# The MLE of the Parameters of a Discrete Competitive System Subject to Environmental Noise

Authors: Huili Xiang; Zhijun Liu
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

A randomized discrete competitive system is investigated and the maximum likelihood estimation (MLE) of the parameters of the system is obtained. Also, a corresponding numerical simulation is offered to support our theoretical results.


---

## Body

## 1. Introduction

It is well known that various discrete competitive systems have received great attention owing to their theoretical and practical significance and there is a large volume of literature relevant to many good results (see [1–6]). Note that the population systems, in the real world, are often perturbed by various types of environmental noises. May [7] also pointed out that due to environmental fluctuation, the birth rate, the death rate, and other parameters usually show random fluctuation to a certain extent. To accurately describe such systems, it is necessary to use stochastic difference equations. In the present contribution, we will consider the maximum likelihood estimation (MLE) of the parameters of a discrete competitive system with environmental noise and our motivation comes from the works in [8–10]. Let us first introduce the following competitive system governed by differential equations:
(1)
x
1
′
(
t
)
=
x
1
(
t
)
b
1
-
a
11
x
1
(
t
)
-
c
2
x
2
(
t
)
1
+
x
2
(
t
)
,
x
2
′
(
t
)
=
x
2
(
t
)
b
2
-
a
22
x
2
(
t
)
-
c
1
x
1
(
t
)
1
+
x
1
(
t
)
,
where b
i,  i
=
1,2, represents the intrinsic growth rate; a
i
i and c
i,  i
=
1,2, stand for intraspecific competing rate and interspecific competing rate, respectively. For the relevant ecology of model (1) we refer to the readers to [11]. Obviously, by a computation, system (1) becomes
(2)
d
log
⁡
x
1
t
d
t
=
b
1
-
a
11
x
1
t
-
c
2
x
2
t
1
+
x
2
t
,
d
log
⁡
x
2
(
t
)
d
t
=
b
2
-
a
22
x
2
(
t
)
-
c
1
x
1
(
t
)
1
+
x
1
(
t
)
.
Considering system (2) with the equidistant points in time 0
,
Δ
t
,
2
Δ
t
,
…
,
n
Δ
t for some Δ
t
>
0, we obtain
(3)
log
⁡
⁡
x
1
,
k
=
log
⁡
⁡
x
1
,
k
-
1
+
b
1
-
a
11
x
1
,
k
-
1
-
c
2
x
2
,
k
-
1
1
+
x
2
,
k
-
1
Δ
t
,
log
⁡
⁡
x
2
,
k
=
log
⁡
⁡
x
2
,
k
-
1
+
b
2
-
a
22
x
2
,
k
-
1
-
c
1
x
1
,
k
-
1
1
+
x
1
,
k
-
1
Δ
t
.
Suppose that the parameter b
i,  i
=
1,2, is stochastically perturbed in the following way:
(4)
b
i
⟶
b
i
+
σ
i
ε
i
,
k
,
where ε
i
,
k
~
N
(
0,1
) and σ
i
2 stands for the noise intensity. Then system (3) can be described by the randomized equations:
(5)
log
⁡
⁡
x
1
,
k
=
log
⁡
⁡
x
1
,
k
-
1
+
b
1
-
a
11
x
1
,
k
-
1
-
c
2
x
2
,
k
-
1
1
+
x
2
,
k
-
1
Δ
t
+
σ
1
Δ
t
ε
1
,
k
,
log
⁡
⁡
x
2
,
k
=
log
⁡
⁡
x
2
,
k
-
1
+
b
2
-
a
22
x
2
,
k
-
1
-
c
1
x
1
,
k
-
1
1
+
x
1
,
k
-
1
Δ
t
+
σ
2
Δ
t
ε
2
,
k
.
Here we choose initial value x
(
0
)
=
(
x
1
(
0
)
,
x
2
(
0
)
)
∈
R
+
2.The main aim of this paper is to investigate the MLE of the parameters of system (5). To the best of our knowledge, there are few published papers concerned with system (5). The rest of this paper is organized as follows. Section 2 focuses on the maximum likelihood estimation of the parameters of system (5). Numerical simulation and discussion are presented in Section 3.


## 2. The MLE of the Parameters

In this section, we focus on the MLE of the parameters of system (5). Suppose that {
(
x
1
,
k
,
x
2
,
k
)
,
k
=
1,2
,
…
,
n
} are the real observed values from system (5). For the sake of simplicity, let u
i
,
k
=
log
⁡
⁡
x
i
,
k  (
i
=
1,2
;
k
=
1,2
,
…
,
n
). Suppose that (
U
1
,
0
,
U
2
,
0
)
,
(
U
1
,
1
,
U
2,1
)
,
…
,
(
U
1
,
n
,
U
2
,
n
) are the corresponding observed quantity and ρ is the correlation coefficient of U
1
,
l and U
2
,
l,  l
=
1,2
,
…
,
n. Then the following notations are used throughout this paper:
(6)
α
1
=
σ
2
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
1
,
k
-
1
)
2
-
σ
2
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
(
2
u
1
,
k
-
1
)
,
α
2
=
σ
2
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
1
,
k
-
1
)
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
2
,
k
-
1
)
1
+
exp
⁡
⁡
(
u
2
,
k
-
1
)
-
σ
2
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
1
,
k
-
1
+
u
2
,
k
-
1
)
1
+
exp
⁡
⁡
(
u
2
,
k
-
1
)
,
α
3
=
ρ
σ
1
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
2
,
k
-
1
)
1
+
exp
⁡
⁡
(
u
2
,
k
-
1
)
-
ρ
σ
1
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
1
,
k
-
1
)
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
1
,
k
-
1
)
1
+
exp
⁡
⁡
(
u
1
,
k
-
1
)
,
α
4
=
ρ
σ
1
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
1
,
k
-
1
+
u
2
,
k
-
1
)
-
ρ
σ
1
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
2
,
k
-
1
)
,
β
1
=
σ
2
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
∑
k
=
1
n
‍
exp
⁡
(
u
1
,
k
)
-
σ
2
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
+
u
2
,
k
-
1
1
+
exp
⁡
⁡
(
u
2
,
k
-
1
)
,
β
2
=
σ
2
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
2
-
σ
2
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
2
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
2
,
β
3
=
ρ
σ
1
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
+
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
-
ρ
σ
1
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
2
,
k
-
1
)
1
+
exp
⁡
⁡
u
2
,
k
-
1
,
β
4
=
ρ
σ
1
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
2
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
-
ρ
σ
1
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
,
γ
1
=
ρ
σ
2
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
2
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
-
ρ
σ
2
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
,
γ
2
=
ρ
σ
2
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
+
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
-
ρ
σ
2
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
,
γ
3
=
σ
1
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
2
-
σ
1
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
2
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
2
,
γ
4
=
σ
1
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
σ
1
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
+
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
,
η
1
=
ρ
σ
2
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
+
u
2
,
k
-
1
-
ρ
σ
2
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
,
η
2
=
ρ
σ
2
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
2
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
-
ρ
σ
2
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
,
η
3
=
σ
1
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
u
2
,
k
-
1
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
-
σ
1
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
+
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
,
η
4
=
σ
1
Δ
t
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
2
-
σ
1
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
2
u
2
,
k
-
1
,
ζ
1
=
σ
2
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
exp
⁡
⁡
u
1
,
k
-
1
-
σ
2
u
1
,
n
-
u
1
,
0
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
+
ρ
σ
1
u
2
,
n
-
u
2
,
0
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
-
ρ
σ
1
∑
k
=
1
n
‍
u
2
,
k
-
u
2
,
k
-
1
exp
⁡
⁡
u
1
,
k
-
1
,
ζ
2
=
σ
2
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
-
σ
2
u
2
,
n
-
u
2
,
0
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
+
ρ
σ
1
u
2
,
n
-
u
2
,
0
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
-
ρ
σ
1
∑
k
=
1
n
‍
u
2
,
k
-
u
2
,
k
-
1
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
,
ζ
3
=
σ
1
∑
k
=
1
n
‍
(
u
2
,
k
-
u
2
,
k
-
1
)
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
-
σ
1
u
2
,
n
-
u
2
,
0
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
+
ρ
σ
2
u
1
,
n
-
u
1
,
0
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
-
ρ
σ
2
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
,
ζ
4
=
σ
1
∑
k
=
1
n
‍
u
2
,
k
-
u
2
,
k
-
1
exp
⁡
⁡
u
2
,
k
-
1
-
σ
1
u
2
,
n
-
u
2
,
0
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
+
ρ
σ
2
u
1
,
n
-
u
1
,
0
n
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
-
ρ
σ
2
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
exp
⁡
⁡
u
2
,
k
-
1
,
A
1
=
1
Δ
t
1
-
ρ
2
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
-
M
1,2
(
k
)
Δ
t
2
,
A
2
=
1
Δ
t
(
1
-
ρ
2
)
∑
k
=
1
n
‍
u
2
,
k
-
u
2
,
k
-
1
-
M
2,1
(
k
)
Δ
t
2
,
B
=
ρ
Δ
t
(
1
-
ρ
2
)
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
-
M
1,2
(
k
)
Δ
t
·
u
2
,
k
-
u
2
,
k
-
1
-
M
2,1
(
k
)
Δ
t
,
M
1
,
2
(
k
)
=
b
1
-
a
11
exp
⁡
(
u
1
,
k
-
1
)
-
c
2
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
,
M
2
,
1
(
k
)
=
b
2
-
a
22
exp
⁡
(
u
2
,
k
-
1
)
-
c
1
exp
⁡
⁡
(
u
1
,
k
-
1
)
1
+
exp
⁡
⁡
(
u
1
,
k
-
1
)
.
Then we have the following main result.Theorem 1.
The maximum likelihood estimation of the parameters of system (5) can be expressed as
(7)
a
i
i
^
=
D
i
i
D
,
c
i
^
=
D
i
D
,
(
i
=
1,2
)
,
σ
1
^
2
=
A
1
A
2
-
B
A
1
A
2
n
A
2
,
σ
2
^
2
=
A
1
A
2
-
B
A
1
A
2
n
A
1
,
b
1
^
=
1
n
Δ
t
u
1
,
n
-
u
1
,
0
+
D
11
D
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
+
D
2
D
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
,
b
2
^
=
1
n
Δ
t
u
2
,
n
-
u
2
,
0
+
D
22
D
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
2
,
k
-
1
)
+
D
1
D
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
,
where
(8)
D
=
α
1
α
2
α
3
α
4
β
1
β
2
β
3
β
4
γ
1
γ
2
γ
3
γ
4
η
1
η
2
η
3
η
4
,
D
11
=
ζ
1
α
2
α
3
α
4
ζ
1
β
2
β
3
β
4
ζ
1
γ
2
γ
3
γ
4
ζ
1
η
2
η
3
η
4
,
D
22
=
α
1
α
2
α
3
ζ
1
β
1
β
2
β
3
ζ
2
γ
1
γ
2
γ
3
ζ
3
η
1
η
2
η
3
ζ
4
,
D
1
=
α
1
α
2
ζ
1
α
4
β
1
β
2
ζ
2
β
4
γ
1
γ
2
ζ
3
γ
4
η
1
η
2
ζ
4
η
4
,
D
2
=
α
1
ζ
1
α
3
α
4
β
1
ζ
2
β
3
β
4
γ
1
ζ
3
γ
3
γ
4
η
1
ζ
4
η
3
η
4
.Proof.
Making the change of variableu
i
,
k
=
log
⁡
x
i
,
k,  (
i
=
1,2
), we can rewrite system (5) as
(9)
u
1
,
k
=
u
1
,
k
-
1
+
b
1
-
a
11
exp
⁡
⁡
u
1
,
k
-
1
-
c
2
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
Δ
t
+
σ
1
Δ
t
ε
1
,
k
,
u
2
,
k
=
u
2
,
k
-
1
+
b
2
-
a
22
exp
⁡
⁡
u
2
,
k
-
1
-
c
1
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
Δ
t
+
σ
1
Δ
t
ε
1
,
k
.
Suppose that U
1
,
0
,
U
2
,
0
,
U
1
,
1
,
U
2,1
,
…
,
(
U
1
,
n
,
U
2
,
n
) are the observed quantity and (
u
1
,
0
,
u
2
,
0
)
,
(
u
1
,
1
,
u
2
,
1
)
,
…
,
(
u
1
,
n
,
u
2
,
n
) are the corresponding observed values of system (9). The information flow is given by F
k
-
1
=
σ
(
(
U
1
,
l
,
U
2
,
l
),  l
≤
k
-
1
). Denote by ρ the correlation coefficient of U
1
,
l and U
2
,
l. Then, for given F
k
-
1, the conditional density function of (
U
1
,
k
,
U
2
,
k
) can be expressed as
(10)
f
u
1
,
k
,
u
2
,
k
∣
F
k
-
1
=
1
2
π
σ
1
σ
2
Δ
t
1
-
ρ
2
·
exp
⁡
-
1
2
1
-
ρ
2
u
1
,
k
-
u
1
,
k
-
1
-
M
1
,
2
(
k
)
Δ
t
σ
1
Δ
t
2
·
u
1
,
k
-
u
1
,
k
-
1
-
M
1
,
2
(
k
)
Δ
t
σ
1
Δ
t
2
-
2
ρ
u
1
,
k
-
u
1
,
k
-
1
-
M
1,2
k
Δ
t
σ
1
Δ
t
·
u
2
,
k
-
u
2
,
k
-
1
-
M
2
,
1
k
Δ
t
σ
2
Δ
t
+
u
2
,
k
-
u
2
,
k
-
1
-
M
2,1
(
k
)
Δ
t
σ
2
Δ
t
2
,
where
(11)
M
1
,
2
(
k
)
=
b
1
-
a
11
exp
⁡
⁡
u
1
,
k
-
1
-
c
2
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
,
M
2
,
1
k
=
b
2
-
a
22
exp
⁡
⁡
u
2
,
k
-
1
-
c
1
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
.
Then, for given F
0, the joint conditional density function of (
U
1
,
1
,
U
2
,
1
)
,
(
U
1
,
2
,
U
2
,
2
)
,
…
,
(
U
1
,
n
,
U
2
,
n
) is
(12)
f
u
1,1
,
u
2,1
,
u
1,2
,
u
2,2
,
…
,
u
1
,
n
,
u
2
,
n
∣
F
0
=
1
2
π
Δ
t
1
-
ρ
2
σ
1
σ
2
n
·
∏
k
=
1
n
‍
exp
⁡
⁡
-
1
2
(
1
-
ρ
2
)
+
u
2
,
k
-
u
2
,
k
-
1
-
M
2,1
(
k
)
Δ
t
σ
2
Δ
t
2
·
u
1
,
k
-
u
1
,
k
-
1
-
M
1
,
2
(
k
)
Δ
t
σ
1
Δ
t
2
-
2
ρ
u
1
,
k
-
u
1
,
k
-
1
-
M
1,2
(
k
)
Δ
t
σ
1
Δ
t
·
u
2
,
k
-
u
2
,
k
-
1
-
M
2
,
1
(
k
)
Δ
t
σ
2
Δ
t
+
u
2
,
k
-
u
2
,
k
-
1
-
M
2,1
(
k
)
Δ
t
σ
2
Δ
t
2
.
With the constants omitted, the logarithmic likelihood function can be written as
(13)
L
n
(
b
1
,
b
2
,
a
11
,
a
22
,
c
1
,
c
2
,
σ
1
2
,
σ
2
2
)
=
-
n
2
log
⁡
σ
1
2
+
log
⁡
⁡
σ
2
2
-
1
2
σ
1
2
Δ
t
(
1
-
ρ
2
)
·
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
-
M
1
,
2
(
k
)
Δ
t
2
+
ρ
Δ
t
1
-
ρ
2
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
-
M
1
,
2
k
Δ
t
σ
1
×
u
2
,
k
-
u
2
,
k
-
1
-
M
2
,
1
k
Δ
t
σ
2
-
1
2
σ
2
2
Δ
t
(
1
-
ρ
2
)
∑
k
=
1
n
‍
u
2
,
k
-
u
2
,
k
-
1
-
M
2,1
(
k
)
Δ
t
2
.
Taking partial derivative with respect to parameters b
1, b
2, a
11, a
22, c
1, c
2, σ
1, σ
2 in (13), respectively, one obtains the following likelihood equations:
(14)
n
ρ
Δ
t
σ
1
b
2
-
n
σ
2
b
1
Δ
t
=
ρ
σ
1
(
u
2
,
n
-
u
2
,
0
)
-
σ
2
(
u
1
,
n
-
u
1
,
0
)
+
c
1
ρ
Δ
t
σ
1
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
1
+
exp
⁡
⁡
u
1
,
k
-
c
2
Δ
t
σ
2
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
1
+
exp
⁡
⁡
u
2
,
k
+
a
22
ρ
Δ
t
σ
1
∑
k
=
1
n
‍
exp
⁡
u
2
,
k
-
1
-
a
11
Δ
t
σ
2
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
,
(15)
n
ρ
Δ
t
σ
2
b
1
-
σ
1
b
2
=
ρ
σ
2
(
u
1
,
n
-
u
1
,
0
)
-
σ
1
(
u
2
,
n
-
u
2
,
0
)
+
c
2
ρ
Δ
t
σ
2
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
1
+
exp
⁡
⁡
u
2
,
k
-
c
1
Δ
t
σ
1
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
1
+
exp
⁡
⁡
u
1
,
k
+
a
11
ρ
Δ
t
σ
2
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
-
a
22
Δ
t
σ
1
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
,
(16)
σ
2
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
-
M
1,2
k
Δ
t
exp
⁡
⁡
(
u
1
,
k
-
1
)
=
ρ
σ
1
∑
k
=
1
n
‍
u
2
,
k
-
u
2
,
k
-
1
-
M
2
,
1
k
Δ
t
exp
⁡
⁡
(
u
1
,
k
-
1
)
,
(17)
σ
2
∑
k
=
1
n
u
1
,
k
-
u
1
,
k
-
1
-
M
1
,
2
k
Δ
t
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
‍
=
ρ
σ
1
∑
k
=
1
n
‍
u
2
,
k
-
u
2
,
k
-
1
-
M
2
,
1
k
Δ
t
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
,
(18)
σ
1
∑
k
=
1
n
u
2
,
k
-
u
2
,
k
-
1
-
M
2,1
k
Δ
t
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
‍
=
ρ
σ
2
∑
k
=
1
n
u
1
,
k
-
u
1
,
k
-
1
-
M
1
,
2
k
Δ
t
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
,
‍
(19)
σ
1
∑
k
=
1
n
u
2
,
k
-
u
2
,
k
-
1
-
M
2,1
k
Δ
t
exp
⁡
⁡
u
2
,
k
-
1
‍
=
ρ
σ
2
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
-
M
1
,
2
k
Δ
t
exp
⁡
⁡
u
2
,
k
-
1
,
(20)
-
n
2
σ
1
2
+
1
2
σ
1
4
Δ
t
1
-
ρ
2
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
-
M
1
,
2
k
Δ
t
2
=
ρ
2
σ
1
3
σ
2
Δ
t
1
-
ρ
2
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
-
M
1
,
2
k
Δ
t
·
u
2
,
k
-
u
2
,
k
-
1
-
M
2
,
1
k
Δ
t
,
(21)
-
n
2
σ
2
2
+
1
2
σ
2
4
Δ
t
(
1
-
ρ
2
)
∑
k
=
1
n
‍
u
2
,
k
-
u
2
,
k
-
1
-
M
2
,
1
k
Δ
t
2
=
ρ
2
σ
1
σ
2
3
Δ
t
1
-
ρ
2
∑
k
=
1
n
‍
u
1
,
k
-
u
1
,
k
-
1
-
M
1
,
2
k
Δ
t
·
u
2
,
k
-
u
2
,
k
-
1
-
M
2
,
1
k
Δ
t
.
By (14) and (15), one has
(22)
b
1
=
1
n
Δ
t
u
1
,
n
-
u
1
,
0
+
a
11
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
1
,
k
-
1
)
+
c
2
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
(
u
2
,
k
-
1
)
1
+
exp
⁡
⁡
u
2
,
k
-
1
,
(23)
b
2
=
1
n
Δ
t
u
2
,
n
-
u
2
,
0
+
a
22
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
+
c
1
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
.
It then follows from (22), (23), and (16)–(19) that
(24)
α
1
a
11
+
α
2
c
2
+
α
3
c
1
+
α
4
a
22
=
σ
2
∑
k
=
1
n
u
1
,
k
-
u
1
,
k
-
1
exp
⁡
⁡
u
1
,
k
-
1
‍
-
σ
2
u
1
,
n
-
u
1
,
0
n
∑
k
=
1
n
exp
⁡
⁡
u
1
,
k
-
1
‍
+
ρ
σ
1
u
1
,
n
-
u
1
,
0
n
∑
k
=
1
n
exp
⁡
⁡
u
1
,
k
-
1
‍
-
ρ
σ
1
∑
k
=
1
n
u
2
,
k
-
u
2
,
k
-
1
exp
⁡
⁡
u
1
,
k
-
1
‍
,
(25)
β
1
a
11
+
β
2
c
2
+
β
3
c
1
+
β
4
a
22
=
σ
2
∑
k
=
1
n
u
1
,
k
-
u
1
,
k
-
1
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
‍
-
σ
2
u
2
,
n
-
u
2
,
0
n
∑
k
=
1
n
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
+
ρ
σ
1
u
2
,
n
-
u
2
,
0
n
∑
k
=
1
n
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
-
ρ
σ
1
∑
k
=
1
n
u
2
,
k
-
u
2
,
k
-
1
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
‍
,
(26)
γ
1
a
11
+
γ
2
c
2
+
γ
3
c
1
+
γ
4
a
22
=
σ
1
∑
k
=
1
n
u
2
,
k
-
u
2
,
k
-
1
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
-
σ
1
u
2
,
n
-
u
2
,
0
n
∑
k
=
1
n
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
+
ρ
σ
2
u
1
,
n
-
u
1
,
0
n
∑
k
=
1
n
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
-
ρ
σ
2
∑
k
=
1
n
u
1
,
k
-
u
1
,
k
-
1
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
‍
,
(27)
η
1
a
11
+
η
2
c
2
+
η
3
c
1
+
η
4
a
22
=
σ
1
∑
k
=
1
n
‍
u
2
,
k
-
u
2
,
k
-
1
exp
⁡
⁡
u
2
,
k
-
1
-
σ
1
u
2
,
n
-
u
2
,
0
n
∑
k
=
1
n
exp
⁡
⁡
u
2
,
k
-
1
+
ρ
σ
2
u
1
,
n
-
u
1
,
0
n
∑
k
=
1
n
exp
⁡
⁡
u
2
,
k
-
1
-
ρ
σ
2
∑
k
=
1
n
u
1
,
k
-
u
1
,
k
-
1
exp
⁡
⁡
u
2
,
k
-
1
‍
.
So, the Gramer rule implies that
(28)
a
i
i
^
=
D
i
i
D
,
c
i
^
=
D
i
D
,
(
i
=
1,2
)
.
Further, from (22) and (23), one achieves the fact that
(29)
b
1
^
=
1
n
Δ
t
u
1
,
n
-
u
1
,
0
+
D
11
D
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
+
D
2
D
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
1
+
exp
⁡
⁡
u
2
,
k
-
1
,
b
2
^
=
1
n
Δ
t
u
2
,
n
-
u
2
,
0
+
D
22
D
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
2
,
k
-
1
+
D
1
D
Δ
t
∑
k
=
1
n
‍
exp
⁡
⁡
u
1
,
k
-
1
1
+
exp
⁡
⁡
u
1
,
k
-
1
.
It is easy to see from (20), (21), (23), and (24) that
(30)
-
n
σ
1
2
+
A
1
σ
1
4
-
B
σ
1
3
σ
2
=
0
,
-
n
σ
2
2
+
A
2
σ
2
4
-
B
σ
2
3
σ
1
=
0
.
Then one achieves the fact that
(31)
σ
1
^
2
=
A
1
A
2
-
B
A
1
A
2
n
A
2
,
σ
2
^
2
=
A
1
A
2
-
B
A
1
A
2
n
A
1
.
This completes the proof.


## 3. Numerical Simulations

In this section, we give some simulation results of the above estimators to compare the true values and the estimators. In Tables1–4, we fix the true values of parameters as follows: a
11
=
2, a
22
=
1, b
1
=
1, b
2
=
1.5, c
1
=
1.2, c
2
=
1.8, ρ
=
0.65, σ
1
2
=
0.09, and σ
2
2
=
0.16, the number of the sample “size n” increases from 500 to 2000, the data of the columns named Average are obtained by the average of 10 MLEs from the data coming from (5) with x
1
(
0
)
=
0.5, x
2
(
0
)
=
0.44, a
11
=
2, a
22
=
1, b
1
=
1, b
2
=
1.5, c
1
=
1.2, c
2
=
1.8, ρ
=
0.65, σ
1
2
=
0.09, σ
2
2
=
0.16, and Δ
t
=
0.1. The columns named AE show the absolute error of MLE. In Tables 1–4, it is easy to see that there is a correlation between the absolute error of MLE and the sample size. Generally speaking, with the increase in the number of the sample, the absolute error of MLE will decrease. Therefore, it is reasonable to estimate the parameters of system (5) by MLE.Table 1
The simulation values of the MLE of(
a
11
,
a
22
) when Δ
t
=
0.1.
True
(
a
11
,
a
22
)
Size
n
Average
AE
a
11-MLE
a
22-MLE
a
11
a
22
(
2,1
)
500
1.991630
1.014662
0.008370
0.014662
1000
2.004029
0.992371
0.004029
0.007629
2000
2.006013
1.005748
0.006013
0.005748Table 2
The simulation values of the MLE of(
b
1
,
b
2
) when Δ
t
=
0.1.
True
(
b
1
,
b
2
)
Size
n
Average
AE
b
1-MLE
b
2-MLE
b
1
b
2
(
1,1.5
)
500
1.013570
1.508662
0.013570
0.008662
1000
0.993755
1.504625
0.006245
0.004625
2000
1.004742
1.496252
0.004742
0.003748Table 3
The simulation values of the MLE of(
c
1
,
c
2
) when Δ
t
=
0.1.
True
(
c
1
,
c
2
)
Size
n
Average
AE
c
1-MLE
c
2-MLE
c
1
c
2
(
1.2,1.8
)
500
1.225380
1.785378
0.025380
0.014622
1000
1.209025
1.808455
0.009025
0.008455
2000
1.207742
1.854748
0.007742
0.054748Table 4
The simulation values of the MLE of(
σ
1
2
,
σ
2
2
) when Δ
t
=
0.1.
True
(
σ
1
2
,
σ
2
2
)
Size
n
Average
AE
σ
1
2-MLE
σ
2
2-MLE
σ
1
2
σ
2
2
(
0.09,0.16
)
500
0.093120
0.164635
0.003120
0.004635
1000
0.086975
0.163322
0.003025
0.003322
2000
0.093742
0.163342
0.003742
0.003342



---

*Source: 305624-2015-02-23.xml*