# Large Deviations in Testing Squared Radial Ornstein-Uhleneck Model

Authors: Shoujiang Zhao; Qiaojing Liu
Journal: Journal of Probability and Statistics
 (2011)

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## Abstract

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.


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## Body

## 1. Introduction

Let us consider the hypothesis testing for the following squared radial Ornstein-Uhleneck model:(1.1)dXt=(δ+2αXt)dt+2XtdWt,X0=0,
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:(1.2)H0:α=α0,H1:α=α1,
where α0,α1<0. The hypothesis testing is based on a partition of (1.3)ΩT={dPα1δdPα0δ|F[0,T]∈R}
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 probabilitye1(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, (1.4)e1(T)=Pα0δ(BT),e2(T)=Pα1δ(BTc).
By the Neyman-Pearson lemma (cf. [1]), the optional decision region BT has the following form: (1.5){1Tlog⁡dPα1δdPα0δ|F[0,T]≥c},
where ℱ[0,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.
Leta(T) be a positive function satisfying
(2.1)a(T)T⟶0,a(T)T⟶∞asT⟶∞.
For any a>0, set
(2.2)BT={1a(T)log⁡dPα1δdPα0δ|F[0,T]≥T4a(T)(α1-α0)2δα0-|α12-α02|2α0aδ-α0}.
Then
(2.3)lim⁡T→∞Ta2(T)log⁡Pα0δ(BT)=-a,lim⁡T→∞Ta2(T)log⁡Pα1δ(BTc)=-∞.Theorem 2.2.
Ifα1<α0<0, then for each  a>0, there exists a  ξ(a)∈ℝ, such that
(2.4)lim⁡T→∞1Tlog⁡Pα0δ(1Tlog⁡dPα1δdPα0δ|F[0,T]≥ξ(a))=-a,
and when a∈(0,zα1),
(2.5)liminf⁡T→∞1Tlog⁡Pα1δ(1Tlog⁡dPα1δdPα0δ|F[0,T]<ξ(a))≤-Iα1(ξ(a)),
when a∈(zα1,+∞),
(2.6)liminf⁡T→∞1Tlog⁡(1-Pα1δ(1Tlog⁡dPα1δdPα0δ|F[0,T]<ξ(a)))≤-Iα1(ξ(a)),
where
(2.7)zα1=-(α0-α1)2δ4α1,Iα1(z)={α02-α12z+(α1-α0)δ/2(δ4-α1(z+(α1-α0)δ/2)α12-α02)2,z<(α0-α1)δ2,+∞,otherwise.Theorem 2.3.
Ifα0<α1<0, then for each  a>0, there exists a  ξ̃(a)∈ℝ, such that
(2.8)lim⁡T→∞1Tlog⁡Pα0δ(1Tlog⁡dPα1δdPα0δ|F[0,T]≥ξ̃(a))=-a,
and when a∈(0,ẑα1),
(2.9)liminf⁡T→∞1Tlog⁡Pα1δ(1Tlog⁡dPα1δdPα0δ|F[0,T]<ξ̃(a))≤-Îα1(ξ̃(a)),
when a∈(ẑα1,+∞),
(2.10)liminf⁡T→∞1Tlog⁡(1-Pα1δ(1Tlog⁡dPα1δdPα0δ|F[0,T]<ξ̃(a)))≤-Îα1(ξ̃(a)),
where
(2.11)ẑα1=-(α0-α1)2δ4α1,Îα1(z)={α02-α12z+(α1-α0)δ/2(δ4-α1(z+(α1-α0)δ/2)α12-α02)2,z>(α0-α1)δ2,+∞,otherwise.


## 3. Moderate Deviations in Testing Squared Radial Ornstein-Uhleneck Model

In this section, we will prove Theorem2.1. Let us introduce the log-likelihood ratio process of squared radial Ornstein-Uhleneck model and study the moderate deviations of the log-likelihood ratio process.By [7], the log-likelihood ratio process has the representation (3.1)log⁡dPα1δdPα0δ|F[0,T]=12(α1-α0)(Xt-δt)-α12-α022∫0tXs2ds.The following Lemma (cf. [9]) plays an important role in this paper.Lemma 3.1.
The law ofXt under  Pαδ is  γ(δ/2,α/e2tα-1), where  γ(a,b) denotes the Gamma distribution:
(3.2)γ(a,b)(dx)=baxa-1Γ(a)e-bx(dx),x>0.
Moreover, for any θ∈ℝ,
(3.3)Eαδ(eθXt)=(1-θα(e2tα-1))-δ/2.Lemma 3.2.
For any closed subsetF⊂ℝ,
(3.4)limsup⁡T→∞Ta2(T)log⁡Pα0δ(1a(T)(log⁡dPα1δdPα0δ|F[0,T]-(α1-α0)2δT4α0)∈F)≤-inf⁡z∈F-4α03z2(α12-α02)2δ,
and for any open subset G⊂ℝ,
(3.5)liminf⁡T→∞Ta2(T)log⁡Pα0δ(1a(T)(log⁡dPα1δdPα0δ|F[0,T]-(α1-α0)2δT4α0)∈G)≥-inf⁡z∈G-4α03z2(α12-α02)2δ.Proof.
Let(3.6)ΛT(y)=log⁡Eα0δexp⁡{a(T)yT(log⁡dPα1δdPα0δ|F[0,T]-(α1-α0)2δT4α0)}.
By (3.1), for any φ<0, we have
(3.7)ΛT(y)=-a(T)yδ(α1-α0)24α0+log⁡Eα0δexp⁡{λ(XT-δT)+u∫0TXt2ds}=-a(T)yδ(α1-α0)24α0+log⁡Eφδ[dPα0δdPφδexp⁡{λ(XT-δT)+u∫0TXt2ds}]=-a(T)yδ(α12-α02)4α0+log⁡Eφδ[(u-12α02+12φ2)∫0TXs2dsexp⁡{(u-12α02+12φ2)∫0TXs2ds(λ+α0-φ2)XT-α0-φ2δT+(u-12α02+12φ2)∫0TXs2ds}],
where
(3.8)λ=a(T)y(α1-α0)2T,u=-a(T)y(α12-α02)2T.
For T large enough, α02-2u>0, we can choose φ=-α02-2u, then φ<0 and
(3.9)ΛT(y)=-a(T)yδ(α12-α02)4α0-α0-φ2δT+log⁡Eφδ[exp⁡{(λ+α0-φ2)XT}].
By Lemma3.1, we have
(3.10)lim⁡T→∞Ta2(T)log⁡Eφδ[exp⁡{(λ+α0-φ2)XT}]=lim⁡T→∞-Tδ2a2(T)log⁡(1-1φ(λ+α0-φ2)(e2Tφ-1))=0.
Therefore,
(3.11)Λ(y):=lim⁡T→∞Ta2(T)ΛT(y)=lim⁡T→∞Ta2(T)(-a(T)yδ(α12-α02)4α0-α0-φ2δT)=lim⁡T→∞Ta2(T)(-a(T)yδ(α12-α02)4α0-α0δT2(1-1+a(T)y(α12-α02)Tα02))=y216(α12-α02)2δ-α03.
Finally, the Gärtner-Ellis theorem (cf. [10]) implies the conclusion of Lemma 3.2.Noting that(3.12)log⁡dPα1δdPα0δ|F[0,T]=-log⁡dPα0δdPα1δ|F[0,T],
we also have the following result.Lemma 3.3.
For any closed subsetF⊂ℝ,
(3.13)limsup⁡T→∞Ta2(T)log⁡Pα1δ(1a(T)(log⁡dPα1δdPα0δ|F[0,T]+(α1-α0)2δT4α1)∈F)≤-inf⁡z∈F-4α13z2(α12-α02)2δ,
and for any open subset G⊂ℝ,
(3.14)liminf⁡T→∞Ta2(T)log⁡Pα1δ(1a(T)(log⁡dPα1δdPα0δ|F[0,T]+(α1-α0)2δT4α1)∈G)≥-inf⁡z∈G-4α13z2(α12-α02)2δ.Proof of Theorem2.1.
The first claim is a direct conclusion of Lemma3.2. Since
(3.15)Ta(T)(α1-α0)2δα0+Ta(T)(α1-α0)2δα1⟶-∞,asT⟶∞,
by Lemma 3.3, we see that the second one also holds.


## 4. Large Deviations in Testing Fractional Ornstein-Uhleneck Model

In this section, we will prove Theorems2.2 and 2.3. We first study the large deviations of the log-likelihood ratio process.Lemma 4.1.
Assumeα1<α0<0. Then for any closed subset  F⊂ℝ,
(4.1)limsup⁡T→∞1Tlog⁡Pα0δ(1Tlog⁡dPα1δdPα0δ|F[0,T]∈F)≤-inf⁡z∈FIα0(z),
and for any open subset G⊂ℝ,
(4.2)liminf⁡T→∞1Tlog⁡Pα0δ(1Tlog⁡dPα1δdPα0δ|F[0,T]∈G)≥-inf⁡z∈GIα0(z),
where
(4.3)Iα0(z)={α02-α12z+(α1-α0)δ/2(δ4-α0(z+(α1-α0)δ/2)α12-α02)2,z<(α0-α1)δ2,+∞,otherwise.Proof.
Let(4.4)ΛT(y)=log⁡Eα0δexp⁡{ylog⁡dPα1δdPα0δ|F[0,T]}.
Then for φ<0, we have
(4.5)ΛT(y)=log⁡Eα0δexp⁡{λ(XT-δT)+u∫0TXt2ds}=log⁡Eφδ[dPα0δdPφδexp⁡{λ(XT-δT)+u∫0TXt2ds}]=log⁡Eφδ[exp⁡{(λ+α0-φ2)(XT-δT)+(u-12α02+12φ2)∫0TXs2ds}],
where λ=(α1-α0)y/2,u=y(α02-α12)/2.
Sinceα02-2u>0, for y>(α02/α02-α12), we can choose φ=-α02-2u, for each y>(α02/α02-α12); then φ<0 and
(4.6)ΛT(y)=-δT(λ+α0-φ2)+log⁡Eφδ[e(λ+(α0-φ)/2)XT].
By Lemma 3.1, we get
(4.7)lim⁡T→∞1Tlog⁡Eφδ[exp⁡{(λ+α0-φ2)XT}]=0.
Therefore,
(4.8)Λ(y)=:lim⁡T→∞1TΛT(y)=-δ2((α1-α0)y+α0+α02+(α12-α02)y).
SinceΛ(y) is a strictly convex differentiable function on 𝒟Λ=(α-,+∞) with
(4.9)α-=α02α02-α12<0,lim⁡y→α-Λ′(y)=+∞,
where 𝒟Λ is the effective domain of Λ, we see that Λ(y) is steep. Finally, by
(4.10)sup⁡y∈R{zy-Λ(y)}={α02-α12z+(α1-α0)δ/2(δ4-α0(z+(α1-α0)δ/2)α12-α02)2,z<(α0-α1)δ2,+∞,otherwise,
and Gärtner-Ellis theorem, we complete the proof of this lemma.Similarly, whenα0<α1<0, we have (4.11)Λ(y)=:lim⁡T→∞1TΛT(y)=-δ2((α1-α0)y+α0+α02+(α12-α02)y).
Since Λ(y) is a strictly convex differentiable function on 𝒟Λ=(-∞,α+) with (4.12)α+=α02α02-α12>0,
and limy→α+Λ′(y)=+∞, we can see that Λ(y) is steep. By Gärtner-Ellis theorem, we also have the following result.Lemma 4.2.
Assumeα0<α1<0. Then for any closed subset  F⊂ℝ,
(4.13)limsup⁡T→∞1Tlog⁡Pα0δ(1Tlog⁡dPα1δdPα0δ|F[0,T]∈F)≤-inf⁡z∈FÎα0(z),
and for any open subset G⊂ℝ,
(4.14)liminf⁡T→∞1Tlog⁡Pα0δ(1Tlog⁡dPα1δdPα0δ|F[0,T]∈G)≥-inf⁡z∈GÎα0(z),
where
(4.15)Îα0(z)={α02-α12z+((α1-α0)δ/2)(δ4-α0(z+(α1-α0)δ/2)α12-α02)2,z>(α0-α1)δ2,+∞,otherwise.
Note that
(4.16)log⁡dPα1δdPα0δ|F[0,T]=-log⁡dPα0δdPα1δ|F[0,T].
Then we have the following Lemma.Lemma 4.3.
Assumeα1<α0<0. Then for any closed subset  F⊂ℝ,
(4.17)limsup⁡T→∞1Tlog⁡Pα1δ(1Tlog⁡dPα1δdPα0δ|F[0,T]∈F)≤-inf⁡z∈FÎα1(z),
and for any open subset G⊂ℝ,
(4.18)liminf⁡T→∞1Tlog⁡Pα1δ(1Tlog⁡dPα1δdPα0δ|F[0,T]∈G)≥-inf⁡z∈GÎα1(z),
where
(4.19)Iα1(z)={α02-α12z+(α1-α0)δ/2(δ4-α1(z+(α1-α0)δ/2)α12-α02)2,z<(α0-α1)δ2,+∞,otherwise.Lemma 4.4.
Assumeα0<α1<0. Then for any closed subset  F⊂ℝ,
(4.20)limsup⁡T→∞1Tlog⁡Pα1δ(1Tlog⁡dPα1δdPα0δ|F[0,T]∈F)≤-inf⁡z∈FIα1(z),
and for any open subset G⊂ℝ,
(4.21)liminf⁡T→∞1Tlog⁡Pα1δ(1Tlog⁡dPα1δdPα0δ|F[0,T]∈G)≥-inf⁡z∈GIα1(z),
where
(4.22)Îα1(z)={α02-α12z+(α1-α0)δ/2(δ4-α1(z+(α1-α0)δ/2)α12-α02)2,z>(α0-α1)δ2,+∞,otherwise.By the expression ofIα0(z), Iα1(z), Îα0(z), and Îα1(z), the following lemma is.Lemma 4.5.
(i)(4.23)Iα0(z)=0iffzα0=(α0-α1)2δ4α0,Iα1(z)=0iffzα1=-(α1-α0)2δ4α1,
for all z<(α0-α1)δ/2,
(4.24)Iα0(z)=Iα1(z)+z;
(ii)(4.25)Îα0(z)=0iffẑα0=(α0-α1)2δ4α0,Îα1(z)=0iffẑα1=-(α1-α0)2δ4α1,
for all z>(α0-α1)δ/2,
(4.26)Îα0(z)=Îα1(z)+z.Proof of Theorems2.2 and 2.3.
Since the proofs of the two theorems are similar, we only prove Theorem2.2. 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 ξ(a)∈(zα0,(α0-α1)δ/2)) such that
(4.27)lim⁡T→∞1Tlog⁡Pα0δ(1Tlog⁡dPα1δdPα0δ|F[0,T]≥ξ(a))=-a.
It is clear thatξ(a) is increasing for a>0, and by Lemma 4.5, we get Iα0(zα1)=zα1, which implies ξ(zα1)=zα1. Hence for a∈(0,zα1), we have ξ(a)∈(0,zα1), and since Iα1(z) is nonincreasing for z≤ξ(a), therefore we get
(4.28)Iα1(ξ(a))=inf⁡{Iα1(z):z≤ξ(a)}.
Similarly, for a∈(zα1,+∞), we have ξ(a)∈(zα1,(α0-α1)δ/2), and since Iα1(z) is nondecreasing for z≥ξ(a), therefore we get
(4.29)Iα1(ξ(a))=inf⁡{Iα1(z):z≥ξ(a)},
which complete the proof of Theorem 2.2.



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*Source: 741701-2011-09-11.xml*