup
θ∈T
Pn
θ
{
sup
u∈C(l)
z+
n (θ, u) > ε
}
= O
(
n−1+γm/m̃0
)
.
Let us consider the case l = N0. Clearly,
Pn
θ
{
sup
u∈C(N0)
z+
n (θ, u) > ε
}
≤
(2.19) ≤ Pn
θ
{
sup
|u|0<n−γm/m̃0
∣∣∣S∗+
β (u)− S∗+
β (0)− En
θ S∗+
β (u)
∣∣∣ > ε
}
.
Let us rewrite the expression standing under the sign of supremum in (2.19) in the form
of ν1(θ, u) + ν2(θ, u) + ν3(θ, u), where
ν1(θ, u) = d−1
n (θ)
∑
(∇f(j, u)−∇f(j, 0))(χ{Xj ∗ f(j, u)} − β),
ν2(θ, u) = d−1
n (θ)
∑
∇f(j, 0)(χ{Xj ∗ f(j, u)} − χ{εj ∗ 0}),
ν3(θ, u) = d−1
n (θ)
∑
∇f(j, u)(P(f(j, u)− f(j, 0))− β).
PARAMETER ESTIMATORS OF NONLINEAR QUANTILE REGRESSION 89
It is easy to show that, for |u|0 < n−γm/m̃0,
(2.20) |ν1(θ, u)| ≤ β′n
1
2
(
q∑
i=1
d−2
in (θ)Φ(i)
2n(u, 0)
)1/2
≤ k1n
1
2−γm
m̃0 ,
(2.21) |ν3(θ, u)| ≤ p0Φ
1
2
2n(u, 0)
(
q∑
i=1
d2
in(θ + n1/2d−1
n (θ)u)
d2
in(θ)
)1/2
≤ k2n
1
2−γm
m̃0 ,
where k1 and k2 are the same as in (2.7) and (2.8), correspondingly.
If γ > 1
2 , then the exponents in (2.20) and (2.21) are negative for n > n0. That is, for
ε′ < ε, it remains to estimate the probability
Pn
θ
{
sup
|u|0<n−γm/m̃0
|ν2(θ, u)| > ε′
}
≤ Pn
θ
⎧⎨⎩
(
q∑
i=1
(
d−1
in (θ) max
1≤j≤n
|gi(j, θ)|
)2
)1/2∑
χ̃j > ε′
⎫⎬⎭ ,
(2.22) ≤ Pn
θ
{
k4n
−1/2
∑
χ̃j > ε′
}
,
χ̃j = χ
{
inf
|u|0≤n−γm/m̃0
f(j, u)− f(j, 0) ≤ εj ≤ sup
|u|0≤n−γm/m̃0
f(j, u)− f(j, 0)
}
.
From the conditions of the theorem,∑
En
θ χ̃j ≤ k5n
−γm/m̃0, j = 1, . . . , n.
Hence, instead of (2.22), it is sufficient to estimate, for any ε′′ > 0, the probability
Pn
θ
{
n−1/2
∑
(χ̃j − En
θ χ̃j) > ε′′
}
≤ (ε′′)2 k5n
−γm/m̃0.
Taking into account the fact that all the bounds are uniform in θ ∈ T , we obtain that
the lemma is proved for z+
n (θ, u). The case of z−n (θ, u) is investigated similarly. �
Let us set
En
θ S±
β (θ̂n) = (En
θ S±
β (τ))τ=θ̂n
.
Lemma 2. Under the conditions of the theorem, for any ε > 0,
(2.23) sup
θ∈T
Pn
θ
{
|S±
β (θ) + En
θ S±
β (θ̂n)| > ε
}
−→
n→∞ 0.
Proof. Let us introduce the events
A±
i (θ) = {S±
iβ(θ) + En
θ S±
iβ(θ̂n)− S±
iβ(θ̂n) ≥ −ε(1 + |En
θ S±
β (θ̂n)|)},
i = 1, . . . , q.
It follows from (1.11) and the previous lemma that
(2.24) inf
θ∈T
Pn
θ {A±
i (θ)} −→
n→∞ 1, i = 1, ..., q.
For the events {|θ̂n − θ| < r}, r < r0, S±
β (θ̂n) ≥ 0. Therefore, relation (2.24) is true for
the events
B±
i (θ) = {S±
iβ(θ) + En
θ S±
iβ(θ̂n) ≥ −ε(1 + |En
θ S±
β (θ̂n)|)} ⊃ A±
i (θ).
90 A. V. IVANOV AND I. V. ORLOVSKY
On the other hand,
S+
iβ(θ) + S−
iβ(θ) =
∑
|gi(j, θ)|χ{εj = 0} = 0 (mod Pn
θ ),
and the events B−
i (θ) are equally like to the events
C+
i (θ) = {S+
iβ(θ) + En
θ S+
iβ(θ̂n) ≤ ε(1 + |En
θ S+
β (θ̂n)|)}.
Furthermore, for ε < q−1, the events D+
i (θ) = B+
i (θ) ∩ C+
i (θ), i = 1, ..., q,
(2.25) D+
i (θ) =
{∣∣∣S+
iβ(θ) + En
θ S+
iβ(θ̂n)
∣∣∣ ≤ ε(1 + |En
θ S+
β (θ̂n)|)
}
,
q⋂
i=1
D+
i (θ) ⊆
{∣∣∣S+
β (θ) + En
θ S+
β (θ̂n)
∣∣∣ ≤ qε(1 + |En
θ S+
β (θ̂n)|)
}
⊆
{∣∣∣En
θ S+
β (θ̂n)
∣∣∣ ≤ (1− qε)−1(qε + |S+
β (θ)|)
}
= X+(θ),
i.e.,
(2.26) inf
θ∈T
Pn
θ {X+(θ)} −→
n→∞ 1.
Let us note that
(2.27) Pn
θ {|En
θ S+
β (θ̂n)| > M} ≤ Pn
θ {X+(θ)}+ Pn
θ {|S+
β (θ)| > M(1− qε)− qε},
where X+(θ) is a complement of the event X+(θ). Let us denote
ηj = χ{εj < 0} − β, j ≥ 1,
Iin(θ) = {1, . . . , n} ∩ {j : gi(j, θ) > 0}.
Then Pn
θ - a.s.
S+
β (θ)− d−1
in (θ)
∑
gi(j, θ)ηj = d−1
in (θ)
∑
j∈Iin(θ)
gi(j, θ)χ{εj = 0} = 0.
Therefore, by the Chebyshev inequality,
Pn
θ {|S+
β (θ)| > M(1− qε)− qε} ≤ q(M(1 − qε)− qε)−2 −→
M→∞
0,
i.e., the vector S+
β (θ) is bounded in probability. It follows from (2.26) and (2.27) that
the vector En
θ S+
β (θ̂n) is also bounded in probability uniformly in θ ∈ T .
According to (2.25),
sup
θ∈T
Pn
θ
{
|S+
β (θ) + En
θ S+
β (θ̂n)| > ε
(
1 + |En
θ S+
β (θ̂n)|
)}
−→
n→∞ 0.
Therefore, (2.23) holds. We remark that the boundedness in probability of the r.v.
En
θ S+
β (θ̂n) can also be obtained immediately from condition C, the explicit form of
En
θ S+
β (θ̂n), and from the conditions of the theorem. �
Lemma 3. Under the conditions of the theorem, for any ε > 0,
(2.28) Pn
θ
{
|En
θ S+
β (θ̂n)− p(0)I(θ)dn(θ)(θ̂n − θ)| > ε
}
−→
n→∞ 0.
Proof. If the quantity n−1/2|dn(θ)(θ̂n − θ)| is small, then it follows from inequality (2.6)
and the boundedness of the r.v. En
θ S+
β (θ̂n) in probability that the norm of the vector
dn(θ)(θ̂n−θ) is bounded in probability. The statement of Lemma 3 follows from condition
C and inequalities (2.3)-(2.5). �
PARAMETER ESTIMATORS OF NONLINEAR QUANTILE REGRESSION 91
3. Proof of the theorem
Relations (2.23) and (2.28) show that, for any ε > 0,
(3.1) Pn
θ
{
|(p(0))−1Λ(θ)S+
β (θ) + dn(θ)(θ̂n − θ)| > ε
}
−→
n→∞ 0.
As was noted above,
S+
β (θ) = d−1
n (θ)
∑
∇g(j, θ)ηj (mod Pn
θ ).
Let us apply Corollary 17.2 in ([5], p. 165) to the random vectors
ξjn = n1/2d−1
n (θ)∇g(j, θ)ηj , j = 1, . . . , n.
It follows from (1.5) that
n−1
∑
En
θ |ξjn|3 ≤ q1/2
q∑
i=1
n−1
∑
d−3
in (θ)|gi(j, θ)|3n3/2 ≤ k10 <∞
uniformly in θ ∈ T . Then
(3.2) sup
θ∈T
sup
C∈Cq
∣∣∣Pn
θ
{
I−1/2(θ)S+
β (θ) ∈ C
}
− Φ(C)
∣∣∣ = O(n−1/2).
Let us find the correlation matrix of S+
β (θ). Clearly, ES+
β (θ) = 0. Then, taking into
account A1, we get
En
θ S+
iβ(θ)S+
lβ(θ) = d−1
in (θ)d−1
ln (θ)
∑
gi(j, θ)gl(j, θ)Eη2
j , i, l = 1, ..., q.
It follows from the form of ηj that Eη2
j = β(1− β). Then
(3.3) En
θ S+
β (θ)(S+
β (θ))T = β(1 − β)I(θ).
Relations (3.1)-(3.3) yield that, for any ε > 0 and C ∈ Cq,
(3.4) −Δn + Φ(C−ε) ≤ Pn
θ
{
p(0)√
β(1− β)
I1/2(θ)dn(θ)(θ̂n − θ) ∈ C
}
≤ Δn + Φ(Cε),
where C−ε and Cε are the exterior and interior sets parallel to C, and Δn −→
n→∞ 0
uniformly in θ ∈ T and C ∈ Cq. The statement of the theorem follows from (3.4) and
the theorem from Section 3 in [6] which state that, for any ε > 0,
sup
C∈Cq
|Φ(C±ε)− Φ(C)| ≤ kε,
where k is a constant that does not depend on ε.
Bibliography
1. Koenker, R. and Bassett, G., Regression quantile, Econometrica 46 (1978), 33-50.
2. Orlovsky I.V., Consistency of Koenker and Basset estimators in nonlinear regression models,
Scintific News of National Technical University of Ukraine ”Kyiv Politekhn. Inst.” 4(42) (2005),
140-147.
3. Ivanov A.V., Asymptotic Theory of Nonlinear Regression, Kluwer, Dordrecht, 1997.
4. Huber P.J., Robust Statistics, Wiley, New York, 1981.
5. Huber P.J., The Behaviour of Maximum Likelihood Estimates under Nonstandard Conditions,
Proc. of the 5th Berkeley Symp. on Mathematical Statistics and Probability. I, University of
California Press: Berkeley, (1967), 221-234.
6. Bhattacharya R.N. and Ranga Rao R., Normal Approximation and Asymptotic Expantions,
Wiley, New York, 1976.
E-mail : ivanov@paligora.kiev.ua, avalon@ln.ua
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