Asymptotic properties of Lp-estimators
Some sufficient conditions for consistency and asymptotic normality of a non-linear regression parameter Lp-estimator are presented for a continuous time regression model with Gaussian stationary noise possessing the long-range dependence or weak dependence property.
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irk-123456789-45362009-11-26T12:00:42Z Asymptotic properties of Lp-estimators Ivanov, A.V. Some sufficient conditions for consistency and asymptotic normality of a non-linear regression parameter Lp-estimator are presented for a continuous time regression model with Gaussian stationary noise possessing the long-range dependence or weak dependence property. 2008 Article Asymptotic properties of Lp-estimators / A.V. Ivanov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 60–68. — Бібліогр.: 14 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4536 519.21 en Інститут математики НАН України |
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Some sufficient conditions for consistency and asymptotic normality of a non-linear regression parameter Lp-estimator are presented for a continuous time regression model with Gaussian stationary noise possessing the long-range dependence or weak dependence property. |
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Ivanov, A.V. |
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Ivanov, A.V. Asymptotic properties of Lp-estimators |
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Ivanov, A.V. |
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Ivanov, A.V. |
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Asymptotic properties of Lp-estimators |
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Asymptotic properties of Lp-estimators |
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Asymptotic properties of Lp-estimators |
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Asymptotic properties of Lp-estimators |
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Asymptotic properties of Lp-estimators |
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asymptotic properties of lp-estimators |
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Інститут математики НАН України |
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2008 |
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Asymptotic properties of Lp-estimators / A.V. Ivanov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 60–68. — Бібліогр.: 14 назв.— англ. |
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AT ivanovav asymptoticpropertiesoflpestimators |
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2025-07-02T07:45:27Z |
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Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 60–68
UDC 519.21
ALEXANDER V. IVANOV
ASYMPTOTIC PROPERTIES OF LP -ESTIMATORS
Some sufficient conditions for consistency and asymptotic normality of a non-linear
regression parameter Lp-estimator are presented for a continuous time regression
model with Gaussian stationary noise possessing the long-range dependence or weak
dependence property.
Introduction
Consider a regression model
X(t) = g(t, θ) + ε(t), t ≥ 0,
where g : [0,∞)× Θc → R1 is a continuous function, Θc is a closure in Rm of an open
bounded convex set Θ, θ ∈ Θ. It is supposed that
A1. ε(t), t ∈ R1 is a real measurable mean-square continuous stationary Gaussian
process defined on the complete probability space (Ω,F , P ), Eε(0) = 0.
Definition. Any random variable (r.v.) θ̂T having a property
QpT (θ̂T ) = inf
τ∈Θc
QpT (τ), QpT (τ) =
∫ T
0
|X(t)− g(t, τ)|pdt, 1 ≤ p <∞
is said to be an Lp-estimator of the unknown θ ∈ Θ.
It follows from [1–3] that our assumptions provide the existence of the Lp-estimator.
Lp-estimators belong to a wide class of M -estimators [4] and use the loss function
ρ(x) = |x|p. Least squares estimators (p = 2) and least moduli estimators (p = 1) are
the most studied Lp-estimators [5,6]. The discription of the asymptotic properties of
Lp-estimators for p ∈ (1, 2) is a challenging theoretical problem. For linear and nonlinear
regression models with discrete time and independent identically distributed observation
errors, the consistency and asymptotic normality of lp-estimators were considered in [4,
6–10].
1. Consistency of Lp-estimators
Suppose g(t, ·) ∈ C1(Θc); gi(t, θ) =
∂
∂θi
g(t, θ);
d2
iT (θ) =
∫ T
0
g2
i (t, θ)dt, i = 1, . . . ,m; d2
T (θ) = diag
(
d2
iT (θ)
)m
i=1
;
lim
T→∞
T−1d2
iT (θ) > 0, i = 1, . . . ,m.
2000 AMS Mathematics Subject Classification. Primary 62J02; Secondary 62J99.
Key words and phrases. Lp-estimator, regression model.
60
ASYMPTOTIC PROPERTIES OF LP -ESTIMATORS 61
Let UT (θ) = T− 1
2 dT (θ)(Θ − θ); ûT = T− 1
2 dT (θ)(θ̂T − θ); f(t, u) = g(t, θ + T
1
2 d−1
T (θ)u);
fi(t, u) = gi(t, θ + T
1
2 d−1
T (θ)u), ΦpT (u1, u2) =
∫ T
0
|f(t, u1)− f(t, u2)|pdt,
Q̃pT (u) = QpT (θ + T
1
2 d−1
T (θ)u), u ∈ U c
T (θ);
v(r) = {u ∈ R
m : ‖u‖ < r}, μp = E|ε(0)|p.
B1. For any R > 0, there exist ki(R) < +∞, i = 1, . . . ,m such that
sup
u∈Uc
T (θ)∩vc(R)
sup
t∈[0,T ]
|gi(t, θ + T
1
2 d−1
T (θ)u)|d−1
iT (θ) ≤ ki(R)T−1/2.
C1 (contrast condition). For any r > 0, there exists Δ(r) > 0 such that
(1) inf
u∈Uc
T (θ)\v(r)
T− 1
pEQ̃
1
p
pT (u) ≥ T− 1
pEQ̃
1
p
pT (0) + Δ(r),
and Δ(R0) = ρ0μ
1
p
p + Δ0 for some R0 > 0, where ρ0 > 2 and Δ0 > 0 are some numbers.
A2. ε(t), t ∈ R1, is a strongly dependent process, namely: B(t) = Eε(t)ε(0) =
L(|t|)
|t|α , 0 < α < 1, where L(t), t ∈ [0,∞) is a function slowly varying at infinity,
B(0) = 1.
A3. B ∈ L1(R1), B(0) = 1.
Theorem 1. For any r > 0 as T →∞:
1) under assumptions A1, A2, B1, and C1,
(2) P{‖ûT‖ ≥ r} = O(B(T ));
2) under assumptions A1, A3, B1, and C1,
(3) P{‖ûT‖ ≥ r} = O(T−1).
We will give an outline of the proof of statement (2). The proof of (3) is similar. Let
hT (θ, u) = Q̃
1
p
pT (u)− EQ̃
1
p
pT (u).
By the definition of Lp-estimator,
Q̃
1
p
pT (ûT ) ≤ hT (θ, 0) + EQ̃
1
p
pT (0) a.s.
Therefore, by condition C1 for γ ∈ (0, 1), one has
P {‖ûT‖ ≥ r} = P
{
‖ûT ‖ ≥ r, Q̃
1
p
pT (ûT ) ≤ hT (θ, 0) + EQ̃
1
p
pT (0)
}
≤
≤ P
{
inf
u∈Uc
T (θ)\v(r)
T− 1
p Q̃
1
p
pT (u) ≤ hT (θ, 0) + EQ̃
1
p
pT (0)
}
≤
≤ P
{
− inf
u∈Uc
T (θ)\v(r)
T− 1
phT (θ, u) + T− 1
phT (θ, 0) ≥ Δ(r)
}
≤
≤ P
{
sup
u∈Uc
T (θ)\v(r)
T− 1
p |hT (θ, u)| ≥ γΔ(r)
}
+
+ P
{
T− 1
phT (θ, 0) ≥ (1− γ)Δ(r)
}
=
= P1 + P2.(4)
62 ALEXANDER V. IVANOV
To estimate P2,, we set
ξ(t) = |ε(t)|p − μp, ηT = T−1
∫ T
0
ξ(t)dt.
Using the expansion of the function |x|p in the Hilbert space L2(R1, ϕ(x)dx), ϕ(x) =
(2π)−
1
2 e−
x2
2 , in Hermite polynomials, one can obtain the inequality (see, for example,
[5, 11])
(5) Eη2
T ≤ Dξ(0)
1
T 2
∫ T
0
∫ T
0
B2(t− s)dtds.
Applying the standard argument [11, 12], it can be shown from A2 and (5) that
ηT →
T→∞
0 a.s. If so, then
(6) ζT = T− 1
p
(∫ T
0
|ε(t)|pdt
) 1
p
→
T→∞
μ
1
p
p a.s.
On the other hand, Eζp
T = μp for any T. Therefore ([13], p. 105),
(7) EζT = ET− 1
p Q̃
1
p
pT (0) →
T→∞
μ
1
p
p ,
and, for T > T0 and some 0 < C0 < (1− γ)Δ(r),
P2 = {ζT ≥ (1 − γ)Δ(r) + EζT } ≤
{
ζT ≥ (1− γ)Δ(r) + μ
1
p
p − C0
}
=
=
{
ηT ≥
(
μ
1
p
p + (1− γ)Δ(r) − C0
)p
− μp
}
= O(B2(T )),(8)
as follows from (5).
To estimate P1, one obtains, by the triangle inequality,
(9) Φ
1
p
pT (0, u)− Q̃
1
p
pT (0) ≤ Q̃
1
p
pT (u) ≤ Φ
1
p
pT (0, u) + Q̃
1
p
pT (0),
and, taking the expectations,
(10) −EQ̃
1
p
pT (0)− Φ
1
p
pT (0, u) ≤ −EQ̃
1
p
pT (u) ≤ EQ̃
1
p
pT (0)− Φ
1
p
pT (0, u).
The addition of inequalities (9) and (10) leads to the majorant
|h(θ, u)| ≤ Q̃
1
p
pT (0) + EQ̃
1
p
pT (0).
Therefore,
(11) P1 ≤ P {ζt + EζT ≥ γΔ(r)} .
Having taken in (11) r = R0 from condition C1 and γ =
2
ρ0
, we arrive at the inequality
(12) P1 ≤ P
{
ζT ≥
(
μ
1
p
p − EζT
)
+ μ
1
p
p +
2Δ0
ρ0
}
.
Relation (6) shows that, for T > T0,
(13) P1 ≤ P
{
ζT ≥ μ
1
p
p +
Δ0
ρ0
}
= P
{
ηT ≥
(
μ
1
p
p +
Δ0
ρ0
)p
− μp
}
= O(B2(T )).
Taking bound (8) for r = R0 and bound (13) into account, one has, for any r ∈ (0, R0),
(14)
P {‖ûT‖ ≥ r} ≤ P {R0 ≥ ‖ûT ‖ ≥ r} + P {‖ûT‖ ≥ R0}
= P {R0 ≥ ‖ûT ‖ ≥ r} +O(B2(T )).
ASYMPTOTIC PROPERTIES OF LP -ESTIMATORS 63
As far as
(15) inf
u∈Uc
T (θ)∩(vc(R0)\v(r))
T− 1
pEQ̃
1
p
pT (u) ≥ inf
u∈Uc
T (θ)\v(r))
T− 1
pEQ̃
1
p
pT (u),
condition C1 is fulfilled also for the left-hand side of inequality (15). So, as previously,
we obtain an inequality similar to (4) for γ′ ∈ (0, 1):
P {R0 ≥ ‖ûT‖ ≥ r} ≤ P
{
− inf
u∈Uc
T (θ)∩(vc(R0)\v(r))
T− 1
phT (θ, u) ≥ γ′Δ(r)
}
+
+ P
{
T− 1
phT (θ, 0) ≥ (1− γ′)Δ(r)
}
≤ P3 +O(B2(T )),(16)
P3 = P
{
sup
u∈Uc
T (θ)∩vc(R0)
T− 1
p |hT (θ, u)| ≥ γ′Δ(r)
}
.
For any ε > 0, R > 0, condition B1 yields the existence of δ = δ(ε,R) > 0 such that
(17) sup
u1,u2∈Uc
T (u)∩vc(R), ‖u1−u2‖<δ
T−1ΦpT (u1, u2) < ε.
Let F (1), . . . , F (l) be closed sets of diameters less than δ that corresponds to the
number R = R0 and ε =
(
c1Δ(r)γ′
2
)p
from inequality (17), and let c1 ∈ (0, 1) be some
number,
l⋃
i=1
F (i) = vc(R0). If the points ui ∈ F (i) ∩U c
T (θ), i = 1, . . . , l0, l0 ≤ l are fixed,
then
(18)
P3 ≤
l0∑
i=1
P
{
sup
u′,u′′∈F (i)∩Uc
T (θ)
T− 1
p |hT (θ, u′)− hT (θ, u′′)|+ T− 1
p |hT (θ, ui)| ≥ γ′Δ(r)
}
.
For u′, u′′ ∈ F (i), one has, by inequality (17),
T− 1
p |hT (θ, u′)− hT (θ, u′′)| ≤
≤ T− 1
p
∣∣∣∣Q̃ 1
p
pT (u′)− Q̃
1
p
pT (u′′)
∣∣∣∣+ T− 1
pE
∣∣∣∣Q̃ 1
p
pT (u′)− Q̃
1
p
pT (u′′)
∣∣∣∣ ≤
≤ 2T− 1
p Φ
1
p
pT (u′, u′′) < c1γ
′Δ(r)
and
(19) P3 ≤
l0∑
i=1
P
{
T− 1
p |hT (θ, ui)| ≥ (1− c1)γ′Δ(R)
}
.
For any u ∈ vc(R0), one obtains further
(20) |hT (θ, u)| ≤
∣∣∣∣Q̃ 1
p
pT (u)−
(
EQ̃pT (u)
) 1
p
∣∣∣∣+(EQ̃pT (u)
) 1
p −EQ̃
1
p
pT (u) = a1(u)+ a2(u).
Taking the expectation of both parts of the inequality
(21)
∣∣∣∣EQ̃ 1
p
pT (u)− Q̃pT (u)
∣∣∣∣ 1p ≥ (EQ̃pT (u)
) 1
p − Q̃
1
p
pT (u),
we derive the bound
(22) T− 1
p a2(u) ≤ T− 1
pE
∣∣∣∣EQ̃ 1
p
pT (u)− Q̃pT (u)
∣∣∣∣ 1p ≤ (T−2DQ̃pT (u)
) 1
2p
.
64 ALEXANDER V. IVANOV
Let us use the notation
Δf(t, u) = f(t, 0)− f(t, u), ξ(t, u) = |ε(t) + Δf(t, u)|p.
Then B1 yields
(23) sup
u∈Uc
T (θ)∩vc(R0)
sup
t∈[0,T ]
|Δf(t, u)| ≤ R0‖k(R0)‖,
k(R0) =
(
k1(R0), . . . , kq(R0)
)
, and consequently,
Eξ2(t, u) ≤ 22p−1
(
μ2p + (R0‖k(R0)‖)2p
)
= c2 <∞.
Therefore,
(24) cov (ξ(t, u), ξ(s, u)) =
∞∑
m=1
Cm(t, u)Cm(s, u)
m!
Bm(t− s)
with
Cm(t, u) =
∫ ∞
−∞
|x+ Δf(t, u)|pHm(x)ϕ(x)dx,
where Hm(x), m ≥ 1, are Hermite polynomials.
With regard for the relation
(25)
∞∑
m=1
C2
m(t)
m!
= Dξ(t, u) ≤ c2,
we arrive at the bound [11]
T−2DQ̃pT (u) = T−2
∫ T
0
∫ T
0
cov (ξ(t, u), ξ(s, u)) dtds ≤
≤
∞∑
m=1
1
m!
(
T−2
∫ T
0
∫ T
0
C2
m(t, u)Bm(t− s)dtds
)
≤
≤ c2T−2
∫ T
0
∫ T
0
B(t− s)dtds = O(B(T )),(26)
and
(27) T− 1
p a2(u) = O(B
1
2p (T )).
On the other hand,
(28) T− 1
p a1(u) ≤ T− 1
p
∣∣∣Q̃pT (u)− EQ̃pT (u)
∣∣∣ 1p .
Due to (26)-(28) for any number 0 < c3 < (1 − c1)γ′Δ(r) and u ∈ vc(R0) for T > T0,
P
{
T− 1
p |hT (θ, u)| ≥ (1− c1)γ′Δ(r)
}
≤ P
{
T−1
∣∣∣Q̃pT (u)− EQ̃pT (u)
∣∣∣ ≥ cp3} ≤
c−2p
3 T−2DQ̃pT (u) = O(B(T ))(29)
hence
(30) P3 = O(B(T )).
Relations (16) and (30) yield (2). �
Sometimes, it is sufficient to check a simpler modification of condition C1. For exam-
ple, if
(31) sup
t≥0
sup
τ1,τ2∈Θc
|g(t, τ1)− g(t, τ2)| ≤ g0 <∞,
ASYMPTOTIC PROPERTIES OF LP -ESTIMATORS 65
then, to obtain (2) and (3) instead of (1), one can use the contrast inequality
(32) inf
u∈Uc
T (θ)\v(r)
T− 1
p
(
EQ̃pT (u)
) 1
p ≥ μ
1
p
p + Δ(r).
Assuming
diT (θ) � T 1
2 , i = 1, . . . ,m,
one can take the normalization
T− 1
2 dT (θ) = Im
without loss of generality. Then UT (θ) = Θ− θ, Q̃pT (u) = QpT (θ + u) and so on.
Instead of the differentiability of g and assumption B1, we suppose
B2. Inequality (31) is valid, and for any ε > 0, there exists δ = δ(ε) such that
sup
τ1,τ2∈Θc: ‖τ1−τ2‖<δ
1
T
∫ T
0
|g(t, τ1)− g(t, τ2)|pdt < ε.
Instead of C1, we assume
C2 (contrast condition). For any r > 0, there exists Δ(r) > 0 such that
inf
u∈(Θ−θ)\v(r)
T−1
∫ T
0
(g(t, θ + u)− g(t, θ))2dt ≥ Δ(r).
Theorem 2. If Θ is a bounded set, then under assumptions A1, A2, B2, and C2 for
any r > 0,
P{‖θ̂T − θ‖ ≥ r} = O(B(T )) as T →∞.
A similar statement can be formulated for the process ε(t), t ∈ R
1 with integrated
covariance function.
To prove the theorem, one has to check contrast conditions C1 or (32). They can be
written now in the form of the following assumption:
For any r > 0, there exists Δ∗(r) > 0 such that
inf
τ∈Θc: ‖τ−θ‖≥r
T−1EQpT (τ) ≥ μp + Δ∗(r).
Write
g0(t) = |g(t, θ)− g(t, τ)|.
The validity of C1 follows from the inequalities
(33) T−1EQpT (τ) − μp ≥ p
2
T−1
∫ T
0
g2
0(t)
∫ ∞
g0(t)
xpϕ(x)dxdt ≥ p
2
G0Δ(r) = Δ∗(r) > 0,
where ‖τ − θ‖ ≥ r, Δ(r) is taken from C2,
G0 =
∫ ∞
g0
xpϕ(x)dx, ϕ(x) =
1√
2π
e−
x2
2 ,
and g0 is defined in (31).
In fact, inequality (33) is true for any bounded, even continuously differentiable density
function on R1 which is non-decreasing on (−∞, 0], and μp <∞ [6].
Suppose
(34) g(t, θ) =
m∑
i=1
gi(t)θi.
Then d2
iT =
∫ T
0
g2
i (t)dt, i = 1, . . . ,m, dT = diag(diT ). Condition B1 is transformed into
66 ALEXANDER V. IVANOV
B3. For some ki < +∞, i = 1, . . . ,m,
max
t∈[0,T ]
|gi(t)|d−1
iT ≤ kiT−1/2.
Set
J il
T = d−1
iT d
−1
lT
∫ T
0
gi(t)gl(t)dt, i, l = 1, . . . ,m;
JT =
(
J il
T
)m
i,l=1
, andλmin(JT ) is the least eigenvalue of a positive definite matrix JT .
B4. λmin(JT ) ≥ λ∗ > 0.
Theorem 3. Let the regression function g be of the form (34) and satisfy assumptions
B3 and B4. Then, for any r > 0 as T →∞:
1) P{‖ûT‖ ≥ r} = O(B(T )), if the process ε(t), t ∈ R1, is subjected to A1, A2;
2) P{‖ûT‖ ≥ r} = O(T−1), if the process ε(t), t ∈ R1, is subjected to A1, A3.
Outline the proof of 1). By the triangle inequality,
(35) T− 1
pEQ̃
1
p
pT (u) ≥ T− 1
p Φ
1
p
pT (u, 0)− T− 1
pEQ̃
1
p
pT (0).
Using (7), we conclude that condition C1 will be fulfilled if
(i) there exists R0 > 0 such that, for ‖u‖ ≥ R0 and T > T0,
(36) T− 1
p Φ
1
p
pT (u, 0) ≥ 2μ
1
p
p + Δ(R0),
where Δ(R0) has the same property as that in C1;
(ii) for any 0 < r < R0 and r ≤ ‖u‖ < R0,
(37) T− 1
pEQ̃
1
p
pT (u) ≥ μ
1
p
p + Δ(r,R0)
for some Δ(r,R0) > 0.
To check (36), we will use the representation
(38) T−1ΦpT (u, 0) = T−1
T∫
0
∣∣∣∣ m∑
i=1
gi(t)T
1
2 d−1
iT ui
∣∣∣∣2∣∣∣∣ m∑
i=1
gi(t)T
1
2 d−1
iT ui
∣∣∣∣2−p dt.
It follows from B3 that
(39)
∣∣∣∣∣
m∑
i=1
gi(t)T
1
2 d−1
iT ui
∣∣∣∣∣
2−p
≤
(
max
1≤i≤m
ki
)2−p
m
2−p
2 ‖u‖2−p.
On the other hand, we have, by B4,
(40) T−1
T∫
0
∣∣∣∣∣
m∑
i=1
gi(t)T
1
2 d−1
iT ui
∣∣∣∣∣
2
dt =
m∑
i,l=1
J il
T uiul ≥ λ∗‖u‖2,
and, therefore,
(41) T− 1
p Φ
1
p
pT (u, 0) ≥ c4‖u‖,
where
c4 = λ
1
p∗
(
max
1≤i≤m
ki
) p−2
p
·m p−2
2p .
It is clear from (41) that inequality (36) can be satisfied by the proper choice of ‖u‖.
ASYMPTOTIC PROPERTIES OF LP -ESTIMATORS 67
As follows from (7) and (27), condition (37) will be fulfilled for R0 > ‖u‖ ≥ r0, if
(42) T− 1
p
(
EQ̃pT (u)
) 1
p ≥ μ
1
p
p + Δ1(r,R0)
or
(43) T−1EQ̃pT (u) ≥ μp + Δ2(r,R0),
where Δ1(r,R0) and Δ2(r,R0) are some positive constants.
Similarly to (8),
(44) T−1EQ̃pT (u)− μp ≥ p
2
T−1
T∫
0
Δ2f(t, u)
∞∫
|Δf(t,u)|
xpϕ(x)dxdt.
If ‖u‖ < R0, then we have, by inequality (23),
(45)
∞∫
|Δf(t,u)|
xpϕ(x)dx ≥
∞∫
R0‖k(R0)‖
xpϕ(x)dx = G0 > 0.
Thus, (44), (45), and (40) yield
(46) T−1EQ̃pT (u)− μp ≥ p
2
G0λ∗r2 = Δ2(r,R0) > 0.
2. Asymptotic uniqueness of the
solution to a system of normal equations
If ρ(x) = |x|p, then ρ′(x) = ψ(x) = p|x|p−1sgnx, ρ′′ = ψ′ = p(p− 1)|x|p−2, x �= 0, and
ψ′(0) = +∞.
The Lp-estimator θ̂T is a solution to the system of ”normal” equations
(47) grad
(
γT−1QpT (τ)
)
= 0, γ = (Eψ′(ε(0)))−1
> 0
or
(48) grad
(
γT−1Q̃pT (u)
)
= 0, u = T−1
2 dT (θ)(τ − θ).
Assume Θ ⊂ Rm to be an open bounded set and g(t, ·) ∈ C2(Θc). Write
gil(t, θ) =
∂2
∂τi∂τl
g(t, θ), d2
il,T (θ) =
∫ T
0
g2
il(t, θ)dt, i, l = 1, . . . ,m.
B5:
1) sup
t∈[0,T ]
sup
τ∈Θc
|gi(t, τ)|d−1
iT (θ) ≤ kiT− 1
2 ;
2) sup
t∈[0,T ]
sup
τ∈Θc
|gil(t, τ)|d−1
il,T (θ) ≤ kilT− 1
2 ;
3) sup
τ∈Θc
dil,T (τ)d−1
iT (θ)d−1
lT (θ) ≤ k̃ilT−1
2 ;
4) Td−2
iT (θ)d−2
lT (θ)
T∫
0
(
gil(t, θ + T
1
2 d−1
T (θ)u)− gil(t, θ)
)2
dt ≤ kil‖u‖2, i, l = 1, . . . ,m.
Theorem 4. Suppose p ∈ (3
2 , 2). Then, under assumptions A1, A2, B4, B5, and C1,
the system of equations (47) (or (48)) has a unique solution with probability 1−O(B(T ))
as T →∞.
The idea of the proof consists in the comparison of two matrices
HT (u) = Hessian
(
γT−1Q̃pT (u)
)
and JT (θ).
68 ALEXANDER V. IVANOV
Using the inequality for symmetric matrices [14]
|λmin(HT (u))− λmin(JT (θ))| ≤ m · max
1≤i,l≤m
∣∣Hil
T (u)− J il
T (θ)
∣∣ ,
one can prove that HT (u) is a positive definite matrix in some neighborhood of zero with
probability 1−O(B(T )) as T →∞.
3. Asymptotic normality of Lp-estimators
Assume further that there exist the limits Λ(θ) = lim
T→∞
J−1
T (θ) and
σ(θ) = lim
T→∞
D−1
T (θ)
(∫ 1
0
∫ 1
0
∇g(tT, θ)∇∗g(sT, θ)
|t− s|α
)
D−1
T (θ),
D2
T (θ) = T−1d2
T (θ).
It follows from Theorem 4 that one can apply the Brouwer fixed-point theorem to
prove
Theorem 5. Under assumptions of Theorem 4, the normalized Lp-estimator
B− 1
2 (T )T− 1
2 dT (θ)(θ̂T − θ)
is asymptotically normal N(0,Λ(θ)Σ(θ)Λ(θ)) r.v.
The details of the proof can be found in [11].
The results similar to Theorems 4 and 5 can be obtained for the process ε(t), t ∈ R1
satisfying the weak dependence condition.
Bibliography
1. Jennrich R.I., Asymptotic Properties of Non-Linear Least Squares Estimators, Ann. Math.
Statist. 40 (1969), 633-643.
2. Pfanzagl J., On the measurability and consistency of minimum contrast estimates, Metrika 14
(1969), 249-272.
3. Schmetterer L., Einfuhrung in die Mathematische Statistik, Springer, Berlin, 1966.
4. Huber P., Robust Statistics, Wiley, New York, 1981.
5. Ivanov A.V., Leonenko N.N., Statistical Analysis of Random Fields, Kluwer, Dordrecht, 1989.
6. Ivanov A.V., Asymptotic Theory of Nonlinear Regression, Kluwer AP, Dordrecht, 1997.
7. Ronner A.E., Asymptotic Normality of p-Norm Estimators in Multiple Regression, Z. Wahrsch.
verw. Gebiete 66 (1984), 613-620.
8. Ivanov A.V., On consistency of lα-estimators of regression function parameter, Probability
Theory and Mathematical Statistics 42 (1990), 42-48. (in Russian)
9. Bardadym T.O., Ivanov A.V., Asymptotic normality of lα-estimators of nonlinear regression
model parameter, DAN USSR A (1988), no. 8, 68-70. (in Russian)
10. Bardadym T.O., Ivanov A.V., On asymptotic normality of lα-estimators of nonlinear regres-
sion model parameter, Probability Theory and Mathematical Statistics 60 (1999), 1-10. (in
Ukrainian)
11. Ivanov A.V., Orlovsky I.V., Lp-estimates in nonlinear regression with long-range dependence,
Theory of Stochastic Processes 7(23) (2002), no. 3-4, 38-49.
12. Cramer H., Leadbetter M.R., Stationary and Related Stochastic Processes, Wiley, New York,
1967.
13. Gikhman I.I., Skorokhod A.V., Introduction to the Theory of Random Processes, Nauka, Mos-
cow, 1965. (in Russian)
14. Wilkinson J.H., The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965.
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