Hindawi Publishing Corporation
Journal of Probability and Statistics
Volume 2011, Article ID 741701, 11 pages
doi:10.1155/2011/741701
Research Article
Large Deviations in Testing Squared Radial
Ornstein-Uhleneck Model
Shoujiang Zhao and Qiaojing Liu
College of Science, China Three Gorges University, Yichang 443002, China
Correspondence should be addressed to Shoujiang Zhao, shjzhao@yahoo.com.cn
Received 12 May 2011; Accepted 10 July 2011
Academic Editor: Mohammad Fraiwan Al-Saleh
Copyright q 2011 S. Zhao and Q. Liu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study the large deviations and moderate deviations of hypothesis testing for squared radial
Ornstein-Uhleneck model. Large deviation principles for the log-likelihood ratio are obtained, by
which we give negative regions in testing squared radial Ornstein-Uhleneck model and get the
decay rates of the error probabilities.
1. Introduction
Let us consider the hypothesis testing for the following squared radial Ornstein-Uhleneck
model:
dXt δ 2αXt dt 2 Xt dWt ,
X0 0,
1.1
where α < 0 is the unknown parameter to be tested on the basis of continuous observation of
the process {Xt , t ≥ 0} on the time interval 0,T, W is a standard Brownian motion and, δ > 0
is known. We denote the distribution of the solution 1.1 by Pαδ .
We decide the two hypothesis:
H0 : α α0 ,
H1 : α α1 ,
1.2
where α0 , α1 < 0. The hypothesis testing is based on a partition of
⎧
⎨ dP δ α1 ΩT ⎩ dPαδ 0
F0,T ⎫
⎬
∈R
⎭
1.3
2
Journal of Probability and Statistics
of the outcome process on 0, T into two decision regions BT and its compliment BTc , and
we decide that H0 is true or false according to the outcome X· ∈ BT or X· ∈ BTc .
The probability e1 T of accepting H1 when H0 is actually true is called the error
probability of the first kind. The probability e2 T of accepting H0 when H1 is actually true is
called the error probability of the second kind. That is,
e1 T Pαδ0 BT ,
e2 T Pαδ1 BTc .
1.4
By the Neyman-Pearson lemma cf. 1, the optional decision region BT has the following
form:
⎧
⎨1
dPαδ1 log
⎩T
dPαδ0 ⎫
⎬
F0,T ≥c ,
⎭
1.5
where F0,T is the σ-algebra generated by the outcome process on 0, T .
The research of hypothesis testing problem has started in the 1930s cf. 1. Since the
optional decision region BT has the above form, we are interested in the calculation or
approximation of the constant c, and the hypothesis testing problem can be studied by large
deviations cf. 2–6. In those papers, some large deviation estimates of the error probabilities for some i.i.d. sequences, Markov chains, stationary Gaussian processes, stationary
diffusion processes, Ornstein-Uhlenbeck processes are obtained. In this paper, we study the
large deviations and moderate deviations for the hypothesis testing problem of squared
radial Ornstein-Uhleneck model; by large deviation principle, we obtain that the decay of
the error probability of the second kind approaches to 0 or 1 exponentially fast depending on
the fixed exponent of the decay of the error probability of the first kind; we also give negative
regions and get the decay rates of the error probabilities by moderate deviation principle. The
large and moderate deviations for parameter estimators of squared radial Ornstein-Uhleneck
model were studied in 7, 8.
2. Main Results
In this section, we state our main results.
Theorem 2.1. Let aT be a positive function satisfying
aT −→ 0,
T
aT √ −→ ∞
T
as T −→ ∞.
2.1
For any a > 0, set
⎧
⎫
2
2
2
⎨ 1
dPαδ1 α
−
α
aδ ⎬
T α1 − α0 δ
0
1
log
BT ≥
−
.
⎩ aT α0
2α0
−α0 ⎭
dPαδ0 F0,T 4aT 2.2
Journal of Probability and Statistics
3
Then
lim
T
T → ∞ a2 T log Pαδ0 BT −a,
2.3
T
lim 2
log Pαδ1 BTc −∞.
T → ∞ a T Theorem 2.2. If α1 < α0 < 0, then for each a > 0, there exists a ξa ∈ R, such that
⎞
⎛
δ dP
1
1
α
1
lim log Pαδ0 ⎝ log
≥ ξa⎠ −a,
T →∞T
T
dPαδ0 F0,T 2.4
⎞
⎛
dPαδ1 1
δ ⎝1
lim inf log Pα1
log
< ξa⎠ ≤ −Iα1 ξa,
T →∞ T
T
dPαδ0 F0,T 2.5
and when a ∈ 0, zα1 ,
when a ∈ zα1 , ∞,
⎛
⎞⎞
⎛
δ dP
1
1
α1 lim inf log⎝1 − Pαδ1 ⎝ log
< ξa⎠⎠ ≤ −Iα1 ξa,
T →∞ T
T
dPαδ0 F0,T 2.6
where
zα1 −
Iα1 z ⎧
⎪
⎪
⎨
α20 − α21
z α1 − α0 δ/2
⎪
⎪
⎩∞,
α0 − α1 2 δ
,
4α1
δ α1 z α1 − α0 δ/2
−
4
α21 − α20
2
,
z<
α0 − α1 δ
,
2
2.7
otherwise.
Theorem 2.3. If α0 < α1 < 0, then for each a > 0, there exists a ξ a ∈ R, such that
⎞
⎛
δ dP
1
1
α
1
⎠ −a,
lim log Pαδ0 ⎝ log
≥ ξa
T →∞T
T
dPαδ0 F0,T 2.8
and when a ∈ 0, zα1 ,
⎞
⎛
dPαδ1 1
δ ⎝1
lim inf log Pα1
log
T →∞ T
T
dPαδ0 ⎠ ≤ −Iα ξa
,
< ξa
1
F0,T 2.9
4
Journal of Probability and Statistics
when a ∈ zα1 , ∞,
⎛
⎞⎞
⎛
dPαδ1 1
1
δ
⎠⎠ ≤ −Iα ξa
,
lim inf log⎝1 − Pα1 ⎝ log
<
ξa
1
T →∞ T
T
dPαδ0 F0,T 2.10
where
zα1 −
Iα1 z ⎧
⎪
⎪
⎨
α20 − α21
z α1 − α0 δ/2
⎪
⎪
⎩∞,
α0 − α1 2 δ
,
4α1
δ α1 z α1 − α0 δ/2
−
4
α21 − α20
2
,
z>
α0 − α1 δ
,
2
2.11
otherwise.
3. Moderate Deviations in Testing Squared Radial
Ornstein-Uhleneck Model
In this section, we will prove Theorem 2.1. Let us introduce the log-likelihood ratio process
of squared radial Ornstein-Uhleneck model and study the moderate deviations of the loglikelihood ratio process.
By 7, the log-likelihood ratio process has the representation
dPαδ1 log
dPαδ 0
F0,T α2 − α20
1
α1 − α0 Xt − δt − 1
2
2
t
0
3.1
Xs2 ds.
The following Lemma cf. 9 plays an important role in this paper.
Lemma 3.1. The law of Xt under Pαδ is γδ/2, α/e2tα−1 , where γa, b denotes the Gamma distribution:
γa, bdx ba xa−1 −bx
e dx,
Γa
3.2
x > 0.
Moreover, for any θ ∈ R,
Eδα
e
θXt
−δ/2
θ 2tα
e −1
1−
.
α
3.3
Lemma 3.2. For any closed subset F ⊂ R,
⎛
⎞
⎞
⎛
δ 2
dP
−4α30 z2
1
T
−
α
δT
α
α1 1
0
⎠ ∈ F ⎠ ≤ −inf ⎝log
log Pαδ0 ⎝
−
lim sup 2
,
z∈F α2 − α2 2 δ
aT 4α0
dPαδ0 F0,T T → ∞ a T 0
1
3.4
Journal of Probability and Statistics
5
and for any open subset G ⊂ R,
⎞
⎛
⎞
⎛
δ 2
dP
−4α30 z2
1
T
−
α
δT
α
α1 1
0
⎠ ∈ G⎠ ≥ −inf ⎝log
lim inf 2
log Pαδ0 ⎝
−
.
T → ∞ a T z∈G α2 − α2 2 δ
aT 4α0
dPαδ0 F0,T 0
1
3.5
Proof. Let
⎧
⎞⎫
⎛
δ 2
⎬
⎨ aT y
dP
−
α
δT
α
α
1
0
1 ⎠ .
⎝log
ΛT y log Eδα0 exp
−
⎭
⎩ T
4α0
dPαδ0 F0,T 3.6
By 3.1, for any ϕ < 0, we have
T
aT yδα1 − α0 2
δ
2
ΛT y −
log Eα0 exp λXT − δT u
Xt ds
4α0
0
δ
T
dPα0
aT yδα1 − α0 2
δ
2
log Eϕ
exp λXT − δT u
Xt ds
−
4α0
dPϕδ
0
aT yδ α21 − α20
α0 − ϕ
α0 − ϕ
δ
XT −
δT
log Eϕ exp
−
λ
4α0
2
2
T
1 2 1 2
2
u − α0 ϕ
Xs ds ,
2
2
0
3.7
where
aT yα1 − α0 λ
,
2T
aT y α21 − α20
u−
.
2T
3.8
For T large enough, α20 − 2u > 0, we can choose ϕ − α20 − 2u, then ϕ < 0 and
!
"#
aT yδ α21 − α20
α0 − ϕ
α0 − ϕ
δ
δT log Eϕ exp λ −
ΛT y −
XT .
4α0
2
2
3.9
By Lemma 3.1, we have
"#
α0 − ϕ
XT
2
α0 − ϕ 2T ϕ
1
Tδ
lim − 2
log 1 −
−1
0.
λ
e
T →∞
ϕ
2
2a T T
log Eδϕ exp
T → ∞ a2 T lim
!
λ
3.10
6
Journal of Probability and Statistics
Therefore,
Λ y : lim
T
T → ∞ a2 T T
lim 2
T → ∞ a T ΛT y
aT yδ α21 − α20
α0 − ϕ
δT
−
−
4α0
2
$
⎛
⎛
⎞⎞
2
%
2
%
−
α
aT
α
y
T ⎝ aT yδ α21 − α20
α0 δT ⎝
0
1
⎠⎠
1 − &1 lim 2
−
−
T → ∞ a T 4α0
2
T α20
3.11
2
y2 α21 − α20 δ
.
16
−α30
Finally, the Gärtner-Ellis theorem cf. 10 implies the conclusion of Lemma 3.2.
Noting that
dPαδ1 log
dPαδ 0
F0,T dPαδ0 − log
dPαδ 1
3.12
,
F0,T we also have the following result.
Lemma 3.3. For any closed subset F ⊂ R,
⎞
⎛
⎞
⎛
δ 2
dP
−4α31 z2
1
T
−
α
δT
α
α
1
0
1
⎠ ∈ F ⎠ ≤ −inf ⎝log
log Pαδ1 ⎝
lim sup 2
,
z∈F α2 − α2 2 δ
aT 4α1
dPαδ0 F0,T T → ∞ a T 0
1
3.13
and for any open subset G ⊂ R,
⎛
⎞
⎞
⎛
δ 2
dP
−4α31 z2
1
T
−
α
δT
α
α1 1
0
⎠ ∈ G⎠ ≥ −inf ⎝log
lim inf 2
log Pαδ1 ⎝
.
T → ∞ a T z∈G α2 − α2 2 δ
aT 4α1
dPαδ0 F0,T 0
1
3.14
Proof of Theorem 2.1. The first claim is a direct conclusion of Lemma 3.2. Since
T α1 − α0 2 δ
T α1 − α0 2 δ
−→ −∞,
aT α0
aT α1
by Lemma 3.3, we see that the second one also holds.
as T −→ ∞,
3.15
Journal of Probability and Statistics
7
4. Large Deviations in Testing Fractional Ornstein-Uhleneck Model
In this section, we will prove Theorems 2.2 and 2.3. We first study the large deviations of the
log-likelihood ratio process.
Lemma 4.1. Assume α1 < α0 < 0. Then for any closed subset F ⊂ R,
⎛
⎞
dPαδ1 1
δ ⎝1
log
lim sup log Pα0
∈ F ⎠ ≤ −inf Iα0 z,
z∈F
T
dPαδ0 F0,T T →∞ T
4.1
and for any open subset G ⊂ R,
⎞
⎛
δ dP
1
1
α
1
∈ G⎠ ≥ −inf Iα0 z,
lim inf log Pαδ0 ⎝ log
T →∞ T
z∈G
T
dPαδ0 F0,T 4.2
where
Iα0 z ⎧
⎪
⎪
⎨
α20 − α21
z α1 − α0 δ/2
⎪
⎪
⎩∞,
δ α0 z α1 − α0 δ/2
−
4
α21 − α20
2
,
z<
α0 − α1 δ
,
2
4.3
otherwise.
Proof. Let
⎧
⎫
δ ⎨
⎬
dP
α1 ΛT y log Eδα0 exp y log
.
⎩
dPαδ0 F0,T ⎭
4.4
Then for ϕ < 0, we have
T
δ
2
ΛT y log Eα0 exp λXT − δT u
Xt ds
0
log Eδϕ
dPαδ0
T
exp λXT − δT u
dPϕδ
0
T
α0 − ϕ
1 2 1 2
δ
2
Xs ds ,
log Eϕ exp
λ
XT − δT u − α0 ϕ
2
2
2
0
Xt2 ds
4.5
where λ α1 − α0 y/2, u yα20 − α21 /2.
Since α20 − 2u > 0, for y > α20 /α20 − α21 , we can choose ϕ − α20 − 2u, for each y >
α20 /α20 − α21 ; then ϕ < 0 and
'
(
α0 − ϕ
log Eδϕ eλα0 −ϕ/2XT .
ΛT y −δT λ 2
4.6
8
Journal of Probability and Statistics
By Lemma 3.1, we get
1
log Eδϕ exp
T →∞T
!
λ
lim
"#
α0 − ϕ
0.
XT
2
4.7
Therefore,
1 δ
Λ y : lim ΛT y −
α1 − α0 y α0 α20 α21 − α20 y .
T →∞T
2
4.8
Since Λy is a strictly convex differentiable function on DΛ α− , ∞ with
α20
α− < 0,
α20 − α21
lim Λ y ∞,
4.9
y → α−
where DΛ is the effective domain of Λ, we see that Λy is steep. Finally, by
)
*
sup zy − Λ y y∈R
⎧
⎪
⎪
⎨
α20 − α21
z α1 − α0 δ/2
⎪
⎪
⎩∞,
δ α0 z α1 − α0 δ/2
−
4
α21 − α20
2
,
z<
α0 − α1 δ
,
2
otherwise,
4.10
and Gärtner-Ellis theorem, we complete the proof of this lemma.
Similarly, when α0 < α1 < 0, we have
2
1 δ
2
2
Λ y : lim ΛT y −
α1 − α0 y α0 α0 α1 − α0 y .
T →∞T
2
4.11
Since Λy is a strictly convex differentiable function on DΛ −∞, α with
α α20
α20 − α21
4.12
> 0,
and limy → α Λ y ∞, we can see that Λy is steep. By Gärtner-Ellis theorem, we also
have the following result.
Lemma 4.2. Assume α0 < α1 < 0. Then for any closed subset F ⊂ R,
⎞
⎛
dPαδ1 1
δ ⎝1
log
lim sup log Pα0
T
dPαδ0 T →∞ T
∈ F ⎠ ≤ −inf Iα0 z,
F0,T z∈F
4.13
Journal of Probability and Statistics
9
and for any open subset G ⊂ R,
⎞
⎛
δ dP
1
1
α1 lim inf log Pαδ0 ⎝ log
∈ G⎠ ≥ −inf Iα0 z,
T →∞ T
z∈G
T
dPαδ0 F0,T 4.14
where
Iα0 z ⎧
⎪
⎪
⎨
α20 − α21
z α1 − α0 δ/2
⎪
⎪
⎩∞,
δ α0 z α1 − α0 δ/2
−
4
α21 − α20
2
,
z>
α0 − α1 δ
,
2
4.15
otherwise.
Note that
dPαδ1 log
dPαδ 0
F0,T dPαδ0 − log
dPαδ 1
4.16
.
F0,T Then we have the following Lemma.
Lemma 4.3. Assume α1 < α0 < 0. Then for any closed subset F ⊂ R,
⎛
dPαδ1 1
δ ⎝1
lim sup log Pα1
log
T
dPαδ0 T →∞ T
⎞
∈ F ⎠ ≤ −inf Iα1 z,
4.17
⎞
⎛
δ dP
1
1
α1 lim inf log Pαδ1 ⎝ log
∈ G⎠ ≥ −inf Iα1 z,
T →∞ T
z∈G
T
dPαδ0 F0,T 4.18
z∈F
F0,T and for any open subset G ⊂ R,
where
Iα1 z ⎧
⎪
⎪
⎨
α20 − α21
z α1 − α0 δ/2
⎪
⎪
⎩∞,
δ α1 z α1 − α0 δ/2
−
4
α21 − α20
2
,
z<
α0 − α1 δ
,
2
4.19
otherwise.
Lemma 4.4. Assume α0 < α1 < 0. Then for any closed subset F ⊂ R,
⎞
⎛
dPαδ1 1
δ ⎝1
lim sup log Pα1
log
T
dPαδ0 T →∞ T
∈ F ⎠ ≤ −inf Iα1 z,
F0,T z∈F
4.20
10
Journal of Probability and Statistics
and for any open subset G ⊂ R,
⎞
⎛
dPαδ1 1
δ ⎝1
log
lim inf log Pα1
∈ G⎠ ≥ −inf Iα1 z,
T →∞ T
z∈G
T
dPαδ0 F0,T 4.21
where
Iα1 z ⎧
⎪
⎪
⎨
α20 − α21
z α1 − α0 δ/2
⎪
⎪
⎩∞,
δ α1 z α1 − α0 δ/2
−
4
α21 − α20
2
,
z>
α0 − α1 δ
,
2
4.22
otherwise.
By the expression of Iα0 z, Iα1 z, Iα0 z, and Iα1 z, the following lemma is.
Lemma 4.5. i
Iα0 z 0
Iα1 z 0
iff zα0 iff zα1
α0 − α1 2 δ
,
4α0
α1 − α0 2 δ
−
,
4α1
4.23
for all z < α0 − α1 δ/2,
Iα0 z Iα1 z z;
4.24
ii
α0 − α1 2 δ
,
Iα0 z 0 iff zα0 4α0
Iα1 z 0
iff zα1
α1 − α0 2 δ
−
,
4α1
4.25
for all z > α0 − α1 δ/2,
Iα0 z Iα1 z z.
4.26
Proof of Theorems 2.2 and 2.3. Since the proofs of the two theorems are similar, we only prove
Theorem 2.2. Since Iα0 z is increasing on zα0 , α0 − α1 δ/2 and Iα0 zα0 0, Therefore, for
a > 0, by Lemma 4.1, we can choose a ξa ∈ zα0 , α0 − α1 δ/2 such that
⎞
⎛
dPαδ1 1
δ ⎝1
lim log Pα0
log
T →∞T
T
dPαδ0 ≥ ξa⎠ −a.
F0,T 4.27
Journal of Probability and Statistics
11
It is clear that ξa is increasing for a > 0, and by Lemma 4.5, we get Iα0 zα1 zα1 ,
which implies ξzα1 zα1 . Hence for a ∈ 0, zα1 , we have ξa ∈ 0, zα1 , and since Iα1 z is
nonincreasing for z ≤ ξa, therefore we get
Iα1 ξa inf{Iα1 z : z ≤ ξa}.
4.28
Similarly, for a ∈ zα1 , ∞, we have ξa ∈ zα1 , α0 − α1 δ/2, and since Iα1 z is nondecreasing for z ≥ ξa, therefore we get
Iα1 ξa inf{Iα1 z : z ≥ ξa},
4.29
which complete the proof of Theorem 2.2.
Acknowledgments
The authors would like to express their gratitude to Professor F. Q. Gao and the reviewer for
their valuable comments.
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Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Optimization
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014