Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
Research Article
Nonparametric robust function estimation for
integrated diffusion processes
Yunyan Wang, Mingtian Tang∗
School of Science, Jiangxi University of Science and Technology, No. 86, Hongqi Ave., Ganzhou, 341000, Jiangxi, P. R. China.
Communicated by R. Saadati
Abstract
This paper considers local M-estimation of the unknown drift and diffusion functions of integrated
diffusion processes. We show that under appropriate conditions, the proposed estimators for drift and
diffusion functions in the integrated process are consistent, and the conditions that ensure the asymptotic
normality of these local M-estimators are also stated. The simulation studies show that the proposed
c
estimators perform better than the kernel estimator in robustness. 2016
All rights reserved.
Keywords: Integrated diffusion process, local linear estimator, M-estimation, robust estimation.
2010 MSC: 62G35, 62G20.
1. Introduction
Rt
In this paper, we consider the stochastic process Yt = 0 Xs ds where X is a one-dimensional diffusion
process given by
dXt = µ(Xt )dt + σ(Xt )dBt ,
(1.1)
where {Bt , t ≥ 0} is a standard Brownian motion (or Wiener process), and µ(·) and σ(·), the drift and
diffusion of the process {Xt }, are functions only of the contemporaneous value of Xt . The integrated
diffusion process (Yt , Xt ) solves the following second-order stochastic differential equation:
(
dYt = Xt dt,
(1.2)
dXt = µ(Xt )dt + σ(Xt )dBt .
∗
Corresponding author
Email addresses: yywang@zju.edu.cn (Yunyan Wang), mttang_csu@yahoo.com (Mingtian Tang)
Received 2015-10-09
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
3049
This integrated diffusion processes can model integrated and differentiated diffusion processes, and at
the same time overcome the difficulties associated with the nondifferentiability of the Brownian motion, so
these models play an important role in the financial and economic field.
As we all know, the diffusion models (1.1) represent a widely accepted class of processes in finance
and economics. Estimation and inference about infinitesimal mean function µ(·) and infinitesimal variance
function σ(·) in (1.1) has been an important area in modern econometrics, especially the estimators proposed
and studied based on a discrete time observation of the sample path. For example, one can view [3, 4] for
parametric estimation and [18] for nonparametric estimation based on low-frequency data, as for the highfrequency data, one can view [2, 6, 14] and so on.
However, all sample functions of a diffusion process (1.1) driven by a Brownian motion are of unbounded
variation and nowhere differentiable, so it can not model integrated and differentiated diffusion processes.
On the other hand, integrated diffusion processes play an important role in the field of finance, engineering
and physics. For example, Xt may represent the velocity of a particle and Yt its coordinate, see e.g. [11, 23].
P. D. Ditlevsen and et al. in [10] used integrated diffusion process to model and analyze the ice-core data.
Statistical inference for discretely observed integrated diffusion processes has been considered recently.
For example, parametric inference for integrated diffusion processes has been addressed by [11, 15, 16, 17].
As for the nonparametric inferences, [22] obtained the Nadaraya-Waston estimators for drift and diffusion
functions, [24] studied the local linear estimators for the two functions, and [25] developed the re-weighted
estimator of the diffusion function.
All the above nonparametric estimators are based on local polynomial regression smoothers. However,
local polynomial regression smoothers are not robust, so there is a growing literature on the robust methods.
One popular robust technique is the so-called M-estimator, which is the easiest to cope with as far as
asymptotic theory is concerned as pointed out by [20]. Therefore, M-estimation has been studied by many
authors such as [8, 9, 19] and references therein. Furthermore, some modified M-estimators were proposed,
such as local M-estimator, which is a combination of the local linear smoothing technique and the Mestimation technique. The local M-estimators inherit the nice properties from not only M-estimators but
also local linear estimators. For example, [13] proposed local linear M-estimator with variable bandwidth for
regression function, [21] developed a robust estimator of the regression function by using local polynomial
regression techniques.
So in the present paper, we study robust nonparametric statistical inference for drift and diffusion functions of integrated diffusions. We wish to construct local M-estimators of the drift and diffusion coefficients
in integrated diffusion processes (1.2) based on local linear smoothing technique and the M-estimation
technique.
The remainder of the paper is organized as follows. Section 2 introduces the local M-estimators of drift
and diffusion coefficients in integrated diffusion model and develops the asymptotic results of the estimators
under some mild conditions. Simulation studies are developed in Section 3. Some useful lemmas and all
mathematical proofs are presented in Section 4.
2. Local M-estimator and Asymptotic Results
2.1. Local M-estimators
Just as [22, 25] pointed out that there is a difficulty to estimate the integrated diffusion processes, i.e.,
the value of X in model (1.2) at time ti is impossible to obtain, and at the same time the estimation of
model (1.2) can not be based on the observations {Yti , i = 1, 2, · · · }.
Rt
To simplify we use the notation ti = i∆ ,where ∆ = ti − ti−1 , by Yt = 0 Xu du, we have
Yi∆ − Y(i−1)∆
1
=
∆
∆
Z
i∆
Z
Xu du −
0
(i−1)∆
!
Xu du
0
1
=
∆
Z
i∆
Xu du,
(i−1)∆
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
Y
3050
−Y
when ∆ tends to zero, the values of Xi∆ , X(i−1)∆ and i∆ ∆(i−1)∆ become nearly to each other, so our
estimation procedure will be based on the following equivalent set of data,
X̃i∆ =
Yi∆ − Y(i−1)∆
.
∆
The estimation of drift and diffusion functions in the integrated diffusion model (1.2) depends on the
following equations:
!
X̃(i+1)∆ − X̃i∆ E
(2.1)
F(i−1)∆ = µ(X(i−1)∆ ) + o(1), ∆ → 0,
∆
!
2
− X̃i∆ ) F(i−1)∆ = σ 2 (X(i−1)∆ ) + o(1),
∆
3
2 (X̃(i+1)∆
E
∆ → 0,
(2.2)
where Ft = σ{Xs , s ≤ t}. Eqs. (2.1) and (2.2) can be obtained from Lemma 4.1 in Section 4, readers
can also refer to [7] or [22] for more details about them.
By (2.1), the local linear estimator for µ(x) is defined as the solution to the following problem: Choose
a1 and b1 to minimize the following weighted sum
!2
!
n
X
X̃(i−1)∆ − x
X̃(i+1)∆ − X̃i∆
,
− a1 − b1 (X̃i∆ − x) K
∆
h
i=1
and by (2.2) the local linear estimator for σ 2 (x) is defined as the solution to the following problem: Choose
a2 and b2 to minimize the following weighted sum
!2
!
2
n
3
X
(
X̃
−
X̃
)
X̃
−
x
i∆
(i+1)∆
(i−1)∆
2
− a2 − b2 (X̃i∆ − x) K
,
∆
h
i=1
where K(·) is the kernel function and h = hn is bandwidth.
However, the above local linear estimators are not robust. Therefore, let us choose a1 and b1 to minimize
!
!
n
X
X̃(i+1)∆ − X̃i∆
X̃(i−1)∆ − x
ρ1
− a1 − b1 (X̃i∆ − x) K
∆
h
i=1
and a2 and b2 to minimize
n
X
ρ2
3
2 (X̃(i+1)∆
∆
i=1
2
− X̃i∆ )
!
X̃(i−1)∆ − x
h
− a2 − b2 (X̃i∆ − x) K
!
,
or to satisfy the following equations:
n
X
i=1
ψ1
!
X̃(i+1)∆ − X̃i∆
− a1 − b1 (X̃i∆ − x) K
∆
X̃(i−1)∆ − x
h
!
1
!
=
X̃i∆ −x
h
0
0
0
0
,
(2.3)
,
(2.4)
and
n
X
i=1
ψ2
3
2 (X̃(i+1)∆
∆
− X̃i∆ )
2
!
− a2 − b2 (X̃i∆ − x) K
X̃(i−1)∆ − x
h
!
1
X̃i∆ −x
h
!
=
where ρ1 (·) and ρ2 (·) are given functions and ψ1 (·) and ψ2 (·) are the derivatives of ρ1 (·) and ρ2 (·), respectively.
The local M-estimators of µ(x) and µ0 (x) are denoted as µ̂(x) = â1 and µ̂0 (x) = b̂1 , which are the
solutions of (2.3), the local M-estimators of σ 2 (x) and (σ 2 (x))0 are denoted as σ̂ 2 (x) = â2 and (σ̂ 2 (x))0 = b̂2 ,
which are the solutions of (2.4).
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
3051
2.2. Assumptions and Asymptotic Results
Suppose x0 is a given point, the asymptotic results for our local M-estimators of drift and diffusion
functions in integrated diffusion processes need the following conditions.
(A1). ([22])
Rz
(i) Let interval D = (l, r) be the state space of X, s(z) = exp{− z0 2µ(x)
dx} is the scale density
σ 2 (x)
function (z0 is an arbitrary point inside D). For x ∈ D, l < x1 < x < x2 < r, let
Z x
Z x2
s(u)du = ∞, S[x, r) = lim
s(u)du = ∞;
S(l, x] = lim
x1 →l
(ii)
Rr
l
x2 →r
x1
x
m(x)dx < ∞, where m(x) = (σ 2 (x)s(x))−1 is a speed density function;
(iii) X0 = x has distribution P 0 , where P 0 is the invariant distribution of the ergodic process
(Xt )t∈[0,∞) .
(A2). Let interval D = (l, r) be the state space of X, we assume that
µ(x) σ 0 (x)
µ(x) σ 0 (x)
lim sup
−
< 0, lim sup
−
> 0;
σ(x)
2
σ(x)
2
x→r
x→l
Remark 2.1. (A1) assures that X is stationary ([1]), the stationary density of X denotes as p(x) in this
paper. (A2) assures that the process X is α−mixing. Under assumptions (A1) and (A2), we know that
{X̃i∆ , i = 0, 1, · · · } is stationary and α−mixing.
Let α(k) be the mixing coefficient of process X, we assume that α(k) satisfy the following condition.
P a
(A3). The mixing coefficient α(k) of process X satisfies
k (α(k))γ/(2+γ) < ∞ for some a > γ/(2 + γ),
k≥1
where γ is given in assumption (A10).
The kernel function K(·) and bandwidth h satisfy the following assumptions (A4) and (A5).
(A4). The kernel K(·) is a continuously differentiable, symmetric density function compactly supported on
[−1, 1].
(A5). ∆ → 0, h → 0 and nh2 → ∞ as n → ∞.
Remark 2.2. To simplify our purpose, we consider at this stage only positive and symmetrical kernels which
are the most classical ones. As for how to choose bandwidth the book of [12] is recommended.
(A6).
(i) µ(x) and σ(x) have continuous derivative of order 4 and satisfy |µ(x)| ≤ C(1 + |x|)λ and |σ(x)| ≤
C(1 + |x|)λ for some λ > 0;
(ii) E[X0r ] < ∞, where r = max{4λ, 1 + 3λ, −1 + 5λ, −2 + 6λ}.
Remark 2.3. Assumption (A6) guarantees that Lemma 4.1 can be used properly throughout the paper.
(A7). The density function p(x) of the process X is continuous at x0 , and p(x0 ) > 0. Furthermore, the joint
density of Xi∆ and Xj∆ are bounded for all i, j.
(A8).
(i) lim h1 E(|K 0 (ξni )|2 ) < ∞;
h→0
(ii) lim h1 E(|K 0 (ξni )|4 ) < ∞, where
h→0
ξni = θ((X(i−1)∆ − x)/h) + (1 − θ)((X̃(i−1)∆ − x)/h), 0 ≤ θ ≤ 1.
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
(A9).
(i) E[ ψ1 (ui∆ )| X(i−1)∆ = x] = o(1) with ui∆ =
(ii) E[ ψ2 (vi∆ )| X(i−1)∆ = x] = o(1) with vi∆ =
(A10).
X̃(i+1)∆ −X̃i∆
∆
3052
− µ(X(i−1)∆ );
2
3
(X̃(i+1)∆ −X̃i∆ )
2
∆
− σ 2 (X(i−1)∆ ).
(i) The function ψ1 (·) is continuous and has a derivative ψ10 (·) almost everywhere. Furthermore, it
is assumed that functions
2
E[ ψ10 (ui∆ ) X(i−1)∆ = x] > 0, E[ ψ12 (ui∆ ) X(i−1)∆ = x] > 0, E[ ψ10 (ui∆ ) X(i−1)∆ = x] > 0
and continuous at the point x0 , and there exists a constant γ > 0 such that
2+γ E[ |ψ1 (ui∆ )|2+γ X(i−1)∆ = x], E[ ψ10 (ui∆ ) X(i−1)∆ = x]
are bounded in a neighborhood of x0 ;
(ii) The function ψ2 (·) is continuous and has a derivative ψ20 (·) almost everywhere.
Furthermore,
0 (v )| X
2 (v ) X
it is assumed
that
functions
E[
ψ
=
x]
>
0,
E[
ψ
(i−1)∆
(i−1)∆ = x] > 0,
2 i∆
2 i∆
2
E[ ψ20 (vi∆ ) X(i−1)∆ = x] > 0, and continuous at the point x0 , and there exists a constant
γ > 0 such that E[ |ψ2 (vi∆ )|2+γ X(i−1)∆ = x], E[ |ψ20 (vi∆ )|2+γ X(i−1)∆ = x] are bounded in a
neighborhood of x0 .
(A11).
(i) For any i, j,
E[ ψ12 (ui∆ ) + ψ12 (uj∆ ) X(i−1)∆ = x, X(j−1)∆ = y],
2
2
E[ ψ10 (ui∆ ) + ψ10 (uj∆ ) X(i−1)∆ = x, X(j−1)∆ = y]
are bounded in the neighborhood of x0 ;
(ii) For any i, j,
E[ ψ22 (vi∆ ) + ψ22 (vj∆ ) X(i−1)∆ = x, X(j−1)∆ = y],
2
2
E[ ψ20 (vi∆ ) + ψ20 (vj∆ ) X(i−1)∆ = x, X(j−1)∆ = y]
are bounded in the neighborhood of x0 .
(A12).
(i) The function ψ10 (·) satisfies
E[ sup ψ10 (ui∆ + z) − ψ10 (ui∆ ) X(i−1)∆ = x] = o(1),
|z|≤δ
E[ sup ψ1 (ui∆ + z) − ψ1 (ui∆ ) − ψ10 (ui∆ )z X(i−1)∆ = x] = o(δ),
|z|≤δ
as δ → 0 uniformly in x in a neighborhood of x0 ;
(ii) The function ψ20 (·) satisfies
E[ sup ψ20 (vi∆ + z) − ψ20 (vi∆ ) X(i−1)∆ = x] = o(1),
|z|≤δ
E[ sup ψ2 (vi∆ + z) − ψ2 (vi∆ ) − ψ20 (vi∆ )z X(i−1)∆ = x] = o(δ),
|z|≤δ
as δ → 0 uniformly in x in a neighborhood of x0 .
Remark 2.4. The conditions in assumptions (A9)-(A12) imposed on ψ1 (·) and ψ2 (·) are mild and satisfied
for many applications. Particularly, they are fulfilled for Huber’s ψ(·) function. In this paper, we do not
need the monotonicity and boundedness of ψi (·), i = 1, 2 and the convexity of the function ρi (·), i = 1, 2.
For more details about these conditions we refer to [5] or [13].
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
3053
(A13). Assume that there exists a sequence of positive integers qn such that qn → ∞, qn = o((nh)1/2 ) and
(n/h)1/2 α(qn ) → 0 as n → ∞.
(A14). There exists τ > 2 + γ, where γ is given in Assumption 10, such that E{|ψ1 (ui∆ )|τ |X(i−1)∆ =
x}, E{|ψ2 (vi∆ )|τ |X(i−1)∆ = x} are bounded for all x in a neighborhood of x0 , and α(n) = O(n−θ ),
where θ ≥ (2 + γ)τ /{2(τ − 2 − γ)}.
(A15). n−γ/4 h(2+γ)/τ −1−γ/4 = O(1), where γ and τ are given in assumptions (A10) and (A14), respectively.
Remark 2.5. Obviously, Assumption 15 is automatically satisfied for γ ≥ 1 and it is also fulfilled for 0 < γ < 1
if τ satisfies γ < τ − 2 ≤ γ/(1 − γ).
Throughout the whole paper, let
Z
Kl = K(u)ul du,
Z
Jl =
K0 K1
U=
, V =
K1 K2
G1 (x) = E[ ψ10 (ui∆ ) X(i−1)∆ = x],
H1 (x) = E[ ψ20 (vi∆ ) X(i−1)∆ = x],
ul K 2 (u)du,
J0 J1
J1 J2
for l ≥ 0.
,
A=
K2
K3
,
G2 (x) = E[ ψ12 (ui∆ ) X(i−1)∆ = x],
H2 (x) = E[ ψ22 (vi∆ ) X(i−1)∆ = x].
Our main results are as follows:
Theorem 2.6. Under assumptions (A1)-(A7) and the conditions (i) of the assumptions (A8)-(A12), there
exist solutions
µ̂(x0 ) and µ̂0 (x0 ) to equation (2.3) such that
P
µ̂(x0 ) − µ(x0 )
→ 0, as n → ∞.
(i)
0
0
h(µ̂ (x0 ) − µ (x0 ))
(ii) Furthermore, if assumptions (A13)-(A15) hold, then
√
h2 µ00 (x0 ) −1
D
µ̂(x0 ) − µ(x0 )
−
nh
U A → N (0, Σ1 ),
h(µ̂0 (x0 ) − µ0 (x0 ))
2
P
D
where “→” means convergence in probability, “→” means convergence in distribution, and
Σ1 =
G2 (x0 )
U −1 V
2
G1 (x0 )p(x0 )
U −1 .
Theorem 2.7. Under assumptions (A1)-(A7) and the conditions (ii) of the assumptions (A8)-(A12), there
exist solutions σ̂ 2 (x0 ) and (σ̂ 2 (x0 ))0 to equation (2.4) such that
P
σ̂ 2 (x0 ) − σ 2 (x0 )
→ 0, as n → ∞.
(i)
h((σ̂ 2 (x0 ))0 − (σ 2 (x0 ))0 )
(ii) Furthermore, if assumptions (A13)-(A15) hold, then
√
h2 (σ 2 (x0 ))00 −1
D
σ̂ 2 (x0 ) − σ 2 (x0 )
nh
−
U A → N (0, Σ2 ),
h[(σ̂ 2 (x0 ))0 − (σ 2 (x0 ))0 ]
2
P
D
where “→” means convergence in probability, “→” means convergence in distribution, and
Σ2 =
H2 (x0 )
U −1 V U −1 .
H12 (x0 )p(x0 )
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
3054
3. Simulations
In this section, we perform a Monte Carlo experiment to show the performance of the local M-estimators
discussed in the paper by comparing the mean square error (MSE) between the new estimators and the
following kernel-type estimators for µ(x) and σ 2 (x)(see [22]):
n
P
X̃(i−1)∆ −x
(X̃(i+1)∆ −X̃i∆ )
K
h
∆
µ̄(x) = i=1
,
n
P
X̃(i−1)∆ −x
K
h
i=1
n
P
σ̄ 2 (x) =
K
i=1
X̃(i−1)∆ −x
h
n
P
i=1
K
2
3
(X̃(i+1)∆ −X̃i∆ )
2
X̃(i−1)∆ −x
h
∆
.
Rt
Our experiment is based on the simulation of Yt = Y0 + 0 Xu du, where X is an ergodic process governed
by the following stochastic differential equation:
q
dXt = −10Xt dt + 0.1 + 0.1Xt2 dBt ,
(3.1)
and we use the Euler-Maruyama method to approximate the numerical solution of our stochastic differential
equation.
Throughout we take Huber’s function ψ1 (z) = max{−c, min(c, z)} with c = 0.135 and we smooth using
a Gauss kernel, and the bandwidth used in our simulation is h = hopt , where hopt is the optimal bandwidth,
which minimize the mean square error (MSE):
n
1X
(µ̂(xi ) − µ(xi ))2 ,
n
i=1
where {xi , i = 1, 2, · · · , n} are chosen uniformly to cover the range of sample path of Xt , and we obtain µ̂(·)
by iteration since it has no explicit expression. For any initial value µ̂0 (x), we have
µ̂t−1 (x)
µ̂t (x)
− [Wn (µ̂t−1 (x), µ̂0t−1 (x))]−1 Ψn (µ̂t−1 (x), µ̂0t−1 (x)),
=
µ̂0t−1 (x)
µ̂0t (x)
where µ̂t−1 (x) and µ̂0t−1 (x) are the tth iteration value of µ̂0 (x) and µ̂(x), and
(a1 ,b1 )
∂Ψn (a1 ,b1 )
Wn (a1 , b1 ) = ∂Ψn∂a
,
,
∂b1
1
!
!
n
X
X̃(i−1)∆ − x
X̃(i+1)∆ − X̃i∆
Ψn (a1 , b1 ) =
ψ1
− a1 − b1 (X̃i∆ − x) K
∆
h
i=1
1
X̃i∆ −x
h
!
.
This procedure terminates when
µ̂t (x)
µ̂t−1 (x) ≤ 1 × 10−4 .
−
0
µ̂0 (x)
µ̂
(x)
t
t−1
In order to illustrate continuous time integrated processes, Figure 1 presents
a simulated path of X which
Rt
is defined by (3.1) and Figure 2 presents a simulated path of Yt = Y0 + 0 Xu du where t ∈ [0, T ] = [0, 10]
and X is governed by (3.1).
In order to compare the robustness of the local M-estimators with that of the Nadaraya-Watson estimators, Table 1 and Table 2 show the performance of the local M-estimator and the Nadaraya-Watson
estimator for drift function µ(·) and diffusion function σ 2 (·) in terms of the MSE, respectively. We can see
that the local M-estimator performs better than the Nadaraya-Watson estimator, and the performances of
the both estimators are improved as the sample size increases.
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
3055
0.3
100.1
0.2
100.08
100.06
0.1
100.02
Yt
Xt
100.04
0
−0.1
100
−0.2
99.98
−0.3
99.96
−0.4
99.94
0
1
2
3
4
5
6
t
7
8
9
10
0
1
2
3
4
t
5
6
7
8
9
10
Figure 1: THE SAMPLE PATH OF THE
PROCESS Xt .
Figure 2:
THE SAMPLE PATH OF THE INTEGRATED
PROCESS Yt .
Table 1: The mean square error of Nadaraya-Watson estimator (MSE11 ) and local M-estimator (MSE12 ) for drift
function µ(·).
Table 2: The mean square error of Nadaraya-Watson estimator (MSE21 ) and local M-estimator (MSE22 ) for diffusion function σ 2 (·).
Sample size n
n = 100
n = 200
n = 400
n = 800
MSE11
0.9084
0.7499
0.5754
0.3742
MSE12
0.1652
0.1214
0.1272
0.1171
Sample size n
n = 100
n = 200
n = 400
n = 800
MSE21
0.0988
0.0891
0.0881
0.0862
MSE22
0.0459
0.0458
0.0390
0.0392
4. Lemmas and Proofs
In order to prove Theorem 2.6 and Theorem 2.7 , we need the following lemmas.
Lemma 4.1 ([22]). Let Z be a d-dimensional diffusion process governed by the stochastic integral equation
t
Z
Zt = Z0 +
Z
µ(Zs )ds +
0
t
σ(Zs )dBs ,
0
where µ(z) = [µi (z)]d×1 is a d × 1 vector, σ(z) = [σij (z)]d×d is a d × d diagonal matrix, and Bt is a d × 1
vector of independent Brownian motions. Assume that µ and σ have continuous partial derivatives of order
2s. Let f (z) be a continuous function defined on Rd with values in Rd and with continuous partial derivative
of order 2s + 2. Then
s
X
∆k
E[f (Zi∆ )|Z(i−1)∆ ] =
Lk f (Z(i−1)∆ )
+ R,
k!
k=0
where L is a second-order differential operator defined as
L=
d
X
µi (z)
i=1
and
Z
i∆
Z
u1
∂
1 2
∂2
∂2
∂2
2
2
+ (σ11
(z) 2 + σ22
(z) 2 + ... + σdd
(z) 2 ),
∂zi 2
∂z1
∂z2
∂zd
Z
u2
Z
us
···
R=
(i−1)∆
(i−1)∆
(i−1)∆
E[ Ls+1 f (Zus+1 ) Z(i−1)∆ ]dus+1 dus · · · du1
(i−1)∆
is a stochastic function of order ∆s+1 .
Lemma 4.2 ([22]). Let
ξni = θ((X(i−1)∆ − x)/h) + (1 − θ)((X̃(i−1)∆ − x)/h), 0 ≤ θ ≤ 1,
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
and
n
ε1,n
!
X̃(i−1)∆ − x
g(X̃(i−1)∆ , X̃i∆ ),
h
1 X
=
K
nh
i=1
n
1 X
K
nh
ε2,n =
i=1
3056
X(i−1)∆ − x
g(X̃(i−1)∆ , X̃i∆ ),
h
where g is a measurable function on R × R. Assume that Assumption 1, Assumption 4, and
one of the following two conditions holds,
√
∆/h → 0. If
(i) E[(g(X̃(i−1)∆ , X̃i∆ ))2 ] < ∞ and Assumption 8 (ii).
(ii) h−1 E[|(X̃(i−1)∆ − X(i−1)∆ )g(X̃(i−1)∆ , X̃i∆ )|2 ] < ∞ and Assumption 8 (i).
Then
P
|ε1n − ε2n | → 0.
Lemma 4.3. Under Assumptions 1-7 and the conditions (i) of the Assumptions 8-12, for any random
sequence {ηj }nj=1 , if max |ηj | = op (1), we have
1≤j≤n
(i)
n
P
i=1
(ii)
n
P
i=1
ψ10 (ui∆ + ηi∆ )K
ψ10 (ui∆
X̃(i−1)∆ −x0
h
+ ηi∆ )R1 (Xi∆ )K
(X̃i∆ − x0 )l = nhl+1 G1 (x0 )p(x0 )Kl (1 + op (1)),
X̃(i−1)∆ −x0
h
0 ) 00
(X̃i∆ − x0 )l = nhl+3 G1 (x
µ (x0 )p(x0 )Kl+2 (1 + op (1)),
2
where R1 (X̃i∆ ) = µ(X̃i∆ ) − µ(x0 ) − µ0 (x0 )(X̃i∆ − x0 ).
Proof. (i) Obviously, we have
n
X
+ ηi∆ )K
ψ10 (ui∆ )K
X̃(i−1)∆ − x0
h
i=1
=
n
X
i=1
+
!
X̃(i−1)∆ − x0
h
ψ10 (ui∆
n
X
(X̃i∆ − x0 )l
!
(X̃i∆ − x0 )l
[ψ10 (ui∆ + ηi∆ ) − ψ10 (ui∆ )]K
i=1
X̃(i−1)∆ − x0
h
!
(X̃i∆ − x0 )l
:= Tn1 + Tn2 .
For Tn1 , by Lemma 4.2, we need to consider
" n
#
X
X(i−1)∆ − x0
l
0
E
ψ1 (ui∆ )K
(X̃i∆ − x0 )
h
i=1
" n
#
X X(i−1)∆ − x0 l
0
=E
K
(X̃i∆ − x0 ) E[ψ1 (ui∆ ) X(i−1)∆ = x0 ]
h
i=1
" n
#
X X(i−1)∆ − x0 l
= G1 (x0 )E
K
(X̃i∆ − x0 ) .
h
i=1
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
3057
Next, we will show that
#
" n
X X(i−1)∆ − x0 l
(X̃i∆ − x0 ) = nhl+1 p(x0 )Kl (1 + o(1)).
E
K
h
i=1
In the same lines of arguments as in Lemma 1 of [5], we have
#
" n
X X(i−1)∆ − x0 E
(X(i−1)∆ − x0 )l = nhl+1 p(x0 )Kl (1 + o(1)),
K
h
i=1
so it suffices to show that
n
n
X(i−1)∆ − x0
X(i−1)∆ − x0
1 X
1 X
P
l
(X̃i∆ − x0 ) −
(X(i−1)∆ − x0 )l → 0.
K
K
l+1
l+1
h
h
nh
nh
i=1
i=1
For (4.1), let
δ1n =
n
X(i−1)∆ − x0
1 X
l
K
[(X̃i∆ − x0 ) − (X(i−1)∆ − x0 )l ],
h
nhl+1
i=1
it suffices to show that
lim E(δ1n ) = 0,
n→∞
lim V ar(δ1n ) = 0.
n→∞
By the stationarity and Lemma 4.1, we have
"
#
n
X(i−1)∆ − x0
1 X
l
E(δ1n ) =E
((X̃i∆ − x0 ) − (X(i−1)∆ − x0 )l )
K
h
nhl+1
i=1
X(i−1)∆ − x0
1
l
l
=E l+1 K
((X̃i∆ − x0 ) − (X(i−1)∆ − x0 ) )
h
h
X(i−1)∆ − x0
1
l−1
l(X(i−1)∆ − x0 ) (X̃i∆ − X(i−1)∆ ) + o(1)
=E l+1 K
h
h
X(i−1)∆ − x0
1
l−1
=E l+1 K
l(X(i−1)∆ − x0 ) E[(X̃i∆ − X(i−1)∆ ) X(i−1)∆ ] + o(1)
h
h
"
#
X(i−1)∆ − x0
X(i−1)∆ − x0 l−1
∆
1
= E
K
l
µ(X(i−1)∆ ) + O(∆2 ),
2h
h
h
h
which implies that lim E(δ1n ) = 0.
n→∞
On the other hand,
"
#
n
X(i−1)∆ − x0
1 X
l
l
V ar(δ1n ) = V ar
K
((X̃i∆ − x0 ) − (X(i−1)∆ − x0 ) )
h
nhl+1
i=1
"
#
n
X(i−1)∆ − x0
X(i−1)∆ − x0 l−1
1 X1
1
V ar √
K
l
(X̃i∆ − X(i−1)∆ ) + o(1)
=
nh2
h
h
h
n
i=1
"
#
n
1
1 X
=
V ar √
fil + o(1),
nh2
n
i=1
X(i−1)∆ −x0
X(i−1)∆ −x0 l−1
l
(X̃i∆ − X(i−1)∆ ), and we have
where fil = h1 K
h
h
#
n
n
n n−1
1 X
1X
2 X X
V ar √
fil =
V ar(fil ) +
Cov(fil , fjl ).
n
n
n
"
i=1
i=1
i=j+1 j=1
(4.1)
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
Next we will show that V ar
√1
n
n
P
3058
fil
< ∞, with the same arguments as [23], we only need to show
i=1
E(fil2 ) < ∞. In fact,
E(fil2 )
!
2l−2
2
2 X(i−1)∆ − x0
l
(X̃i∆ − X(i−1)∆ )
h
!
1 2 X(i−1)∆ − x0 2 X(i−1)∆ − x0 2l−2
2
l
K
E[(X̃i∆ − X(i−1)∆ ) X(i−1)∆ ] ,
h2
h
h
1 2
K
h2
=E
=E
X(i−1)∆ − x0
h
and
2
2 X(i−1)∆ ] .
X(i−1)∆ ] − 2E[X̃i∆ X(i−1)∆ X(i−1)∆ ] + E[X(i−1)∆
E[(X̃i∆ − X(i−1)∆ )2 X(i−1)∆ ] = E[X̃i∆
By Lemma 4.1, for the first term, we have
2 E[X̃i∆
X(i−1)∆ ]
=E
(Yi∆ − Y(i−1)∆ )2 X(i−1)∆
∆2
!
1
2
+ σ 2 (X(i−1)∆ )∆+X(i−1)∆ µ(X(i−1)∆ )∆+R1 ,
= X(i−1)∆
3
where R1 is a stochastic function of order ∆2 . For the second term, we have
Yi∆ − Y(i−1)∆
1
2
E[X̃i∆ X(i−1)∆ X(i−1)∆ ] = E
X(i−1)∆ X(i−1)∆ = X(i−1)∆
+ X(i−1)∆ µ(X(i−1)∆ )∆ + R2 ,
∆
2
where R2 is a stochastic function of order ∆2 . For the third term, we have
2
X(i−1)∆ ] = X 2
E[X(i−1)∆
(i−1)∆ .
Then we have
1
E[(X̃i∆ − X(i−1)∆ )2 X(i−1)∆ ] = σ 2 (X(i−1)∆ )∆ + R,
3
where R = R1 − 2R2 . So we have E(fil2 ) < ∞, and lim V ar(δ1n ) = 0 is immediate from nh2 → ∞.
n→∞
For Tn2 , by Assumption 5 and Assumption 12(i), with the similar arguments as in Lemma 5.1 of [13],
we can get Tn2 = op (nhl+1 ). This completes the proof of lemma.
(ii) This part can be proved by the same arguments with the first part of this lemma, so we omit the
details.
Lemma 4.4. Under Assumptions 1-7 and the conditions (ii) of the Assumptions 8-12, for any random
sequence {ηj }nj=1 , if max |ηj | = op (1), we have
1≤j≤n
(i)
n
P
i=1
(ii)
n
P
i=1
ψ20 (vi∆
ψ20 (vi∆
+ ηi∆ )K
X̃(i−1)∆ −x0
h
+ ηi∆ )R2 (Xi∆ )K
X̃i∆ − x0 )l = nhl+1 H1 (x0 )p(x0 )Kl (1 + op (1)),
X̃(i−1)∆ −x0
h
0)
(X̃i∆ − x0 )l = nhl+3 H1 (x
p(x0 )(σ 2 (x0 ))00 Kl+2 (1 + op (1)),
2
where R2 (X̃i∆ ) = σ 2 (X̃i∆ ) − σ 2 (x0 ) − (σ 2 (x0 ))0 (X̃i∆ − x0 ).
Proof. The proof of this lemma is similar to Lemma 4.3, so we omit the details.
The proofs of the following two lemmas are similar to Theorem 1 in [5], so we omit the details.
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
3059
Lemma 4.5. Under Assumptions 1-7, the conditions (i) of the Assumptions 8-11 and Assumptions 13-15,
we have


n
P
X̃(i−1)∆ −x0
ψ1 (ui∆ )K
h
D
1 
i=1
 → N (0, Σ3 ),
√ 
n


P
X̃(i−1)∆ −x0
X̃i∆ −x0
nh
ψ (u )K
1
i∆
h
i=1
h
where Σ3 = G2 (x0 )p(x0 )V.
Lemma 4.6. Under Assumptions 1-7, the conditions (ii) of the Assumptions 8-11 and Assumptions 13-15,
we have


n
P
X̃(i−1)∆ −x0
ψ2 (vi∆ )K
h
D
1 
i=1
 → N (0, Σ4 ),
√ 
n


P
X̃(i−1)∆ −x0
X̃i∆ −x0
nh
ψ2 (vi∆ )K
h
h
i=1
where Σ4 = H2 (x0 )p(x0 )V.
The proof of Theorem 2.6. (i) We prove the consistency of the local M-estimators of µ(x) and µ0 (x). Let
!
1
r = (a1 , hb1 )T , r0 = (µ(x0 ), hµ0 (x0 ))T , ri∆ = (r − r0 )T
,
X̃i∆ −x0
h
and
Ln (r) =
n
X
i=1
ρ1
!
X̃(i+1)∆ − X̃i∆
− a1 − b1 (X̃i∆ − x0 ) K
∆
X̃(i−1)∆ − x0
h
!
.
Then we have
ri∆ = (r − r0 )T
!
1
X̃i∆ −x0
h
= (a1 − µ(x0 ), hb1 − hµ0 (x0 ))
!
1
X̃i∆ −x0
h
X̃i∆ − x0
h
= a1 − µ(x0 ) + (b1 − µ0 (x0 ))(X̃i∆ − x0 )
= a1 − µ(x0 ) + (hb1 − hµ0 (x0 ))
= a1 + b1 (X̃i∆ − x0 ) − µ(x0 ) − µ0 (x0 )(X̃i∆ − x0 )
= a1 + b1 (X̃i∆ − x0 ) + R1 (X̃i∆ ) − µ(X̃i∆ )
= a1 + b1 (X̃i∆ − x0 ) + R1 (X̃i∆ ) −
X̃(i+1)∆ − X̃i∆
− ui∆
∆
!
.
Let Sδ be the circle centered at r0 with radius δ. We will show that for any sufficiently small δ,
lim P { inf Ln (r) > Ln (r0 )} = 1.
n→∞
(4.2)
r∈Sδ
In fact, for r ∈ Sδ , we have
Ln (r) − Ln (r0 ) =
n
X
i=1
ρ1
!
X̃(i+1)∆ − X̃i∆
− a1 − b1 (X̃i∆ − x0 ) K
∆
X̃(i−1)∆ − x0
h
!
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
3060
n
X
!
!
X̃(i+1)∆ − X̃i∆
X̃(i−1)∆ − x0
0
−
− µ(x0 ) − µ (x0 )(X̃i∆ − x0 ) K
ρ1
∆
h
i=1
!
n
X
X̃(i−1)∆ − x0
[ρ1 (ui∆ + R1 (X̃i∆ ) − ri∆ ) − ρ1 (ui∆ + R1 (X̃i∆ ))]
=
K
h
i=1
!Z
n
ui∆ +R1 (X̃i∆ )−ri∆
X
X̃(i−1)∆ − x0
=
K
ψ1 (t)dt
h
ui∆ +R1 (X̃i∆ )
i=1
!Z
n
ui∆ +R1 (X̃i∆ )−ri∆
X
X̃(i−1)∆ − x0
=
ψ1 (ui∆ )dt
K
h
u
+R
(
X̃
)
1
i∆
i∆
i=1
!Z
n
ui∆ +R1 (X̃i∆ )−ri∆
X
X̃(i−1)∆ − x0
+
ψ10 (ui∆ )(t − ui∆ )dt
K
h
u
+R
(
X̃
)
1
i∆
i∆
i=1
!Z
n
u
+R
(
X̃
1
X
i∆
i∆ )−ri∆
X̃(i−1)∆ − x0
+
[ψ1 (t) − ψ1 (ui∆ ) − ψ10 (ui∆ )(t − ui∆ )]dt
K
h
ui∆ +R1 (X̃i∆ )
i=1
:=Ln1 + Ln2 + Ln3 .
Next, we will show that
Ln1 = op (nhδ),
Ln2 =
(4.3)
nh
(r − r0 )T G1 (x0 )p(x0 )U (1 + op (1))(r − r0 ) + Op (nh3 δ),
2
Ln3 = op (nhδ 2 ).
(4.5)
For (4.3), we have
Ln1 =
n
X
i=1
K
X̃(i−1)∆ − x0
h
!Z
ui∆ +R1 (X̃i∆ )−ri∆
ψ1 (ui∆ )dt
ui∆ +R1 (X̃i∆ )
n
X
!
X̃(i−1)∆ − x0
=
K
ψ1 (ui∆ )(−ri∆ )
h
i=1
!
n
X
X̃
−
x
0
(i−1)∆
= −(r − r0 )T
K
ψ1 (ui∆ )
h
i=1
1
X̃i∆ −x0
h
= −(r − r0 )T Wn ,
where
n
P
X̃(i−1)∆ −x0
ψ
(u
)K
1 i∆
h

i=1
Wn = 
n
 P
X̃(i−1)∆ −x0
X̃i∆ −x0
ψ1 (ui∆ )K
h
h

i=1
By Lemma 4.5, we have E(Wn ) = o(1), and
V ar(Wn ) = nhG2 (x)p(x0 )V (1 + o(1)).
Note that
Wn = E(Wn ) + Op
p
V ar(Wn ) ,
(4.4)


.

!
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
√
so we have Wn = Op ( nh), which implies that (4.3) holds.
For (4.4), we have
!Z
n
ui∆ +R1 (X̃i∆ )−ri∆
X
X̃(i−1)∆ − x0
Ln2 =
[ψ10 (ui∆ )(t − ui∆ )]dt
K
h
ui∆ +R1 (X̃i∆ )
i=1
!
n
X̃(i−1)∆ − x0
1X
2
ψ10 (ui∆ )(ri∆
− 2R1 (X̃i∆ )ri∆ )
=
K
2
h
i=1
!
!
n
X̃i∆ −x0
X
X̃(i−1)∆ − x0
1
1
h
=
ψ10 (ui∆ )(r − r0 )T
K
(r − r0 )
2
(X̃i∆ −x0 )
X̃i∆ −x0
2
h
2
i=1
h
h
!
n
X
X̃(i−1)∆ − x0
ψ10 (ui∆ )R1 (X̃i∆ )ri∆
−
K
h
i=1
:= Ln21 + Ln22 .
From Lemma 4.3 with l = 0, l = 1 and l = 2, respectively, we have
!
!
n
X̃i∆ −x0
X̃(i−1)∆ − x0
1
1X
h
Ln21 =
K
ψ10 (ui∆ )(r − r0 )T
(r − r0 )
2
(X̃i∆ −x0 )
X̃i∆ −x0
2
h
i=1
h
h2
nh
K0 K1
T
(r − r0 ) G1 (x0 )p(x0 )
(1 + op (1))(r − r0 )
=
K1 K2
2
nh
(r − r0 )T G1 (x0 )p(x0 )U (1 + op (1))(r − r0 )
=
2
and
Ln22
n
X
!
X̃(i−1)∆ − x0
=−
ψ10 (ui∆ )R1 (X̃i∆ )ri∆
K
h
i=1
!
!
n
X
X̃
−
x
1
0
(i−1)∆
= −(r − r0 )T
K
ψ10 (ui∆ )R1 (X̃i∆ )
X̃i∆ −x0
h
h
i=1
3
nh
K2
=−
(r − r0 )T G1 (x0 )µ00 (x0 )p(x0 )
(1 + op (1))
K3
2
= Op (nh3 δ).
Therefore
Ln2 = Ln21 + Ln22 =
nh
(r − r0 )T G1 (x0 )p(x0 )U (1 + op (1))(r − r0 ) + Op (nh3 δ).
2
For (4.5), we have
Ln3 =
=
n
X
i=1
n
X
i=1
K
K
X̃(i−1)∆ − x0
h
!Z
X̃(i−1)∆ − x0
h
!Z
ui∆ +R1 (X̃i∆ )−ri∆
[ψ1 (t) − ψ1 (ui∆ ) − ψ10 (ui∆ )(t − ui∆ )]dt
ui∆ +R1 (X̃i∆ )
R1 (X̃i∆ )−ri∆
R1 (X̃i∆ )
[ψ1 (t + ui∆ ) − ψ1 (ui∆ ) − ψ10 (ui∆ )t]dt
3061
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
3062
n
X
!
X̃(i−1)∆ − x0
[ψ1 (zi∆ + ui∆ ) − ψ1 (ui∆ ) − ψ10 (ui∆ )zi∆ ](−ri∆ )
=
K
h
i=1
!
n
X
X̃
−
x
0
(i−1)∆
[ψ1 (zi∆ + ui∆ ) − ψ1 (ui∆ ) − ψ10 (ui∆ )zi∆ ]
= −(r − r0 )T
K
h
i=1
1
X̃i∆ −x0
h
!
,
where the second-to-last equality follows from the integral
mean value theorem and zi∆ lies between R1 (X̃i∆ )
and R1 (X̃i∆ ) − ri∆ , for i = 1, 2, · · · , n. Since X̃i∆ − x0 ≤ h, we have
max |zi∆ | ≤ max R1 (X̃i∆ ) + (r − r0 )T
i
i
1
X̃i∆ −x0
h
!
≤ max R1 (X̃i∆ ) + 2δ,
i
(4.6)
and by Taylor’s expansion,
1 00
2
0
max R1 (X̃i∆ ) = max µ(X̃i∆ ) − µ(x0 ) − µ (x0 )(X̃i∆ − x0 ) = max µ (ξi )(X̃i∆ − x0 ) ≤ Op (h2 ), (4.7)
i
i
i
2
where ξi lies between X̃i∆ and x0 , for i = 1, 2, · · · , n.
T
For any
given η > 0, let Dη = {(δ1∆ , δ2∆ , · · · , δn∆ ) : |δi∆ | ≤ η, ∀i ≤ n}, by Assumption 12(i) and
X̃i∆ − x0 ≤ h, we have
#
!
n
X
X̃
−
x
l
0
(i−1)∆
(X̃i∆ − x0 ) E sup [ψ1 (δi∆ + ui∆ ) − ψ1 (ui∆ ) − ψ10 (ui∆ )δi∆ ]K
h
Dη i=1
!
#
" n
l
X
X̃(i−1)∆ − x0 0
≤E
sup ψ1 (δi∆ + ui∆ ) − ψ1 (ui∆ ) − ψ1 (ui∆ )δi∆ K
X̃i∆ − x0 h
i=1 Dη
" n
!
#
l
X
X̃(i−1)∆ − x0 ≤ aη δE
K
X̃i∆ − x0 h
"
i=1
≤ bη δ,
where aη and bη are two sequences of positive numbers, tending to zero as η → 0. Therefore by (4.6) and
(4.7), we have
!
n
X
X̃(i−1)∆ − x0
0
[ψ1 (zi∆ + ui∆ ) − ψ1 (ui∆ ) − ψ1 (ui∆ )zi∆ ]K
(X̃i∆ − x0 )l = op (nhl+1 δ),
h
i=1
which implies that (4.5) holds.
Let λ be the smallest eigenvalue of the positive definite matrix U . Then, for any r ∈ Sδ , we have
Ln (r) − Ln (r0 ) = Ln1 + Ln2 + Ln3
nh
G1 (x0 )p(x0 )(r − r0 )T U (r − r0 )(1 + op (1)) + Op (nh3 δ)
=
2
nh
≥
G1 (x0 )p(x0 )λδ 2 (1 + op (1)) + Op (nh3 δ).
2
So as n → ∞, we have
P
nh
2
inf Ln (r) − Ln (r0 ) >
G1 (x0 )p(x0 )λδ > 0 → 1,
r∈Sδ
2
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
3063
which implies that (4.2) holds. From (4.2), we know that Ln (r) has a local minimum in the interior
of Sδ , so there exists solutions to equation (2.3). Let (µ̂(x0 ), hµ̂0 (x0 ))T be the closest solutions to r0 =
(µ(x0 ), hµ0 (x0 ))T , then
n
o
2
lim P (µ̂(x0 ) − µ(x0 ))2 + h2 (µ̂0 (x0 ) − µ0 (x0 )) ≤ δ 2 = 1,
n→∞
which implies the consistency of the local M-estimators of µ(x) and µ0 (x).
(ii) We prove the asymptotic normality of the local M-estimators of µ(x) and µ0 (x). Let
η̂i∆ = R1 (X̃i∆ ) − (µ̂(x0 ) − µ(x0 )) − (µ̂0 (x0 ) − µ0 (x0 ))(X̃i∆ − x0 ).
(4.8)
Then we have
X̃(i+1)∆ − X̃i∆
= µ(X̃i∆ ) + ui∆
∆
= ui∆ + µ(X̃i∆ ) − µ(x0 ) − µ0 (x0 )(X̃i∆ − x0 ) + µ(x0 ) + µ0 (x0 )(X̃i∆ − x0 )
= ui∆ + R1 (X̃i∆ ) + µ̂(x0 ) + µ̂0 (x0 )(X̃i∆ − x0 ) + η̂i∆ − R1 (X̃i∆ )
= µ̂(x0 ) + µ̂0 (x0 )(X̃i∆ − x0 ) + ui∆ + η̂i∆ .
Therefore by (2.3), we have
n
X
ψ1 (ui∆ + η̂i∆ )K
i=1
Let
Tn1 =
n
X
ψ1 (ui∆ )K
i=1
Tn2 =
n
X
i=1
Tn3 =
n
X
i=1
X̃(i−1)∆ − x0
h
X̃(i−1)∆ − x0
h
ψ10 (ui∆ )η̂i∆ K
!
[ψ1 (ui∆ + η̂i∆ ) − ψ1 (ui∆ ) − ψ10 (ui∆ )η̂i∆ ]K
= 0.
X̃i∆ −x
h
!
X̃(i−1)∆ − x0
h
!
1
!
1
= Wn ,
X̃i∆ −x
h
!
(4.9)
1
!
X̃i∆ −x
h
X̃(i−1)∆ − x0
h
,
!
1
X̃i∆ −x
h
!
.
Then by (4.9), we have Tn1 + Tn2 + Tn3 = 0. And by (4.8), we have
!
!
n
X
X̃(i−1)∆ − x0
1
0
Tn2 =
ψ1 (ui∆ )R1 (X̃i∆ )K
X̃i∆ −x
h
h
i=1
!
!
n
0 (x ) − µ0 (x ))(X̃
X
X̃
−
x
(µ̂(x
)
−
µ(x
))
+
(µ̂
−
x
)
0
(i−1)∆
0
0
0
0
0
i∆
−
ψ10 (ui∆ )K
X̃i∆ −x
h
[(µ̂(x0 ) − µ(x0 )) + (µ̂0 (x0 ) − µ0 (x0 ))(X̃i∆ − x0 )]
h
i=1
!
!
n
X
X̃(i−1)∆ − x0
1
0
=
ψ1 (ui∆ )R1 (X̃i∆ )K
X̃i∆ −x
h
h
i=1
!
!
n
X̃i∆ −x
X
X̃
−
x
1
0
µ̂(x0 ) − µ(x0 )
(i−1)∆
0
h
−
ψ1 (ui∆ )K
2
(X̃i∆ −x)
X̃i∆ −x
h(µ̂0 (x0 ) − µ0 (x0 ))
h
i=1
h
h2
nh3
K2
=
G1 (x0 )µ00 (x0 )p(x0 )
(1 + op (1))
K3
2
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
− nhG1 (x0 )p(x0 )
K0 K1
K1 K2
(1 + op (1))
3064
µ̂(x0 ) − µ(x0 )
h(µ̂0 (x0 ) − µ0 (x0 ))
nh3 G1 (x0 )µ00 (x0 )p(x0 )
=
A(1 + op (1)) − nhG1 (x0 )p(x0 )U (1 + op (1))
2
µ̂(x0 ) − µ(x0 )
h(µ̂0 (x0 ) − µ0 (x0 ))
:=Tn21 + Tn22 ,
where the third equality
follows
from Lemma 4.3.
Noting that for X̃i∆ − x0 ≤ h, we have
0
0
sup |η̂i∆ | = sup R1 (X̃i∆ ) − (µ̂(x0 ) − µ(x0 )) − (µ̂ (x0 ) − µ (x0 ))(X̃i∆ − x0 )
i
i
≤ sup R1 (X̃i∆ ) + |µ̂(x0 ) − µ(x0 )| + h µ̂0 (x0 ) − µ0 (x0 )
i
= Op (h2 + (µ̂(x0 ) − µ(x0 )) + h(µ̂0 (x0 ) − µ0 (x0 )))
= op (1),
where the last equality follows from the consistency of (µ̂(x0 ), hµ̂0 (x0 )). Then, by the Assumption 12(i) and
the same argument as that in the first part of Theorem 2.6, we have
!
!
n
X
X̃(i−1)∆ − x0
1
0
Tn3 =
[ψ1 (ui∆ + η̂i∆ ) − ψ1 (ui∆ ) − ψ1 (ui∆ )η̂i∆ ]K
X̃i∆ −x
h
h
i=1
= op (nh)[h2 + (µ̂(x0 ) − µ(x0 )) + h(µ̂0 (x0 ) − µ0 (x0 ))]
= op (Tn22 ).
Therefore, by Tn1 + Tn2 + Tn3 = 0, we have
1 −1
h2 00
µ̂(x0 ) − µ(x0 )
−1
−1
=
(x
)p
(x
)U
(1
+
o
(1))W
+
G
µ (x0 )U −1 A(1 + op (1)).
0
0
p
n
h(µ̂0 (x0 ) − µ0 (x0 ))
nh 1
2
It follows that
√
h2 µ00 (x0 ) −1
1
µ̂(x0 ) − µ(x0 )
−
(x0 )p−1 (x0 )U −1 (1 + op (1)) √ Wn .
nh
U A(1 + op (1)) = G−1
0
0
1
h(µ̂ (x0 ) − µ (x0 ))
2
nh
So by Lemma 4.5, Assumption 5 and the Slutsky’s theorem, we have
√
D
µ̂(x0 ) − µ(x0 )
h2 µ00 (x0 ) −1
−1
−1 N (0, Σ )
nh
−
U A → G−1
3
0
0
1 (x0 )p (x0 )U
2
h(µ̂ (x0 ) − µ (x0 ))
2 (x0 )
= N 0, G2G
U −1 V U −1 = N (0, Σ1 ).
(x )p(x )
1
0
0
This completes the proof.
The proof of Theorem 2.7. The proof follows from Theorem 2.6 using Lemmas 4.4 and 4.6, and is omitted
here.
Acknowledgements
This research is supported by the NSFC (No.11461032 and No.11401267) and the SF of Jiangxi Provincial
Education Department (No. GJJ150687).
Y. Wang, M. Tang, J. Nonlinear Sci. Appl. 9 (2016), 3048–3065
3065
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