S Dachian

Yu A Kutoyants

L Yang

On Hypothesis Testing for Poisson Processes. Singular Cases

Keywords: MSC 2010 Classification: 62M02, 62F03, 62F05 Hypothesis testing, inhomogeneous Poisson processes, asymptotic theory, composite alternatives, singular situations

We consider the problem of hypothesis testing in the situation where the first hypothesis is simple and the second one is local onesided composite. We describe the choice of the thresholds and the power functions of different tests when the intensity function of the observed inhomogeneous Poisson process has two different types of singularity: cusp and discontinuity. The asymptotic results are illustrated by numerical simulations.

Introduction

This is the second part of the study devoted to hypothesis testing problems in the case when the observations are inhomogeneous Poisson processes. The first part was concerned with the regular (smooth) case [START_REF] Dachian | On hypothises testing for Poisson processes[END_REF], while this second part deals with non regular (singular) situations. We suppose that the 1 intensity function of the observed inhomogeneous Poisson process depends on the unknown parameter ϑ in a non regular way (for example, the Fisher information is infinite). The basic hypothesis is always simple (ϑ = ϑ 1 ) and the alternative is one-sided composite (ϑ > ϑ 1 ). In the first part we studied the asymptotic behavior of the Score Function test (SFT), of the General Likelihood Ratio test (GLRT), of the Wald test (WT) and of two Bayes tests (BT1 and BT2). It was shown that the tests SFT, GLRT and WT are locally asymptotically uniformly most powerful. In the present work we study the asymptotic behavior of the GLRT, WT, BT1 and BT2 in two non regular situations. More precisely, we study the tests when the intensity functions has a cusp-type singularity or a jump-type singularity. In both cases the Fisher information is infinite. The local alternatives are obtained by the reparametrization ϑ = ϑ 1 + uϕ n , u > 0. The rate of convergence ϕ n → 0 depends on the type of singularity. In the cusp case ϕ n ∼ n -1 2κ+1 , where κ is the order of the cusp, and in the discontinuous case ϕ n ∼ n -1 . Our goal is to describe the choice of the thresholds and the behavior of the power functions as n → ∞. The important difference between regular and singular cases is the absence of the criteria of optimality. This leads to a situation when the comparison of the power functions can be only done numerically. That is why we present the results of numerical simulations of the limit power functions and the comparison of them with the power functions with small and large volumes of observations (small and large n).

Recall that X = (X t , t ≥ 0) is an inhomogeneous Poisson process with intensity function λ (t), t ≥ 0, if X 0 = 0 and the increments of X on disjoint intervals are independent and distributed according to the Poisson law

P {X t -X s = k} = t s λ (t) dt k k! exp - t s λ (t) dt .
We suppose that the intensity function depends on some one-dimensional parameter, that is, λ (t) = λ (ϑ, t). The basic hypothesis is simple: ϑ = ϑ 1 , while the alternative is one-sided composite: ϑ > ϑ 1 .

The hypothesis testing problems for inhomogeneous Poisson processes were studied by many authors (see, for example, [START_REF] Léger | Hypothesis testing for a non-homogeneous Poisson process[END_REF], [START_REF] Fierro | Testing homogeneity for Poisson processes[END_REF], [START_REF] Dachian | On hypothises testing for Poisson processes[END_REF] and the references therein).

Preliminaries

We consider the model of n independent observations of an inhomogeneous Poisson process: X n = (X 1 , . . . , X n ), where X j = (X j (t) , 0 ≤ t ≤ τ ), j = 1, . . . , n, are Poisson processes with E ϑ X j (t) = Λ (ϑ, t) = t 0 λ (ϑ, s) ds.

We use here the same notations as in [START_REF] Dachian | On hypothises testing for Poisson processes[END_REF]. In particular, ϑ is one-dimensional parameter and E ϑ is the mathematical expectation in the case when the true value is ϑ. The intensity function is supposed to be separated from zero on [0, τ ]. The measures corresponding to Poisson processes with different values of ϑ are equivalent and the likelihood function is defined by the equality

L(ϑ, X n ) = exp n j=1 τ 0 ln λ (ϑ, t) dX j (t) -n τ 0 [λ (ϑ, t) -1] dt .
In non-regular situations we do not have a LAUMP test, and it is interesting to compare the power functions of different tests with the power function of the Neyman-Pearson test (N-PT). Let us recall the definition of the N-PT. Suppose that we have two simple hypotheses H 1 : ϑ = ϑ 1 and H 2 : ϑ = ϑ 2 and our goal is to construct a test ψn (X n ) of size ε, that is, a test with given probability of the error of the first kind E ϑ 1 ψn (X n ) = ε. As usually, the test ψn (X n ) is the probability to reject the hypothesis H 1 and, of course, to accept the hypothesis H 2 .

Let us denote the likelihood ratio statistic as

L (ϑ 2 , ϑ 1 , X n ) = L (ϑ 2 , X n ) /L (ϑ 1 , X n ) .
Then, by the Neyman-Pearson Lemma [13], the N-PT is

ψ * n (X n ) =      1, if L (ϑ 2 , ϑ 1 , X n ) > d ε , q ε , if L (ϑ 2 , ϑ 1 , X n ) = d ε , 0, if L (ϑ 2 , ϑ 1 , X n ) < d ε ,
where the constants d ε and q ε are solutions of the equation

P ϑ 1 (L (ϑ 2 , ϑ 1 , X n ) > d ε ) + q ε P ϑ 1 (L (ϑ 2 , ϑ 1 , X n ) = d ε ) = ε.
In this work we consider the construction of the tests in the following hypothesis testing problem

H 1 : ϑ = ϑ 1 , H 2 : ϑ > ϑ 1 , (1)
that is, we have a simple hypothesis against one-sided composite alternative.

The log likelihood ratio function can be written as follows:

ln L(ϑ, ϑ 1 , X n ) = n j=1 τ 0 ln λ (ϑ, t) λ (ϑ 1 , t) dX j (t) -n τ 0 [λ (ϑ, t) -λ (ϑ 1 , t)] dt.
The power function of a test ψn (

X n ) is β ψn , ϑ = E ϑ ψn (X n ), ϑ > ϑ 1 .
We denote K ε the class of tests ψn of asymptotic size ε:

K ε = ψn : lim n→∞ E ϑ 1 ψn (X n ) = ε .
In this work we study several tests which belong to the class K ε . To compare these tests by their power functions we consider, as usual, the approach of close or contiguous alternatives (since for any fixed alternative the power functions of all tests converge to the same value 1). We put ϑ = ϑ 1 + ϕ n u, where ϕ n = ϕ n (ϑ 1 ) > 0. Here ϕ n → 0 and the rate of convergence depends on the type of singularity of the intensity function. Now the initial problem of hypothesis testing can be rewritten as follows:

H 1 : u = 0, H 2 : u > 0. (2)
The considered tests are usually of the form ψn

= 1 {Yn(X n )>cε} + q ε 1 {Yn(X n )=cε} ,
where the constant c ε is defined with the help of the limit random variable Y (suppose that Y n =⇒ Y under hypothesis H 1 ) by

E ϑ 1 ψn = P ϑ 1 {Y n (X n ) > c ε } + q ε P ϑ 1 {Y n (X n ) = c ε } -→ P ϑ 1 {Y > c ε } = ε
if the limit random variable Y is continuous, and by

P ϑ 1 {Y > c ε } + q ε P ϑ 1 {Y = c ε } = ε if Y has distribution function with jumps.
The corresponding power function will be denoted

β ψn , u = E ϑ 1 +ϕnu ψn , u > 0,
and the comparison of the tests will be carried in terms of their limit power functions.

We consider two different non regular models. In both of them, the intensity function λ (ϑ, t) is not differentiable and the Fisher information is infinite. More precisely, we study the behavior of the tests in two situations. The first one is when the intensity function has a cusp-type singularity (it is continuous but not differentiable), and the second one is when it has a jumptype singularity (it is discontinuous). In both cases the intensity functions λ (ϑ, t) has no derivative at the point t = ϑ.

Note that these statistical models were already studied before in the problems of parameter estimation (see [START_REF] Dachian | Estimation of cusp location by Poisson observations[END_REF] for the cusp-type singularity and [START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF] for discontinuous intensity function), so here we concentrate on the properties of the tests. The main tool is, of course, the limit behavior of the normalized likelihood ratio function, which was already established before in the mentioned works but in a slightly different settings. The proofs given in this work are mainly based on the results presented in [START_REF] Dachian | Estimation of cusp location by Poisson observations[END_REF] and [START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF].

Recall that in the non regular cases considered in this work we do not have a LAUMP tests, and that is why a special attention is paid to numerical simulations of the limit power functions.

Cusp-type singularity

Suppose that the intensity function of the observed Poison processes is

λ (ϑ, t) = a |t -ϑ| κ + h (t) , 0 ≤ t ≤ τ, ϑ ∈ Θ = [ϑ 1 , b),
where κ ∈ (0, 1/2), ϑ 1 > 0, b ≤ τ , a = 0, and h (•) is a known positive bounded function.

To study the local alternatives we introduce the normalizing function

ϕ n = n -1 2H Γ -1 H ϑ 1 , Γ 2 ϑ 1 = 2a 2 B (κ + 1, κ + 1) h (ϑ 1 ) 1 cos (πκ) -1 ,
where B (•, •) is the Beta-function and H = κ + 1 2 is the Hurst parameter. As usually, the change of variables ϑ = ϑ 1 + ϕ n u reduces the initial hypothesis testing problem (1) to the problem [START_REF] Dachian | Hypotheses testing : Poisson versus stress-release[END_REF].

We introduce the stochastic process

Z (u) = exp W H (u) - |u| 2H 2 , u ∈ R,
where W H (•) is a fractional Brownian motion. Further, we define the random variable û by the relation

Z (û) = sup v≥0 Z (u) ,
and we introduce h ε and g ε as the solutions of the equations

P (Z (û) > h ε ) = ε and P (û > g ε ) = ε (3)
respectively. Note that (Z (u) , u ≥ 0) is the likelihood ratio of a similar hypothesis testing problem (u = 0 against u > 0) in the case of observations (Y (t) , t ≥ 0) of the following type

dY (t) = 1 {t<u} dt + dW H (t) , t ≥ 0.
The uniformly most powerful test in this problem does not exist, and we do not have a LAUMP tests in our problem.

GLRT

The GLRT is defined by the relations

ψn (X n ) = 1 {Q(X n )>hε} ,
where h ε is the solution of the first of the equations (3),

Q (X n ) = sup ϑ>ϑ 1 L (ϑ, ϑ 1 , X n ) = L θn , ϑ 1 , X n ,
and θn is the maximum likelihood estimator (MLE).

Let us introduce the function

β (u * ) = P sup u>0 W H (u) - |u -u * | 2H 2 > ln h ε - |u * | 2H 2 , u * ≥ 0.
The properties of the GLRT are given in the following Proposition.

Proposition 1. The GLRT ψn (X n ) belongs to K ε and its power function in the case of local alternatives ϑ = ϑ 1 + ϕ n u * , u * > 0, has the following limit:

β ψn , u * -→ β (u * ) .
Proof. Introduce the normalized likelihood ratio process

Z n (u) = L (ϑ 1 + ϕ n u, X n ) = L (ϑ 1 + ϕ n u, X n ) L (ϑ 1 , X n ) , u ∈ U + n = 0, ϕ -1 n (b -ϑ 1 ) ,
and let the function Z n (u) be linearly decreasing to zero on the interval [ϕ -1 n (bϑ 1 ) , ϕ -1 n (bϑ 1 ) + 1] and equal to 0 for all u > ϕ -1 n (bϑ 1 ) + 1. Now the random function Z n (•) is defined on R + .

Let us fix some d ≤ 0 and denote C d = C d (R d ) the space of continuous functions on R d = [d, ∞) with the property lim v→∞ z (v) = 0. Introduce the uniform metric on this space and denote B the corresponding Borel σ-algebra.

When we study the likelihood ratio process under hypothesis H 1 , we take d = 0 and consider the corresponding measurable space (C 0 , B). Under the alternative ϑ = ϑ u * = ϑ 1 + ϕ n u * , u * > 0, we will use this space with d = -u * .

Let Q be the measure induced on the measurable space (C 0 , B) by the stochastic processes (Z (u) , u ≥ 0), and

Q (ϑ)
n be the measure induced (under the true value ϑ) on the same space by the processes (Z n (u) , u ≥ 0). The continuity with probability 1 of the random functions (Z n (u) , u ≥ 0) follows from the inequality (6) below and the Kolmogorov theorem.

Suppose that we already proved the following weak convergence

Q (ϑ 1 ) n =⇒ Q. ( 4
)
Then the distribution of any continuous in the uniform metric functional Φ (Z n ) converge to the distribution of Φ (Z). In particular, if we take

Φ (z) = sup u≥0 z (u) -h ε ,
we obtain

P ϑ 1 sup ϑ>ϑ 1 L (ϑ, ϑ 1 , X n ) > h ε = P ϑ 1 sup u>0 Z n (u) > h ε -→ P sup u>0 Z (u) > h ε = P {Z (û) > h ε } = ε.
Therefore the test ψn ∈ K ε . Let us note, that we do not know an analytic solution of the equation defining the constant h ε , that is why below we turn to numerical simulations (see Section 3.4). Note also that h ε = h ε (H) and does not depend on Γ ϑ 1 .

To study the power function we consider the same likelihood ratio process but under the alternative ϑ u * = ϑ 1 + ϕ n u * . We can write

Z n (u) = L (ϑ 1 + ϕ n u, X n ) L (ϑ 1 , X n ) = L (ϑ u * , X n ) L (ϑ 1 , X n ) L (ϑ 1 + ϕ n u, X n ) L (ϑ u * , X n ) = L (ϑ u * -ϕ n u * , X n ) L (ϑ u * , X n ) -1 L (ϑ u * + (u -u * )ϕ n , X n ) L (ϑ u * , X n ) = Zn (-u * ) -1 Zn (u -u * )
with an obvious notation. The difference between Z n (•) and Zn (•) is that the "reference value" in the first case is fixed (is equal to ϑ 1 ) and in the second case it is "moving" (is equal to ϑ u * = ϑ 1 + ϕ n u * ). The random variable Zn (-u * ) converge in distribution to Z (-u * ). For the stochastic process ( Zn (uu * ) , u ≥ 0) we have a similar convergence, and so, for any fixed u ≥ 0, we have

Zn (-u * ) , Zn (u -u * ) =⇒ (Z (-u * ) , Z (u -u * )) .
Now, let Q be the measure induced on the measurable space (C -u * , B) by the stochastic processes (Z (u) , u ≥ -u * ), and Q(ϑ) n be the measure induced (under the true value ϑ) on the same space by the stochastic processes ( Zn (u) , u ≥ -u * ). Suppose that we already proved the weak convergence

Q(ϑu * ) n =⇒ Q. ( 5
)
Then for the power function we can write

P ϑu * sup u>0 Z n (u) > h ε = P ϑu * Zn (-u * ) -1 sup u>0 L (ϑ u * + (u -u * )ϕ n , X n ) L (ϑ u * , X n ) > h ε -→ P Z (-u * ) -1 sup u>0 exp W H (u -u * ) - |u -u * | 2H 2 > h ε = P sup u>0 -W H (-u * ) + W H (u -u * ) - |u -u * | 2H 2 + |u * | 2H 2 > ln h ε = P sup u>0 W H (u) - |u -u * | 2H 2 > ln h ε - |u * | 2H 2 = β (u * ) .
This limit power function is obtained below with the help of numerical simulations (see Section 3.4).

Let us also note that the limit (under the alternative

ϑ u * = ϑ 1 + ϕ n u * ) of the likelihood ratio process (Z n (u) , u ≥ 0) is the process (Z (u, u * ) , u ≥ 0) defined by Z(u, u * ) = W H (u) - |u -u * | 2H 2 + |u * | 2H 2 .
To finish the proof we need to verify the convergence [START_REF] Fierro | Testing homogeneity for Poisson processes[END_REF]. To do this we follow the proof of the convergence (4) given in [START_REF] Dachian | Estimation of cusp location by Poisson observations[END_REF]. We introduce the following relations.

2. There exists a positive constant C such that

E ϑu * Z1/2 n (u 2 ) -Z1/2 n (u 1 ) 2 ≤ C |u 2 -u 1 | 2H , u 1 , u 2 ≥ -u * . (6)
3. There exists a positive constant c such that

E ϑu * Z1/2 n (u) ≤ exp -c |u -u * | 2H , u ≥ -u * . (7)
The proofs of these relations are slight modifications of the proofs given in [START_REF] Dachian | Estimation of cusp location by Poisson observations[END_REF]. Note that the characteristic function of the vector Zn (u 1 ) , . . . , Zn (u k ) can be written explicitly and the convergence of this characteristic function to the corresponding limit characteristic function can be checked directly (see Lemma 5 of [START_REF] Dachian | Estimation of cusp location by Poisson observations[END_REF]). The inequalities ( 6) and ( 7) follow from the Lemma 6 and Lemma 7 of [START_REF] Dachian | Estimation of cusp location by Poisson observations[END_REF] respectively.

These relations allow us to obtain the weak convergence ( 5) by applying the Theorem 1.10.1 of [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF]. Note that the convergence ( 4) is a particular case of ( 5) with u * = 0.

Wald test

Recall that the MLE θn is defined by the equation

L θn , ϑ 1 , X n = sup ϑ∈Θ L (ϑ, ϑ 1 , X n ) .
The Wald test (WT) has the following form:

ψ • n (X n ) = 1 {ϕ -1 n ( θn-ϑ1)>gε} ,
where g ε is the solution of the second of the equations (3). Introduce as well the random variable û * as solution of the equation

Z (û * ) = sup u≥-u * Z (u) .
Proposition 2. The WT ψ • n (X n ) belongs to K ε and its power function in the case of local alternatives ϑ = ϑ 1 + ϕ n u * , u * > 0, has the following limit:

β (ψ • n , u * ) -→ β • (u * ) = P (û * > g ε -u * ) .
Proof. The MLE (under hypothesis H 1 ) converges in distribution

ϕ -1 n θn -ϑ 1 =⇒ û. Hence ψ • n ∈ K ε .
For the proof see [START_REF] Dachian | Estimation of cusp location by Poisson observations[END_REF]. Recall that this convergence is a consequence of the weak convergence [START_REF] Daley | An Introduction to the Theory of Point Processes[END_REF].

Let us study this estimator under the alternative ϑ u * = ϑ 1 +ϕ n u * , u * > 0. We have

P ϑu * ϕ -1 n θn -ϑ u * < x = P ϑu * sup ϕ -1 n (θ-ϑu * )<x L (θ, ϑ u * , X n ) > sup ϕ -1 n (θ-ϑu * )≥x L (θ, ϑ u * , X n ) = P ϑu * sup -u * ≤u<x Zn (u) > sup u≥x Zn (u) -→ P sup -u * ≤u<x Z (u) > sup u≥x Z (u) = P (û * < x) .
Here, as before,

Zn (u) = L (ϑ u * + ϕ n u, X n ) L (ϑ u * , X n ) , u ≥ -u * .
Now, the limit of the power function of the WT is deduced from this convergence:

β (ψ • n , u * ) = P ϑu * ϕ -1 n θn -ϑ u * + u * > g ε -→ P {û * > g ε -u * } = β • (u * ) ,
which concludes the proof.

Let us note, that we can also give another representation of the limit power function using the process (Z (u, u * ) , u ≥ 0):

β (ψ • n , u * ) -→ P {û * > g ε } = β • (u * ) ,
where û * is solution of the equation

Z (û * ) = sup u≥0 Z (u, u * ) .
The threshold g ε and the power function β • (•) are obtained below by numerical simulations (see Section 3.4).

Bayes tests

Suppose that the parameter ϑ is a random variable with a priori density p (θ), ϑ 1 ≤ θ < b. This function is supposed to be continuous and positive.

We consider two Bayes tests. The first one is based on the Bayes estimator, while the second one is based on the averaged likelihood ratio.

The first test, which we call BT1, is similar to WT, but is based on the Bayes estimator (BE) rather than on the MLE. Suppose that the loss function is quadratic. Then the BE θn is given by the following conditional expectation:

θn = b ϑ 1 θp (θ|X n ) dθ = b ϑ 1 θp (θ) L (θ, X n ) dθ b ϑ 1 p (θ) L (θ, X n ) dθ . We introduce the test BT1 as ψn (X n ) = 1 {ϕ -1 n ( θn-ϑ1)>kε} ,
where the constant k ε is solution of the equation

P (ũ > k ε ) = ε, ũ = ∞ 0 vZ (v) dv ∞ 0 Z (v) dv .
Introduce as well the function

β (u * ) = P (ũ * > k ε -u * ) , ũ * = ∞ -u * vZ (v) dv ∞ -u * Z (v) dv , u * ≥ 0.
Proposition 3. The BT1 ψn (X n ) belongs to K ε and its power function in the case of local alternatives ϑ = ϑ 1 + ϕ n u * , u * > 0, has the following limit:

β ψn , u * -→ β (u * ) .
Proof. The Bayes estimator θn is consistent and has the following limit distribution (under hypothesis H 1 )

ϕ -1 n θn -ϑ 1 =⇒ ũ
(for the proof see [START_REF] Dachian | Estimation of cusp location by Poisson observations[END_REF]). Hence ψn (X n ) ∈ K ε .

For the power function we have

β ψn , u * = P ϑu * ϕ -1 n θn -ϑ 1 > k ε = P ϑu * ϕ -1 n θn -ϑ u * > k ε -u * .
Let us study the normalized difference ũn = ϕ -1 n θnϑ u * . We can write (using the change of variables

θ = ϑ u * + ϕ n v) b ϑ 1 θp (θ) L (θ, ϑ u * , X n ) dθ = ϕ n ϕ -1 n (b-ϑu * ) -u * (ϑ u * + ϕ n v) p (ϑ u * + ϕ n v) L (ϑ u * + ϕ n v, ϑ u * , X n ) dv = ϕ n ϕ -1 n (b-ϑu * ) -u * (ϑ u * + ϕ n v) p (ϑ u * + ϕ n v) Zn (v) dv. Hence ũn = ϕ -1 n (b-ϑu * ) -u * vp (ϑ u * + ϕ n v) Zn (v) dv ϕ -1 n (b-ϑu * ) -u * p (ϑ u * + ϕ n v) Zn (v) dv =⇒ ∞ -u * vZ (v) dv ∞ -u * Z (v) dv = ũ * (since p (ϑ u * + ϕ n v) -→ p (ϑ 1
) > 0 and Zn =⇒ Z). The detailed proof is based on the properties 1-3 of the likelihood ratio (see [START_REF] Dachian | Estimation of cusp location by Poisson observations[END_REF] or [7, Theorem 1.10.2]).

Let us note, that we can also give another representation of the limit power function using the process (Z (u, u * ) , u ≥ 0):

β ψn , u * -→ P (ũ * > k ε ) = β (u * ) , where ũ * = ∞ 0 vZ (v, u * ) dv ∞ 0 Z (v, u * ) dv .
The second test, which we call BT2, is given by

ψ⋆ n (X n ) = 1 {Rn(X n )>mε} , R n (X n ) = Ln (X n ) p (ϑ 1 ) ϕ n . Here Ln (X n ) = b ϑ 1 L (θ, ϑ 1 , X n ) p (θ) dθ,
and m ε is solution of the equation

P ∞ 0 Z (v) dv > m ε = ε.
Introduce as well the function

β⋆ (u * ) = P Z (-u * ) -1 ∞ -u * Z (v) dv > m ε .
Proposition 4. The BT2 ψ⋆ n (X n ) belongs to K ε and its power function in the case of local alternatives ϑ = ϑ 1 + ϕ n u * , u * > 0, has the following limit:

β ψ⋆ n , u * -→ β⋆ (u * ) .
Proof. Let us first recall how this test was obtained. Introduce the mean error of the second kind ᾱ ψn under alternative H 2 of an arbitrary test ψn as

ᾱ ψn = b ϑ 1 E θ ψn (X n ) p (θ) dθ = E ψn ,
where E is the double mathematical expectation, that is, the expectation with respect to the measure

P (X n ∈ A) = b ϑ 1 P θ (X n ∈ A) p (θ) dθ.
If we consider the problem of the minimization of this mean error, we reduce the initial hypothesis testing problem to the problem of testing of two simple hypotheses

H 1 : X n ∼ P ϑ 1 , H 2 : X n ∼ P .
Then, by the Neyman-Pearson Lemma, the most powerful test in the class K ε which minimizes the mean error ᾱ ψn is

ψ * n (X n ) = 1 { Ln(X n )> mε} ,
where the averaged likelihood ratio

Ln (X n ) = d P d P ϑ 1 (X n ) = ϕ n ϕ -1 n (β-ϑ 1 ) 0 Z n (v) p (ϑ 1 + vϕ n ) dv
and mε is chosen from the condition ψ * n ∈ K ε . Now, it is clear that the BT2 ψ⋆ n (X n ) coincides with the test ψ * n (X n ) if we put mε = m ε p (ϑ 1 ) ϕ n . In the proof of the convergence in distribution of the Bayes estimator it is shown (see [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF]Theorem 1.10.2] and [START_REF] Dachian | Estimation of cusp location by Poisson observations[END_REF]) that

ϕ -1 n L (X n ) =⇒ p (ϑ 1 ) ∞ 0 Z (v) dv.
Therefore (under hypothesis H 1 ),

R n (X n ) =⇒ ∞ 0 Z (v) dv
and the test ψ⋆ n (X n ) belongs to the class K ε . Using a similar argument, we can verify the convergence

R n (X n ) =⇒ Z (-u * ) -1 ∞ -u * Z (v) dv
under the alternative ϑ u * , which concludes the proof.

Simulations

Let us consider the following example. We observe n independent realizations X n = (X 1 , . . . , X n ), where X j = (X j (t), t ∈ [0, 2]), j = 1, . . . , n, of an inhomogeneous Poisson process. The intensity function of this processes is

λ(ϑ, t) = 2 -|t -ϑ| 0.4 , 0 ≤ t ≤ 2,
where the parameter ϑ ∈ [0.5, 2). We take ϑ 1 = 1.5 as the value of the basic hypothesis H 1 . Of course it is sufficient to have simulations for the values ϑ ∈ [1.5, 2), but we consider a wider interval to show the behavior of the likelihood ratio on the both sides of the true value. The Hurst parameter is H = 0.9 and the constant Γ 2 [START_REF] Fierro | Testing homogeneity for Poisson processes[END_REF], and its zoom Z n (u), u ∈ [0.1, 0.5], under the hypothesis H 1 are given in Figure 1.

ϑ 1 = B(1.4, 1.4) 1 cos(0.4π) -1 ≈ 1.027. A realization of the normalized likelihood ratio Z n (u), u ∈ [-5,
Here Figure 1 To find the thresholds of the GLRT and of the WT, we need to find the point of maximum and the maximal value of this function. In the case of the chosen intensity function, the maximum is attained at one of the cusps of the likelihood ratio (that is, on one of the events of one of the observed Poisson processes).

It is interesting to note that if the intensity function has the same singularity but with a different sign: λ (ϑ, t) = 0.5 + |t -ϑ| 0.4 , then it is much more difficult to find the maximum (see Figure 2).

Here Figure 2.

The thresholds of the GLRT, of the WT and of the BT1 are presented in Table 1 For example, the thresholds of the GLRT are obtained by simulating M = 10 5 trajectories of Z i (u), u ∈ [0, 20], i = 1, . . . , M (when u > 20 the value of Z i (u) is negligible), calculating for each of them the quantity sup u Z i (u), and taking the (1ε)M-th greatest between them.

For the computation of the power function we calculate the frequency of accepting the alternative hypothesis. For example, for the GLRT we use

β ψn , u ≈ 1 N N i=1 1 sup v>0 Z n,i (v)>hε
.

We can see in Figure 3 that, like in the regular case, for the small values of u the power function of the WT converge more slowly than that of the GLRT, but still more quickly than that of the BT1. When u is large, the power function of the BT1 converge more quickly than that of the WT, and the power function of the GLRT converge the most slowly.

Here Figure 3.

Since analytic expressions for the power functions of these three tests are not yet available, we compare them with the help of numerical simulations. It is equally interesting to compare them to the Neyman-Pearson Test (N-PT) constructed in the following problem of testing of two simple hypotheses.

Let us fix an alternative ϑ u * = ϑ 1 +u * ϕ n > ϑ 1 and consider the hypothesis testing problem

H 1 : u = 0, H 2 : u = u * .
The Neyman-Pearson test is

ψ * n (X n ) = 1 {Zn(u * )>dε}
, where the threshold d ε is the solution of the equation

P (Z (u * ) > d ε ) = ε. Recall that Z n (u * ) =⇒ Z(u * ) and Z (u * ) = exp W H (u * ) - u 2H * 2 .
Hence

P (Z (u * ) > d ε ) = P W H (u * ) - u 2H * 2 > ln d ε = P ζ > ln d ε + u 2H * 2 u H * and d ε = e zεu H * - u 2H * 2
, where ζ ∼ N (0, 1) and

P (ζ > z ε ) = ε.
Of course, it is impossible to use this N-PT in our initial problem, since u * (the value of u under alternative) is unknown. However, as this test is the most powerful in the class K ε , its power (as function of u * ) shows an upper bound for power functions of all the tests, and the distances between it and the power functions of the studied tests provide useful information.

To study the likelihood ratio function under the alternative we write

Z n (u * ) = L (ϑ 1 + u * ϕ n , X n ) L (ϑ 1 , X n ) = L (ϑ 1 + u * ϕ n -u * ϕ n , X n ) L (ϑ 1 + u * ϕ n , X n ) -1
.

So, for the power of the N-PT, we obtain

β(ψ * n , u * ) = P ϑ 1 +u * ϕn (Z n (u * ) > d ε ) -→ P Z (-u * ) -1 > d ε = P exp -W H (-u * ) + u 2H * 2 > d ε = P ζ > ln d ε -u 2H * 2 u H * = P ζ > z ε -u H * .
Here Figure 4.

We can see that the limit power function of the GLRT is the closest one to the limit power function of the N-PT. When u is small, the limit power function of the BT1 is lower than that of the GLRT. It becomes closer to that of the N-PT when u increases. At the same time, the limit power function of the WT become the lowest one. Let as also mention that the limit power function of the BT1 arrives faster to 1 than the others (see Figure 4).

Discontinuous intensity

Here we consider a similar hypothesis testing problem in the case of inhomogeneous Poisson processes with discontinuous intensity function. Suppose that the intensity function λ (ϑ, t), 0 ≤ t ≤ τ , of the observed Poisson processes satisfies the following condition.

S . The intensity function λ(ϑ, t) = λ(t-ϑ), where the unknown parameter

ϑ ∈ Θ = [ϑ 1 , b) ⊂ (0, τ ), the function λ(s), s ∈ [-b, τ -ϑ 1 ]
, is continuously differentiable everywhere except at the point t * ∈ (-ϑ 1 , τb) and this function has a jump at the point t * (and so, the intensity function λ(ϑ, t) has a jump at the point t = t * + ϑ ∈ (0, τ )).

We have to test the hypotheses

H 1 : ϑ = ϑ 1 , H 2 : ϑ > ϑ 1 .
We study the same tests as before (GLRT, WT, BT1 and BT2), and our goal is to chose the thresholds so, that these tests belong to the class K ε . Let us denote λ(t * +) = λ + , λ(t * -) = λ -and ρ = λ - λ + . To compare the power functions of the tests, we consider local alternatives which in this problem are given by ϑ = ϑ 1 + uϕ n , ϕ n = 1 nλ + . The initial problem is thus reduced to the following one

H 1 : u = 0, H 2 : u > 0.
Recall that the normalized likelihood ratio

Z n (u) = L (ϑ 1 + uϕ n , X n ) L (ϑ 1 , X n ) , u ∈ U + n = 0, nλ + (b -ϑ 1 ) ,
under the hypothesis H 1 converges to the process

Z (u) = exp {ln ρ x * (u) -(ρ -1) u} , u ≥ 0,
where (x * (u) , u ≥ 0) is a Poisson process of unit intensity (see [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes, Armenian Academy of Sciences[END_REF]). As we will see below, the limit likelihood ratio under the alternative

ϑ 1 + u * ϕ n , u * > 0, is Z (u, u * ) = exp {ln ρ x * (u, u * ) -(ρ -1) u} , u ≥ 0,
where (x * (u, u * ) , u ≥ 0) is a Poisson process with "switching" intensity function µ (u, u * ) = ρ 1 {u<u * } + 1 {u≥u * } , u ≥ 0.

Note that the limit likelihood ratio of our problem is the likelihood ratio of a similar hypothesis testing problem (u = 0 against u > 0) in the case of observations of a Poisson process (Y (t) , t ≥ 0) with "switching" intensity function µ (t, u) = ρ 1 {t<u} + 1 {t≥u} , t ≥ 0.

Weak convergence

The considered tests (GLRT, WT, BT1 and BT2) are functionals of the likelihood function L (•, X n ). As it was shown above, all these tests can be written as functionals of the normalized likelihood ratio Z n (•). Therefore, as in regular and cusp-type cases, we have to prove the weak convergence of the measures induced by the normalized likelihood ratio under hypothesis (to find the thresholds) and under alternative (to describe the power functions).

Let D 0 be the space of functions z(•) on R + = [0, +∞) which do not have discontinuities of the second kind and which are such that lim v→∞ z(v) = 0.

We suppose that the functions z(•) ∈ D 0 are càdlàg, that is, the left limit z(t-) = lim sրt z(s) exists and the right limit z(t+) = lim sցt z(s) exists and equals to z(t). Introduce the distance between two function z 1 (•) and z 2 (•) as

d(z 1 , z 2 ) = inf ν sup u∈R + z 1 (u) -z 2 ν(u) + sup u∈R + |u -ν(u)| ,
where the inf is taken over all monotone continuous one-to-one mappings ν : R + -→ R + . Let us also denote

∆ h (z) = sup u∈R + sup δ min z(u ′ ) -z(u) , z(u ′′ ) -z(u) + sup |u|>1/h z(u) ,
where the second sup is taken over the intervals

δ = u ′ , u ′′ ⊆ [u -h, u + h) such that u ∈ δ.
Suppose that we have a sequence (Y n ) n≥1 of stochastic processes with Y n = (Y n (u), u ∈ [0, +∞)) and a process Y 0 = (Y 0 (u), u ∈ [0, +∞)) such that the realizations of these processes belong to the space D 0 . Denote Q (ϑ) n and Q (ϑ) the distributions (which we suppose depending on a parameter ϑ ∈ Θ) induced on the measurable space (D 0 , B) by these processes. Here B is the Borel σ-algebra of the metric space D 0 . Theorem 1. If, as n → ∞, the finite dimensional distributions of the process Y n converge to the finite dimensional distributions of the process Y 0 uniformly in ϑ ∈ Θ and for any δ > 0 we have

lim h→0 lim n→∞ sup ϑ∈Θ Q (ϑ) n {∆ h (Y n ) > δ} = 0, ( 8
)
then

Q (ϑ) n =⇒ Q (ϑ) uniformly in ϑ ∈ Θ as n → ∞.
For the proof see [START_REF] Gikhman | Introduction to the Theory of Random Processes[END_REF], Theorem 9.5.2.

Recall that such a weak convergence of the likelihood ratio process Z n (•) for the discussed model of inhomogeneous Poisson process was already established in [8, Sections 4.4 and 5.4.3] (see as well [START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF]Chapter 5] for similar results). The proof given there corresponds to the weak convergence in the space (D 0 , B) of Z n (•) under hypothesis H 1 . The limit process under the alternative ϑ 1 + u * ϕ n is different and we study it below in order to describe the power functions.

Let now Q be the measure induced on the measurable space (D 0 , B) by the stochastic processes (Z (u, u * ) , u ≥ 0), and Q (ϑ) n be the measure induced (under the true value ϑ) on the same space by the processes (Z n (u) , u ≥ 0). Proposition 5. Let the condition S be fulfilled. Then, under the alternative ϑ u * = ϑ 1 + u * ϕ n , we have the convergence

Q (ϑu * ) n =⇒ Q. ( 9
)
The proof is based on several lemmas, where we verify the convergence of the finite-dimensional distributions and the condition [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes, Armenian Academy of Sciences[END_REF]. As in [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes, Armenian Academy of Sciences[END_REF], we follow the main steps of the proof of Ibragimov and Khasminskii [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF] of a similar convergence in the case of i.i.d. observations. Lemma 1. Let the condition S be fulfilled. Then, under the alternative ϑ u * , the finite-dimensional distributions of the process (Z n (u), u ≥ 0) converge to those of the process (Z(u, u * ), u ≥ 0).

Proof. The characteristic function of ln Z n (u) is (see [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes, Armenian Academy of Sciences[END_REF]):

E ϑu * exp {iµ ln Z n (u)} = exp n τ 0 exp iµ ln λ(t -ϑ 1 -uϕ n ) λ(t -ϑ 1 ) -1 λ(t -ϑ 1 -u * ϕ n ) dt- -inµ τ 0 λ(t -ϑ 1 -uϕ n ) -λ(t -ϑ 1 ) dt = exp n τ 0 A n (u, u * , t) dt -inµ τ 0 B n (u, u * , t) dt
where we denoted

A n (u, u * , t) = exp iµ ln λ u λ 0 -1 -iµ ln λ u λ 0 λ u * , B n (u, u * , t) = λ u -λ 0 -λ 0 ln λ u λ 0 + (λ 0 -λ u * ) ln λ u λ 0 and λ v = λ (t -ϑ 1 -vϕ n ) for v ≥ 0.
We consider two cases: u ≤ u * and u > u * . Let u ≤ u * and 0 ≤ t ≤ t * +ϑ 1 . Then the functions λ 0 , λ u and λ u * are continuously differentiable and, using Taylor expansion, we obtain the estimates

|B n (u, u * , t)| dt ≤ Cu 2 (u * -u) n 3 + Cu (u * -u) n 2 .
So, the main contribution comes from the integrals

n t * +ϑ 1 +uϕn t * +ϑ 1 exp iµ ln λ u λ 0 -1 λ u * dt -iµn t * +ϑ 1 +uϕn t * +ϑ 1 [λ u -λ 0 ] dt = u exp iµ ln λ - λ+ -1 λ - λ + -iuµ λ - λ + -1 + o (1) -→ u [exp (iµ ln ρ) -1] ρ -iuµ [ρ -1] ,
and we obtain, for u ≤ u * ,

E ϑu * exp {iµ ln Z n (u)} -→ exp u exp {iµ ln ρ} -1 ρ -iµ (ρ -1) = E exp {iµ ln Z(u, u * )} .
Now we consider the case u ≥ u * . As before, we obtain the convergence,

n t * +ϑ 1 0 + τ t * +ϑ 1 +uϕn |A n (u, u * , t)| + |B n (u, u * , t)| dt -→ 0.
For the intervals (t

* + ϑ 1 , t * + ϑ 1 + u * ϕ n ) and t * + ϑ 1 + u * ϕ n , t * + ϑ 1 + uϕ n ,
we can write

n t * +ϑ 1 +u * ϕn t * +ϑ 1 A n (u, u * , t) -iµB n (u, u * , t) dt -→ u * λ + exp iµ ln λ - λ + -1 λ --iµ λ --λ + = u * exp {iµ ln ρ} -1 ρ -iµ (ρ -1)
and

n t * +ϑ 1 +uϕn t * +ϑ 1 +u * ϕn A n (u, u * , t) -iµB n (u, u * , t) dt -→ u -u * λ + exp iµ ln λ - λ + -1 λ + -iµ λ --λ + .
So, for u > u * , we get

E ϑu * exp {iµ ln Z n (u)} -→ exp u * exp {iµ ln ρ} -1 ρ -iµ (ρ -1) + (u -u * ) exp {iµ ln ρ} -1 -iµ (ρ -1) = E exp {iµ ln Z(u, u * )} .
Therefore the one-dimensional distributions of the stochastic process Z n (•) converge to those of Z (•, u * ).

The convergence of arbitrary finite-dimensional distributions of Z n (•) to those of Z (•, u * ) can be proved in a similar manner. For example, in the case of two-dimensional distributions we can write (for u 1 < u 2 < u * )

E ϑu * exp iµ 1 ln Z n (u 1 ) + iµ 2 ln Z n (u 2 ) -→ exp (u 2 -u 1 ) exp {iµ 2 ln ρ} -1 ρ -iµ 2 (ρ -1) + u 1 exp i (µ 1 + µ 2 ) ln ρ -1 ρ -i (µ 1 + µ 2 ) (ρ -1) = E exp iµ 1 ln Z(u 1 , u * ) + iµ 2 ln Z(u 2 , u * ) .
So, the lemma is proved.

Further, we can write

Z n (u) = Z n (u * ) Z n (u), where Z n (u) = d P ϑ 1 +uϕn d P ϑ 1 +u * ϕn and Z n (u * ) = d P ϑ 1 +u * ϕn d P ϑ 1 .
Note that Z n (u * ) does not depend of u and we have the convergence (under the alternative

ϑ u * = ϑ 1 + uϕ n ) Z n (u * ) =⇒ Z * (u * ) = exp {ln ρ x ρ (u * ) -u * (ρ -1)} , (10)
where (x ρ (u) , u ≥ 0) is a Poisson process of intensity ρ. Therefore, to prove [START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF], it is sufficient to study the convergence of the measures induced by the stochastic process ( Z n (u), u ≥ 0).

Lemma 2. Let the conditions S be fulfilled. Then there exists a constant C > 0, such that

E ϑ 1 +u * ϕn Z 1/2 n (u 1 ) -Z 1/2 n (u 2 ) 2 ≤ C |u 1 -u 2 | (11)
for all u * , u 1 , u 2 ∈ U + n . Proof. According to [9, Lemma 1.1.5], we have (for v 1 > v 2 > 0) E ϑ 1 +u * ϕn Z 1/2 n (u 1 ) -Z 1/2 n (u 2 ) 2 ≤ nτ 0 λ 1/2 (t -ϑ 1 -u 1 ϕ n ) λ 1/2 (t -ϑ 1 -u * ϕ n ) - λ 1/2 (t -ϑ 1 -u 2 ϕ n ) λ 1/2 (t -ϑ 1 -u * ϕ n ) 2 λ(t -ϑ 1 -u * ϕ n ) dt = n τ 0 λ 1/2 (t -ϑ 1 -u 1 ϕ n ) -λ 1/2 (t -ϑ 1 -u 2 ϕ n ) 2 dt = n t * +u 2 ϕn 0 + t * +u 1 ϕn t * +u 2 ϕn + τ t * +u 1 ϕn λ 1/2 u 1 -λ 1/2 u 2 2 dt = n(I 1 + I 2 + I 3 )
with obvious notations. As the functions λ u 1 and λ u 2 are continuously differentiable on the intervals [0, t * + u 2 ϕ n ] and [t * + u 1 ϕ n , τ ], we can write

λ 1 2 (ϑ 1 + u 1 ϕ n , t) -λ 1 2 (ϑ 1 + u 2 ϕ n , t) = (u 1 -u 2 ) ϕ n 2 λ (ϑ v , t) λ 1 2 (ϑ v , t)
, where v is some intermediate point between u 1 and u 2 . Therefore

n (I 1 + I 3 ) ≤ nϕ 2 n (u 1 -u 2 ) 2 4 t * +u 2 ϕn 0 + τ t * +u 1 ϕn λ (ϑ v , t) 2 λ (ϑ v , t) dt ≤ C nλ 2 + |u 1 -u 2 | 2 ≤ C |u 1 -u 2 |
where we took into account the inequality |u 1u 2 | ≤ Cn.

Since the function λ is bounded, we have the estimate

nI 2 ≤ n |u 1 -u 2 | nλ + C = C λ + |u 1 -u 2 | ,
and so the inequality [START_REF] Cam | Asymptotic Methods in Statistical Decision Theory[END_REF] holds with some constant C > 0.

Lemma 3. Let the condition S be fulfilled. Then there exists a constant k * > 0 such that

E ϑ 1 +u * ϕn Z 1/2 n (u) ≤ exp -k * |u -u * | (12)
for all u * , u ∈ U + n . Proof. According to [9, Lemma 1.1.5], we have

E ϑ 1 +u * ϕn Z 1/2 n (u) = exp - n 2 τ 0 λ 1/2 (t -ϑ 1 -uϕ n ) λ 1/2 (t -ϑ 1 -u * ϕ n ) -1 2 λ(t -ϑ 1 -u * ϕ n ) dt = exp - n 2 τ 0 λ 1/2 t -ϑ 1 -uϕ n -λ 1/2 (t -ϑ 1 -u * ϕ n ) 2 dt = exp - n 2 F n (u, u * )
with an obvious notation.

Let us consider separately the cases u ∈ D = {v : |vu * | < δnλ + } and u ∈ D c = {v : |vu * | ≥ δnλ + }. Here δ is some positive constant which will be chosen later. For simplicity we suppose that u > u * .

For the values u ∈ D we have

nF n (u, u * ) ≥ n t * +ϑ 1 +uϕn t * +ϑ 1 +u * ϕn λ 1/2 t -ϑ 1 -uϕ n -λ 1/2 (t -ϑ 1 -u * ϕ n ) 2 dt ≥ |u -u * | λ + inf t * +u * ϕn≤s≤t * +uϕn λ 1/2 s -uϕ n -λ 1/2 (s -u * ϕ n ) 2 ≥ |u -u * | 2λ + λ --λ + 2 = |u -u * | 2 ( √ ρ -1) 2
for sufficiently small δ. Further, note that for any ν > 0 we have

g (ν) = inf |s-s 0 |>ν τ 0 λ (t -ϑ 1 -s) -λ (t -ϑ 1 -s 0 ) 2 dt > 0.
Indeed, if g (ν) = 0 then for some s * we have λ (tϑ 1s * ) = λ (tϑ 1s 0 ) for all t ∈ [0, τ ], but this equality for discontinuous λ (•) and all t is impossible. Hence, for the values u ∈ D c we have

nF n (u, u * ) ≥ ng (δ) ≥ g (δ) |u -u * | C
where we took into account the inequality |uu * | ≤ Cn. So, the inequality ( 12) is proved.

The presented estimates [START_REF] Cam | Asymptotic Methods in Statistical Decision Theory[END_REF], [START_REF] Léger | Hypothesis testing for a non-homogeneous Poisson process[END_REF] and Lemma 1 allow us to finish the proof following the same lines as it was done in [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes, Armenian Academy of Sciences[END_REF], Section 5.4.3.

GLRT

The GLRT is based on the statistic

Q n (X n ) = sup ϑ≥ϑ 1 L (ϑ, ϑ 1 , X n ) = max L θn +, ϑ 1 , X n , L θn -, ϑ 1 , X n
(where θn is the MLE) and is of the form ψn (X n ) = 1 {Qn(X n )>hε} .

The threshold h ε is defined with the help of the convergence (under hypothesis

H 1 ) Q n (X n ) = sup u∈U + n Z n (u) =⇒ sup u>0 Z (u) = Ẑ.
Hence h ε = h ε (ρ) is solution of the equation

P Ẑ > h ε = ε.
Let us fix an alternative ϑ u * = ϑ 1 + u * ϕ n , u * > 0. Then for the power function we have

β ψn , u * = E ϑu * ψn (X n ) = P ϑu * sup u>0 Z n (u) > h ε -→ P sup u>0 Z (u, u * ) > h ε . Putting Y (u) = ln Z (u, u * ) = ln ρ x * (u, u * ) -(ρ -1) u, we can write sup u>0 Y (u) = max sup 0<u<u * Y (u) , Y (u * ) + sup u≥u * [Y (u) -Y (u * )] . Note that the Poisson process x (v) = x * (u * + v, u * ) -x * (u * , u * ), v ≥ 0, is independent from (x * (u, u * ) , 0 ≤ u ≤ u * ).
Hence we can write the following representation of the limit power function:

β ψn , u * -→ P max sup 0<u<u * Z * (u) , Z * (u * ) Z > h ε ,
where the random variable Z = sup v≥0 exp {ln ρ x * (v) -(ρ -1) v} is independent from (Z * (u) , 0 ≤ u ≤ u * ), and the process Z * (•) is defined as in [START_REF] Kutoyants | Introduction to Statistics of Poisson Processes[END_REF]. Let us note that this expression is useful for numerical simulation of the power function. It simplifies the calculations since the simulated values of Z can be used many times for different values of u * .

Wald test

The Wald test is based on the MLE θn . We already know that

ϕ -1 n θn -ϑ 1 =⇒ û,
where û is defined by the equation max

[Z (û+) , Z (û-)] = sup u>0 Z (u) . The Wald test is ψ • n (X n ) = 1 {ϕ -1 n ( θn-ϑ1)>gε} ,
where the threshold g ε = g ε (ρ) is solution of the equation

P {û > g ε } = ε.
For the power function we have (below

ϑ u * = ϑ 1 + u * ϕ n ) β (ψ • n , u * ) = E ϑu * ψ • n (X n ) = P ϑu * ϕ -1 n θn -ϑ 1 > h ε = P ϑu * sup ϕ -1 n (θ-ϑ 1 )>hε L (θ, X n ) > sup ϕ -1 n (θ-ϑ 1 )≤hε L (θ, X n ) = P ϑu * sup ϕ -1 n (θ-ϑ 1 )>hε L (θ, X n ) L (ϑ 1 , X n ) > sup ϕ -1 n (θ-ϑ 1 )≤hε L (θ, X n ) L (ϑ 1 , X n ) = P ϑu * sup u>hε Z n (u) > sup u≤hε Z n (u) -→ P sup u>hε Z (u, u * ) > sup u≤hε Z (u, u * ) = P {û * > h ε } ,
where the random variable û * is defined by the equation

max [Z (û * +, u) , Z (û * -, u)] = sup u≥0 Z (u, u * ) .
Let us note, that we can also give another representation of the power function using the limit (under the alternative ϑ u * ) of the normalized likelihood ratio L(ϑ u * + uϕ n , ϑ u * , X n ), u ≥ -u * . This limit is the stochastic process (Z ⋆ (u) , u ≥ -u * ) defined by

Z ⋆ (u) =    exp ln ρ x * (u) -(ρ -1) u if u ≥ 0, exp -ln ρ x ρ (-u)--(ρ -1) u if -u * ≤ u ≤ 0,
where (x * (u) , u ≥ 0) and (x ρ (u) , u ≥ 0) are Poisson processes of unit intensity and of intensity ρ respectively. Note that in [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes, Armenian Academy of Sciences[END_REF] this limit was established for a fixed value ϑ but, taking into account Section 4.1, it clearly holds for "moving" value ϑ u * . Note also that the positive real axis part (Z ⋆ (u) , u ≥ 0) of the process Z ⋆ (•) is nothing but the process Z (•). Now, we have

β (ψ • n , u * ) = E ϑu * ψ • n (X n ) = P ϑu * ϕ -1 n θn -ϑ u * + u * > h ε -→ P {û * > h ε -u * }
where û * is defined by the equation

max [Z ⋆ (û * +) , Z ⋆ (û * -)] = sup u≥-u * Z ⋆ (u) .

Bayes tests

Suppose that the parameter ϑ is a random variable with known probability density p (θ), ϑ 1 ≤ θ < b. This function is supposed to be continuous and positive.

We consider two Bayes tests. The first one is based on the Bayes estimator, while the second one is based on the averaged likelihood ratio.

The first test, which we call BT1, is similar to WT, but is based on the Bayes estimator (BE) θn rather than on the MLE:

ψn (X n ) = 1 {ϕ -1 n ( θn-ϑ1)>kε} , θn = b ϑ 1 θp (θ) L (θ, ϑ 1 , X n ) dθ b ϑ 1 θL (θ, ϑ 1 , X n ) dθ
.

The properties of the likelihood ratio established in Lemmas 1-3 allow us to justify the limit

E ϑ 1 ψn (X n ) -→ P {ũ > k ε } , ũ = ∞ 0 vZ (v) dv ∞ 0 Z (v) dv
.

The proof follows from the general results concerning the Bayes estimators described in [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF] (see as well [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes, Armenian Academy of Sciences[END_REF]).

For the power function, using the convergence (under the alternative ϑ u * ) of the process Z n (•) to the process Z (•, u * ), we obtain

β ψn , u * = P ϑu * ϕ -1 n θn -ϑ 1 > k ε -→ P ∞ 0 vZ (v, u * ) dv ∞ 0 Z (v, u * ) dv > k ε .
Let us note, that we can also give another representation of the limit power function using the process Z ⋆ (•). We have the convergence (under the alternative ϑ u * )

ϕ -1 n θn -ϑ 1 = ϕ -1 n θn -ϑ u * + u * =⇒ ∞ -u * vZ ⋆ (v) dv ∞ -u * Z ⋆ (v) dv + u * . Hence β ψn , u * -→ P ∞ -u * vZ ⋆ (v) dv ∞ -u * Z ⋆ (v) dv > k ε -u * .
The second test, which we call BT2, is the test which minimizes the mean error of the second kind. We have Here Figure 6.

ϕ -1 n L (X n ) = ϕ -1 n b ϑ 1 p (θ) L (θ, X n ) L (ϑ 1 , X n ) dθ =⇒ p (ϑ 1 ) ∞ 0 Z (
It is interesting to compare the studied tests with the Neyman-Pearson test (N-PT) corresponding to a fixed value u * of u. Of course, it is impossible to use this N-PT in our initial problem, since u * (the value of u under alternative) is unknown. Nevertheless, its power (as function of u * ) shows an upper bound for power functions of all the tests, and the distances between it and the power functions of studied tests provide an important information. Let us fix some value u * > 0 and introduce the N-PT Here x * (u * ) is a Poisson random variable with parameter u * , and so the quantities D ε and q ε can be computed numerically. A similar calculation yields the limit power of the N-PT:

β (ψ * n , u * ) -→ P (x * (u * , u * ) > D ε ) + q ε P (x * (u * , u * ) = D ε ) .
where x * (u * , u * ) is a Poisson random variable with parameter ρu * .

Here Figure 7.

The results of simulations are presented in Figure 7 for two cases: ε = 0.05 and ε = 0.4. In both cases the limit power function of the GLRT is the closest one to the limit power of the N-PT, and the limit power function of the BT1 arrives faster to 1 than the others.

[13] Lehmann, E. and Romano, J., Testing Statistical Hypotheses, Springer, Heidelberg, 2005.

[14] Pflug, G. C., On an argmax-distribution connected to the Poisson process, Proceedings of the Fifth Prague Conference on Asymtotic Statistics, P.Mandl and M. Huskova eds, 1993, 123-130.

[15] Pyke, R., The supremum and infimum of the Poisson process, Ann. Math. Statist., 1959, 30, 568-576.

  ψ * n (X n ) = 1 {Zn(u * )>dε} + q ε 1 {Zn(u * )=dε} ,where d ε and q ε are solutions of the equationP (Z (u * ) > d ε ) + q ε P (Z (u * ) = d ε ) = ε.Denoting D ε = (ln d ε + (ρ -1)u * ) / ln ρ, we can rewrite this equation asP (x * (u * ) > D ε ) + q ε P (x * (u * ) = D ε ) = ε.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 12345 Figure 1: A realization of Z n (•) with λ (ϑ, t) = 2 -|t -1.5| 0.4 and n = 1000

Figure 6 :Figure 7 :

 67 Figure 6: Power functions of GLRT, WT and BT1 in discontinuous case with ρ = 3

Table 1 :

 1 . Thresholds of GLRT, WT and BT1

	ε	0.01	0.05	0.10	0.2	0.4	0.5
	ln h ε 2.959 1.641 1.081 0.559 0.159 0.068
	g ε	3.041 1.996 1.521 0.950 0.333 0.166
	k ε	2.864 2.0776 1.720 1.365 1.005 0.885

Table 2 :

 2 v) dv. Thresholds of GLRT, WT and BT1

	ε	0.01 0.05 0.10 0.20 0.40 0.50
	ln h ε 4.242 2.607 1.922 1.120 0.573 0.191
	g ε	5.990 3.556 2.078 1.045 0.329 0.099
	k ε	6.669 3.937 2.983 2.132 1.402 1.196

The finite-dimensional distributions of ( Zn (u) , u ≥ -u * ) converge to those of (Z (u) , u ≥ -u * ).

Hence the test

with threshold m ε satisfying the equation

belongs to the class K ε and minimizes the mean error of the second kind.

Simulations

We consider n independent realizations X j = (X j (t), t ∈ [0, 4]), j = 1, . . . , n, of a Poisson process of intensity function

We take ϑ 1 = 3 and b = 4, and therefore

Recall that in this case the limit (under H 1 ) of the likelihood ratio is

where (x * (u) , u ≥ 0) is a Poisson process of unit intensity. A realization of this limit likelihood ratio or, more precisely, of the logarithm of its two-sided version Z ⋆ (•) and its zoom are given in Figure 5.

Here Figure 5.

Using this limit we obtain the threshold h ε of the GLRT as solution of the equation

It is convenient for simulations to transform the limit process as follows:

where Π (•) is a Poisson process of intensity γ = ln 3 2 < 1. Hence, the threshold h ε is determined by the equation

The distribution of sup t>0

[Π (t)t] is given by the well-known formula

obtained by Pyke in [15]. Note that there is equally an analytic expression for the distribution of the random variable t = argmax t≥0 [Π (t)t]. This expression was obtained by Pflug in [14] and is given by

where {η k } k∈N * is an i.i.d. sequence with common distribution

ν is a random variable independent of η k , k ∈ N * , and distributed according to geometric law

and we use the convention 0 k=1 η k = 0. Now, for the threshold g ε of the WT we can write

However, the numerical solution of this equation is not easy, and it is simpler to obtain the threshold g ε by Monte Carlo simulations. The thresholds are presented in Table 2.

Acknowledgements

This study was partially supported by Russian Science Foundation (research project No. 14-49-00079). The authors thank the Referee for helpful comments.


S Dachian

Yu A Kutoyants

L Yang

On Hypothesis Testing for Poisson Processes. Regular Case

Keywords: MSC 2010 Classification: 62M02, 62F03, 62F05 Hypothesis testing, inhomogeneous Poisson processes, asymptotic theory, composite alternatives, regular situation

We consider the problem of hypothesis testing in the situation when the first hypothesis is simple and the second one is local onesided composite. We describe the choice of the thresholds and the power functions of the Score Function test, of the General Likelihood Ratio test, of the Wald test and of two Bayes tests in the situation when the intensity function of the observed inhomogeneous Poisson process is smooth with respect to the parameter. It is shown that almost all these tests are asymptotically uniformly most powerful. The results of numerical simulations are presented.

Introduction

The hypothesis testing theory is a well developed branch of mathematical statistics [START_REF] Lehmann | Testing Statistical Hypotheses[END_REF]. The asymptotic approach allows to find satisfactory solutions 1 in many different situations. The simplest problems, like the testing of two simple hypotheses, have well known solutions. Recall that if we fix the first type error and seek the test which maximizes the power, then we obtain immediately (by Neyman-Pearson lemma) the most powerful test based on the likelihood ratio statistic. The case of composite alternative is more difficult to treat and here the asymptotic solution is available in the regular case. It is possible, using, for example, the Score Function test (SFT), to construct the asymptotically (locally) most powerful test. Moreover, the General Likelihood Ratio Test (GLRT) and the Wald test (WT) based on the maximum likelihood estimator are asymptotically most powerful in the same sense. In the non regular cases the situation became much more complex. First of all, there are different non regular (singular) situations. Moreover, in all these situations, the choice of the asymptotically best test is always an open question.

This work is an attempt to study all these situations on the model of inhomogeneous Poisson processes. This model is sufficiently simple to allow us to realize the construction of the well known tests (SFT, GLRT, WT) and to verify that these test are asymptotically most powerful also for this model, in the case when it is regular. In the next paper we study the behavior of these tests in the case when the model is singular. The "evolution of the singularity" of the intensity function is the following: regular case (finite Fisher information, this paper), continuous but not differentiable (cusp-type singularity, [START_REF] Dachian | On hypothesis testing for Poisson processes[END_REF]), discontinuous (jump-type singularity, [START_REF] Dachian | On hypothesis testing for Poisson processes[END_REF]). In all the three cases we describe the tests analytically. More precisely, we describe the test statistics, the choice of the thresholds and the behavior of the power functions for local alternatives.

Note that the notion of local alternatives is different following the type of regularity/singularity. Suppose we want to test the simple hypothesis ϑ = ϑ 1 against the one-sided alternative ϑ > ϑ 1 . In the regular case, the local alternatives are usually given by ϑ = ϑ 1 + u √ n , u > 0. In the case of a cusptype singularity, the local alternatives are introduced by ϑ = ϑ 1 + u n -1 2κ+1 , u > 0. As to the case of a jump-type singularity, the local alternatives are ϑ = ϑ 1 + u n , u > 0. In all these problems, the most interesting for us question is the comparison of the power functions of different tests. In singular cases, the comparison is done with the help of numerical simulations. The main results concern the limit likelihood ratios in the non-regular situations. Let us note, that in many other models of observations (i.i.d., time series, diffusion processes etc.) the likelihood ratios have the same limits as here (see, for example, [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF] and [START_REF] Dachian | On cusp estimation of ergodic diffusion process[END_REF]). Therefore, the results presented here are of more universal nature and are valid for any other (non necessarily Poissonian) model having one of considered here limit likelihood ratios.

We recall that X = (X t , t ≥ 0) is an inhomogeneous Poisson process with intensity function λ (t), t ≥ 0, if X 0 = 0 and the increments of X on disjoint intervals are independent and distributed according to the Poisson law

P {X t -X s = k} = t s λ (t) dt k k! exp - t s λ (t) dt .
In all statistical problems considered in this work, the intensity functions are periodic with some known period τ and depend on some one-dimensional parameter, that is, λ (t) = λ (ϑ, t). The basic hypothesis and the alternative are always the same: ϑ = ϑ 1 and ϑ > ϑ 1 . The diversity of statements corresponds to different types of regularity/singularity of the function λ (ϑ, t).

The case of unknown period τ needs a special study.

The hypothesis testing problems (or closely related properties of the likelihood ratio) for inhomogeneous Poisson processes were studied by many authors (see, for example, Brown [START_REF] Brown | Discrimination of Poisson processes[END_REF], Kutoyants [START_REF] Yu | Asymptotically optimal algorithm of detecting Poisson signals[END_REF], Léger and Wolfson [START_REF] Léger | Hypothesis testing for a non-homogeneous Poisson process[END_REF], Liese and Lorz [START_REF] Liese | Contiguity and LAN-property of sequences of Poisson processes[END_REF], Sung et al. [START_REF] Sung | Asymptotic locally optimal detector for large-scale sensor networks under the Poisson regime[END_REF], Fazli and Kutoyants [START_REF] Fazli | Two simple hypotheses testing for Poisson process[END_REF], Dachian and Kutoyants [START_REF] Dachian | Hypotheses testing : Poisson versus stress-release[END_REF] and the references therein). Note finally, that the results of this study will appear later in the work [START_REF] Kutoyants | Introduction to Statistics of Poisson Processes[END_REF].

Auxiliary results

For simplicity of exposition we consider the model of n independent observations of an inhomogeneous Poisson process: X n = (X 1 , . . . , X n ), where

X j = (X j (t) , 0 ≤ t ≤ τ ), j = 1, . . . , n, are Poisson processes with inten- sity function λ (ϑ, t), 0 ≤ t ≤ τ . Here ϑ ∈ Θ = [ϑ 1 , b), b < ∞, is a one-dimensional parameter. We have E ϑ X j (t) = Λ (ϑ, t) = t 0 λ (ϑ, s) ds
where E ϑ is the mathematical expectation in the case when the true value is ϑ. Note that this model is equivalent to the one, where we observe an inhomogeneous Poisson process X T = (X t , 0 ≤ t ≤ T ) with periodic intensity λ(ϑ, t + jτ ) = λ(ϑ, t), j = 1, 2, . . . , n -1, and T = nτ (the period τ is supposed to be known). Indeed, if we put X j (s) = X s+τ (j-1) -X τ (j-1) , s ∈ [0, τ ], j = 1, . . . , n, then the observation of one trajectory X T is equivalent to n independent observations X 1 , . . . , X n .

The intensity function is supposed to be separated from zero on [0, τ ]. The measures corresponding to Poisson processes with different values of ϑ are equivalent. The likelihood function is defined by the equality (see Liese [START_REF] Liese | Eine informationstheoretische Bedingung fűr die Aquivalenz unbegrenzt teilbarer Punktprozesse[END_REF])

L(ϑ, X n ) = exp n j=1 τ 0 ln λ (ϑ, t) dX j (t) -n τ 0 [λ (ϑ, t) -1] dt
and the likelihood ratio function is

L (ϑ, ϑ 1 , X n ) = L (ϑ, X n ) /L (ϑ 1 , X n ) .
We have to test the following two hypotheses

H 1 : ϑ = ϑ 1 , H 2 : ϑ > ϑ 1 .
A test ψn = ψn (X n ) is defined as the probability to accept the hypothesis H 2 . Its power function is

β ψn , ϑ = E ϑ ψn (X n ), ϑ > ϑ 1 .
Denote K ε the class of tests ψn of asymptotic size ε ∈ [0, 1]:

K ε = ψn : lim n→∞ E ϑ 1 ψn (X n ) = ε .
Our goal is to construct tests which belong to this class and have some proprieties of asymptotic optimality. The comparison of tests can be done by comparison of their power functions. It is known that for any reasonable test and for any fixed alternative the power function tends to 1. To avoid this difficulty, as usual, we consider close or contiguous alternatives. We put ϑ = ϑ 1 + ϕ n u, where u ∈ U + n = 0, ϕ -1 n (bϑ 1 ) , ϕ n = ϕ n (ϑ 1 ) > 0 and ϕ n → 0. The rate of convergence ϕ n → 0 must be chosen so that the normalized likelihood ratio

Z n (u) = L (ϑ 1 + ϕ n u, X n ) L (ϑ 1 , X n ) , u ≥ 0,
has a non degenerate limit. In the regular case this rate is usually

ϕ n = n -1/2 .
Then the initial problem of hypotheses testing can be rewritten as

H 1 : u = 0, H 2 : u > 0.
The power function of a test ψn is now denoted as

β ψn , u = E ϑ 1 +ϕnu ψn , u > 0.
The asymptotic optimality of tests is introduced with the help of the following definition (see [START_REF] Roussas | Contiguity of Probability Measures[END_REF]). Definition 1. We call a test ψ ⋆ n (X n ) ∈ K ε locally asymptotically uniformly most powerful (LAUMP) in the class K ε if its power function β (ψ ⋆ n , u) satisfies the relation: for any test ψn (X n ) ∈ K ε and any K > 0 we have

lim n→∞ inf 0<u≤K β (ψ ⋆ n , u) -β ψn , u ≥ 0.
Below we show that in the regular case many tests are LAUMP. In the next paper [START_REF] Dachian | On hypothesis testing for Poisson processes[END_REF], where we consider some singular situations, a "reasonable" definition of asymptotic optimality of tests is still an open question. That is why we use numerical simulations to compare the tests in [START_REF] Dachian | On hypothesis testing for Poisson processes[END_REF].

We assume that the following Regularity conditions are satisfied.

Smoothness. The intensity function λ (ϑ, t), 0 ≤ t ≤ τ , of the observed Poisson process X n is two times continuously differentiable w.r.t. ϑ, is separated from zero uniformly on ϑ ≥ ϑ 1 , and the Fisher information is positive:

I (ϑ) = τ 0 λ (ϑ, t) 2 λ (ϑ, t) dt, inf ϑ∈Θ I (ϑ) > 0.
Here λ denotes the derivative of λ w.r.t. ϑ and, at the point ϑ 1 , the derivative is from the right.

Distinguishability. For any ν > 0, we have

inf ϑ * ∈Θ inf |ϑ-ϑ * |>ν λ (ϑ, •) -λ (ϑ 1 , •) L 2 > 0.
Here

h (•) 2 L 2 = τ 0 h (t) 2 dt.
In this case, the natural normalization function is ϕ n = n -1/2 and the change of variables is ϑ = ϑ 1 + u √ n . The key propriety of statistical problems in the regular case is the local asymptotic normality (LAN) of the family of measures of corresponding inhomogeneous Poisson processes at the point ϑ 1 . This means that the normalized likelihood ratio

Zn (u) = L ϑ 1 + u √ n , ϑ 1 , X n admits the representation Zn (u) = exp u ∆ n (ϑ 1 , X n ) - u 2 2 I (ϑ 1 ) + r n ,
where (using the central limit theorem) we have

∆ n (ϑ 1 , X n ) = 1 √ n n j=1 τ 0 λ (ϑ 1 , t) λ (ϑ 1 , t) [dX j (t) -λ (ϑ 1 , t) dt] =⇒ ∆ ∼ N (0, I (ϑ 1 ))
(convergence in distribution under ϑ 1 ), and

r n = r n (ϑ 1 , u, X n ) p
-→ 0 (convergence in probability under ϑ 1 ). Moreover, the last convergence is uniform on 0 ≤ u < K for any K > 0.

Let us now briefly recall how this representation was obtained in [START_REF] Yu | Asymptotically optimal algorithm of detecting Poisson signals[END_REF]. Denoting λ 0 = λ (ϑ 1 , t) and λ u = λ ϑ 1 + u √ n , t , with the help of the Taylor series expansion we can write

ln Z n (u) = n j=1 τ 0 ln λ u λ 0 [dX j (t) -λ 0 dt] -n τ 0 λ u -λ 0 -λ 0 ln λ u λ 0 dt = u √ n n j=1 τ 0 λ0 λ 0 [dX j (t) -λ 0 dt] - u 2 2 τ 0 λ2 0 λ 0 dt + r n = u ∆ n (ϑ 1 , X n ) - u 2 2 I (ϑ 1 ) + r n =⇒ ∆ - u 2 2 I (ϑ 1 ) .
In the sequel, we choose reparametrizations which lead to universal in some sense limits. For example, in the regular case, we put

ϕ n = ϕ n (ϑ 1 ) = 1 nI (ϑ 1 ) , u ∈ U + n = 0, ϕ -1 n (b -ϑ 1 )
.

With such change of variables, we have

Z n (u) = L (ϑ 1 + uϕ n , ϑ 1 , X n ) = exp u ∆ n (ϑ 1 , X n ) - u 2 2 + r n , where ∆ n (ϑ 1 , X n ) = 1 I (ϑ 1 ) ∆ n =⇒ ∆ ∼ N (0, 1) .
The LAN families have many remarkable properties and some of them will be used below.

Let us remind here one general result which is valid in a more general situation. We suppose only that the normalized likelihood ratio Z n (u) converges to some limit Z (u) in distribution. Note that this is the case in all our regular and singular problems. The following property allows us to calculate the distribution under local alternative when we know the distribution under the null hypothesis. Moreover, it gives an efficient algorithm for calculating power functions in numerical simulations.

Lemma 1 (Le Cam's Third Lemma). Suppose that (Z n (u) , Y n ) converges in distribution under ϑ 1 :

(Z n (u) , Y n ) =⇒ (Z (u) , Y ) .
Then, for any bounded continuous function g (•), we have

E ϑ 1 +ϕnu [g (Y n )] -→ E [Z (u) g (Y )] .
For the proof see [START_REF] Cam | Asymptotic Methods in Statistical Decision Theory[END_REF]. In the regular case, the limit of Z n (•) is the random function

Z (u) = exp u ∆ - u 2 2 , u ≥ 0.
So, for any fixed u > 0, we have the convergence

Z n (u) =⇒ Z (u) .
According to this lemma, we can write the following relations for the characteristic function of ∆ n = ∆ n (ϑ 1 , X n ):

E ϑ 1 +ϕnu e iµ∆n -→ E Z (u) e iµ∆ = e -u 2 2 E e u∆+iµ∆ = e iµu-µ 2 2 = E e iµ(u+∆) ,
which yields the asymptotic distribution of the statistic ∆ n under the alternative ϑ 1 + ϕ n u:

∆ n (ϑ 1 , X n ) =⇒ u + ∆ ∼ N (u, 1) .

Weak convergence

All the tests considered in this paper are functionals of the normalized likelihood ratio Z n (•). For each of them, we have to evaluate two quantities. The first one is the threshold, which guarantees the desired asymptotic size of the test, and the second one is the limit power function, which has to be calculated under alternative. Our study is based on the weak convergence of the likelihood ratio Z n (•) under hypothesis (to calculate the threshold) and under alternative (to calculate the limit power function). Note that the test statistics of all the tests are continuous functionals of Z n (•). That is why the weak convergence of Z n (•) allows us to obtain the limit distributions of these statistics. We denote P ϑ the distribution that the observed inhomogeneous Poisson processes X n induce on the measurable space of their realizations. The measures in the family {P ϑ , ϑ ≥ ϑ 1 } are equivalent, and the normalized likelihood ratio is

ln Z n (u) = n j=1 τ 0 ln λ (ϑ 1 + ϕ n (ϑ 1 ) u, t) λ (ϑ 1 , y) dX j (t) -n τ 0 [λ (ϑ 1 + ϕ n (ϑ 1 ) u, t) -λ (ϑ 1 , t)] dt, where u ∈ U + n = 0, ϕ -1 n (b -ϑ 1 )
. We define Z n (u) to be linearly decreasing to zero on the interval ϕ

-1 n (b -ϑ 1 ) , ϕ -1 n (b -ϑ 1 ) + 1 and we put Z n (u) = 0 for u > ϕ -1
n (bϑ 1 ) + 1. Now the random function Z n (u) is defined on R + and belongs to the space C 0 (R + ) of continuous on R + functions such that z (u) → 0 as u → ∞. Introduce the uniform metric in this space and denote B the corresponding Borel sigma-algebra. The next theorem describes the weak convergence under the alternative ϑ = ϑ 1 + ϕ n u * (with fixed u * > 0) of the stochastic process (Z n (u) , u ∈ R + ) to the process

Z (u, u * ) = exp u∆ + uu * - u 2 2 , u ∈ R,
in the measurable space (C 0 (R + ) , B). Note that in [START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF] this theorem was proved for a fixed true value ϑ. In the hypothesis testing problems considered here, we need this convergence both under hypothesis H 1 , that is, for fixed true value ϑ = ϑ 1 (u * = 0), and under alternative H 2 with "moving" true value ϑ = ϑ u * = ϑ 1 + ϕ n u * .

Theorem 1. Let us suppose that the Regularity conditions are fulfilled. Then, under alternative ϑ u * , we have the weak convergence of the stochastic process

Z n = (Z n (u) , u ≥ 0) to Z = (Z (u, u * ) , u ≥ 0).
According to [6, Theorem 1.10.1], to prove this theorem it is sufficient to verify the following three properties of the process Z n (•).

1. The finite-dimensional distributions of Z n (•) converge, under alternative ϑ u * , to the finite-dimensional distributions of Z (•, u * ).

The inequality

E ϑu * Z 1/2 n (u 2 ) -Z 1/2 n (u 1 ) 2 ≤ C |u 2 -u 1 | 2
holds for every u 1 , u 2 ∈ U + n and some constant C > 0.

3. There exists d > 0, such that for some n 0 > 0 and all n ≥ n 0 we have the estimate

P ϑu * Z n (u) > e -d|u-u * | 2 ≤ e -d|u-u * | 2 .
Let us rewrite the random function Z n (•) as follows:

Z n (u) = L (ϑ 1 + uϕ n , ϑ 1 , X n ) = L (ϑ 1 + u * ϕ n , ϑ 1 , X n ) L (ϑ 1 + uϕ n , ϑ 1 + u * ϕ n , X n ).
For the first term we have

L (ϑ 1 + u * ϕ n , ϑ 1 , X n ) = L (ϑ u * -u * ϕ n , ϑ u * , X n ) -1 =⇒ exp u * ∆ + u 2 * 2 .
Therefore we only need to check the conditions 2-3 for the term

Z n (u, u * ) = L (ϑ 1 + uϕ n , ϑ 1 + u * ϕ n , X n ) = L (ϑ u * + (u -u * )ϕ n , ϑ u * , X n ) .
Lemma 2. The finite-dimensional distributions of Z n (•) converge, under alternative ϑ u * , to the finite-dimensional distributions of Z (•, u * ).

Proof. The limit process for

Z n (•, u * ) is exp (u -u * ) ∆ - (u -u * ) 2 2 , u ∈ R + .
Hence

Z n (u) =⇒ exp u * ∆ + u 2 * 2 exp (u -u * ) ∆ - (u -u * ) 2 2 = Z (u, u * ) .
For the details see, for example, [START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF].

Lemma 3. Let the Regularity conditions be fulfilled. Then there exists a constant C > 0, such that

E ϑu * Z 1/2 n (u 1 , u * ) -Z 1/2 n (u 2 , u * ) 2 ≤ C |u 1 -u 2 | 2
for all u * , u 1 , u 2 ∈ U + n and sufficiently large values of n.

Proof. According to [8, Lemma 1.1.5], we have:

E ϑu * Z 1/2 n (u 1 , u * ) -Z 1/2 n (u 2 , u * ) 2 ≤ n τ 0 λ 1/2 (ϑ 1 + u 1 ϕ n , t) λ 1/2 (ϑ 1 + u * ϕ n , t) - λ 1/2 (ϑ 1 + u 2 ϕ n , t) λ 1/2 (ϑ 1 + u * ϕ n , t) 2 λ(ϑ 1 + u * ϕ n , t) dt = n τ 0 λ 1/2 (ϑ 1 + u 1 ϕ n , t) -λ 1/2 (ϑ 1 + u 2 ϕ n , t) 2 dt = n 4 ϕ 2 n (u 2 -u 1 ) 2 τ 0 λ (ϑ v , t) 2 λ (ϑ v , t) dt ≤ C (u 2 -u 1 ) 2 ,
where v is some intermediate point between u 1 and u 2 .

Lemma 4. Let the Regularity conditions be fulfilled. Then there exists a constant d > 0, such that

P ϑu * Z n (u, u * ) > e -d|u-u * | 2 ≤ e -d|u-u * | 2 (1)
for all u * , u ∈ U + n and sufficiently large value of n. Proof. Using the Markov inequality, we get

P ϑu * Z n (u, u * ) > e -d|u-u * | 2 ≤ e 1 2 d|u-u * | 2 E ϑu * Z 1/2 n (u, u * ).
According to [8, Lemma 1.1.5], we have

E ϑu * Z 1/2 n (u, u * ) = exp - 1 2 n τ 0 λ 1/2 (ϑ 1 + uϕ n , t) λ 1/2 (ϑ 1 + u * ϕ n , t) -1 2 λ(ϑ 1 + u * ϕ n , t) dt = exp - 1 2 n τ 0 λ 1/2 ϑ 1 + uϕ n , t -λ 1/2 (ϑ 1 + u * ϕ n , t) 2 dt .
Using the Taylor expansion we get

λ 1/2 ϑ 1 + uϕ n , t = λ 1/2 ϑ 1 + u * ϕ n , t + ϕ n (u -u * ) 2 λ (ϑ v , t) λ 1/2 (ϑ v , t) ,
where v is some intermediate point between u * and u. Hence, for sufficiently large n providing ϕ n |uu * | ≤ γ, we have the inequality I (ϑ v ) ≥ 1 2 I (ϑ 1 ), and we obtain

E ϑu * Z 1/2 n (u, u * ) ≤ exp - 1 8I (ϑ 1 ) |u -u * | 2 I (ϑ v ) ≤ exp - |u -u * | 2 16 . (2)
By Distinguishability condition, we can write

g(γ) = inf ϕn|u-u * |>γ τ 0 λ 1/2 ϑ 1 + uϕ n , t -λ 1/2 (ϑ 1 + u * ϕ n , t) 2 dt > 0,
and hence

τ 0 λ 1/2 ϑ 1 + uϕ n , t -λ 1/2 (ϑ 1 + u * ϕ n , t) 2 dt ≥ g(γ) ≥ g(γ) ϕ 2 n (u -u * ) 2 (b -ϑ 1 ) 2
and

E ϑu * Z 1/2 n (u, u * ) ≤ exp - g(γ) |u -u * | 2 2I (ϑ 1 ) (b -ϑ 1 ) 2 .
(3) So, putting

d = 2 3 min 1 16 , g(γ) 2I (ϑ 1 ) (b -ϑ 1 ) 2 ,
the estimate (1) follows from ( 2) and (3).

The weak convergence of Z n (•) now follows from [6, Theorem 1.10.1].

Hypothesis testing

In this section, we construct the Score Function test, the General Likelihood Ratio test, the Wald test and two Bayes tests. For all these tests we describe the choice of the thresholds and evaluate the limit power functions for local alternatives.

Score Function test

Let us introduce the Score Function test (SFT)

ψ ⋆ n (X n ) = 1 {∆n(ϑ 1 ,X n )>zε} ,
where z ε is the (1ε)-quantile of the standard normal distribution N (0, 1) and the statistic ∆

n (ϑ 1 , X n ) is ∆ n (ϑ 1 , X n ) = 1 nI (ϑ 1 ) n j=1 τ 0 λ (ϑ 1 , t) λ (ϑ 1 , t) [dX j (t) -λ (ϑ 1 , t) dt] .
The SFT has the following well-known properties (one can see, for example, [START_REF] Lehmann | Testing Statistical Hypotheses[END_REF]Theorem 13.3.3] for the case of i.i.d. observations).

Proposition 1. The test ψ ⋆ n (X n ) ∈ K ε and is LAUMP. For its power function the following convergence hold:

β (ψ ⋆ n , u * ) -→ β ⋆ (u * ) = P (∆ > z ε -u * ) , ∆ ∼ N (0, 1) .
Proof. The property ψ ⋆ n (X n ) ∈ K ε follows immediately from the asymptotic normality (under hypothesis H 1 )

∆ n (ϑ 1 , X n ) =⇒ ∆.
Further, we have (under alternative

ϑ u * = ϑ 1 + u * ϕ n ) the convergence β (ψ ⋆ n , u * ) -→ P (∆ + u * > z ε ) = β ⋆ (u * ) .
This follows from the Le Cam's Third Lemma and can be shown directly as follows. Suppose that the intensity of the observed Poisson process is λ (ϑ 1 + u * ϕ n , t), then we can write

∆ n (ϑ 1 , X n ) = 1 nI (ϑ 1 ) n j=1 τ 0 λ (ϑ 1 , t) λ (ϑ 1 , t) [dX j (t) -λ (ϑ 1 + u * ϕ n , t) dt] + 1 nI (ϑ 1 ) n j=1 τ 0 λ (ϑ 1 , t) λ (ϑ 1 , t) [λ (ϑ 1 + u * ϕ n , t) -λ (ϑ 1 , t)] dt = ∆ * n (ϑ 1 , X n ) + u * nI (ϑ 1 ) n j=1 τ 0 λ (ϑ 1 , t) 2 λ (ϑ 1 , t) dt + o (1) = ∆ * n (ϑ 1 , X n ) + u * + o (1) =⇒ ∆ + u * .
To show that the SFT is LAUMP, it is sufficient to verify that the limit of its power function coincides (for each fixed value u * > 0) with the limit of the power of the corresponding likelihood ratio (Neyman-Person) test (N-PT) ψ * n (X n ) . Remind that the N-PT is the most powerful for each fixed (simple) alternative (see, for example, Theorem 13.3 in Lehman and Romano [START_REF] Lehmann | Testing Statistical Hypotheses[END_REF]). Of course, the N-PT is not a real test (in our one-sided problem), since for its construction one needs to know the value u * of the parameter u under alternative.

The N-PT is defined by

ψ * n (X n ) = 1 {Zn(u * )>dε} + q ε 1 {Zn(u * )=dε} ,
where the threshold d ε and the probability q ε are chosen from the condition ψ * n (X n ) ∈ K ε , that is,

P ϑ 1 {Z n (u * ) > d ε } + q ε P ϑ 1 {Z n (u * ) = d ε } = ε.
Of course, we can put q ε = 0 because the limit random variable Z (u * ) has continuous distribution function.

The threshold d ε can be found as follows. The LAN of the family of measures at the point ϑ 1 allows us to write

P ϑ 1 (Z n (u * ) > d ε ) = P ϑ 1 u * ∆ n (ϑ 1 , X n ) - u 2 * 2 + r n > ln d ε -→ P u * ∆ - u 2 * 2 > ln d ε = P ∆ > ln d ε u * + u * 2 = ε.
Hence, we have

ln d ε u * + u * 2 = z ε and d ε = exp u * z ε - u 2 * 2 .
Therefore the N-PT

ψ * n (X n ) = 1 Zn(u * )>exp u * zε- u 2 * 2 belongs to K ε .
For the power of the N-PT we have (denoting as usually

ϑ u * = ϑ 1 + u * ϕ n ) β (ψ * n , u * ) = P ϑu * (Z n (u * ) > d ε ) = P ϑu * (u * ∆ n (ϑ 1 , X n ) + r n > u * z ε ) = P ϑu * ∆ n (ϑ 1 , X n ) + r n u * > z ε -→ P (∆ + u * > z ε ) = β ⋆ (u * ) .
Therefore the limits of the powers of the tests ψ ⋆ n and ψ * n coincide, that is, the Score Function test is asymptotically as good as the Neyman-Pearson optimal one. Note that the limits are valid for any sequence of 0 ≤ u * ≤ K. So, for any K > 0, we can choose a sequence ûn ∈

[0, K] such that sup 0≤u * ≤K |β (ψ ⋆ n , u * ) -β (ψ * n , u * )| = |β (ψ ⋆ n , ûn ) -β (ψ * n , ûn )| -→ 0,
which represents the asymptotic coincidence of the two tests and concludes the proof.

GLRT and Wald test

Let us remind that the maximum likelihood estimator (MLE) θn is defined by the equation:

L θn , ϑ 1 , X n = sup ϑ∈[ϑ 1 ,b) L (ϑ, ϑ 1 , X n ) ,
where the likelihood ratio function is

L (ϑ, ϑ 1 , X n ) = exp n j=1 τ 0 ln λ (ϑ, t) λ (ϑ 1 , t) dX j (t) -n τ 0 [λ (ϑ, t) -λ (ϑ 1 , t)] dt , ϑ ∈ [ϑ 1 , b) . The GLRT is ψn (X n ) = 1 {Q(X n )>hε} ,
where

Q (X n ) = sup ϑ∈[ϑ 1 ,b) L (ϑ, ϑ 1 , X n ) = L θn , ϑ 1 , X n and h ε = exp{z 2 ε /2}.
The Wald's test is based on the MLE θn and is defined as follows:

ψ • n (X n ) = 1 { ϕ -1 n ( θn-ϑ1)>zε} .
The properties of these tests are given in the following Proposition.

Proposition 2. The tests ψn (X n ) and ψ • n (X n ) belong to K ε , their power functions β( ψn , u * ) and β (ψ • n , u * ) converge to β ⋆ (u * ), and therefore they are LAUMP.

Proof. Let us put ϑ = ϑ 1 + uϕ n and denote ûn = ϕ -1 n θnϑ 1 . We have

P ϑ 1 sup ϑ∈[ϑ 1 ,b) L (ϑ, ϑ 1 , X n ) > h ε = P ϑ 1 sup u∈U + n L (ϑ 1 + uϕ n , ϑ 1 , X n ) > h ε = P ϑ 1 sup u∈U + n Z n (u) > h ε .
According to Theorem 1 (with u * = 0), we have the weak convergence (under ϑ 1 ) of the measure of the stochastic processes (Z n (u) , u ≥ 0) to those of the process (Z (u) , u ≥ 0). This provides us the convergence of the distributions of all continuous in uniform metric functionals. Hence

Q (X n ) = sup u>0 Z n (u) =⇒ sup u>0 Z (u) = sup u>0 exp u∆ - u 2 2 = exp ∆ 2 2 1 {∆≥0} , which yields (we suppose that ε ≤ 1 2 ) E ϑ 1 ψn (X n ) -→ P ∆1 {∆≥0} > z ε = P {∆ > z ε } = ε.
Using the same weak convergence we obtain the asymptotic normality of the MLE (see [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF] or [START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF]):

ûn = θn -ϑ 1 ϕ n =⇒ û = ∆1 {∆≥0} ,
and hence E ϑ 1 ψ • n -→ ε. So both ψn and ψ • n belong to K ε . Now, let us fix some u * > 0 and study the limit behavior of the power functions of the tests.

Using the weak convergence of the likelihood ratio process under the alternative ϑ u * = ϑ 1 + u * ϕ n , we have

Q (X n ) = sup u>0 Z n (u) =⇒ sup u>0 Z (u, u * ) = sup u>0 exp u∆ + uu * - u 2 2 = exp (∆ + u * ) 2 2 1 {∆+u * ≥0} .
Hence (we suppose again that ε ≤ 1 2 ),

β ψn , u * = P ϑu * {Q (X n ) > h ε } -→ P (∆ + u * ) 1 {∆+u * ≥0} > z ε = P {∆ > z ε -u * } = β ⋆ (u * ) .
Similarly we have

β (ψ • n , u * ) -→ P (∆ + u * ) 1 {∆+u * ≥0} > z ε = β ⋆ (u * )
. Therefore the tests are LAUMP.

Example 1. As the family of measures is LAN and the problem is asymptotically equivalent to the corresponding hypothesis testing problem for a Gaussian model, we propose here a similar test for Gaussian observations.

Suppose that the random variable X ∼ N (u, 1) and we have to test the hypothesis H 1 : u = 0 against H 2 : u > 0. Then the SFT ψ (X) = 1 {X>zε} is the uniformly most powerful in the class of tests of size ε. Its power function is β ψ, u * = P (∆ > z εu * ). The log-likelihood function is

L (u, X) = - 1 2 ln (2π) - 1 2 (X -u) 2
The one-sided MLE û is given by û = argmax u≥0 L(X, u) = max{X, 0}, and it is easy to see that the test ψ (X) and the Wald test ψ • (X) = 1 {û>zε} have identical power functions.

Let us note, that the asymptotic equivalence to the SFT and the optimality is a well known property of these tests in regular statistical experiments (see, for example, [START_REF] Lehmann | Testing Statistical Hypotheses[END_REF] and [START_REF] Kutoyants | Introduction to Statistics of Poisson Processes[END_REF]). We present these properties here in order to compare the asymptotics of these tests in regular and singular situations (see [START_REF] Dachian | On hypothesis testing for Poisson processes[END_REF]). In particular, we will see that the asymptotic properties of these tests in singular situations will be essentially different.

Bayes tests

Suppose now that the unknown parameter ϑ is a random variable with a priori density p (θ), ϑ 1 ≤ θ < b. Here p (•) is a known continuous function satisfying p (ϑ 1 ) > 0. We consider two approaches. The first one is based on the Bayes estimator and the second one on the averaged likelihood ratio function.

The first test, wich we call BT1, is a Wald type test but based on the Bayes estimator (BE) θn :

ψn (X n ) = 1 {ϕ -1 n ( θn-ϑ1)>kε} .
Remind that the BE for quadratic loss function is

θn = b ϑ 1 θ p (θ|X n ) dθ = b ϑ 1 θ p (θ) L (θ, ϑ 1 , X n ) dθ b ϑ 1 p (θ) L (θ, ϑ 1 , X n ) dθ
.

After the change of variables θ = ϑ 1 + vϕ n in the integrals, we obtain the relation

ϕ -1 n θn -ϑ 1 = U + n vp (ϑ 1 + vϕ n ) Z n (v) dv U + n p (ϑ 1 + vϕ n ) Z n (v) dv
.

The properties of Z n (•) established in the proof of Theorem 1 yield the following convergence in distribution under the hypothesis H 1 (see [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF] or [START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF])

ϕ -1 n θn -ϑ 1 =⇒ ũ = ∞ 0 vZ (v) dv ∞ 0 Z (v) dv = ∞ 0 (v -∆) exp -(v-∆) 2 2 dv ∞ 0 exp -(v-∆) 2 2 dv + ∆ = -1 √ 2π exp -(v-∆) 2 2 +∞ v=0 1 √ 2π ∞ 0 exp -(v-∆) 2 2 dv + ∆ = 1 √ 2π exp -∆ 2 2 (1 -F (-∆)) + ∆ = f (∆) F (∆) + ∆,
where f (•) and F (•) are the density and the distribution function of the standard normal Gaussian random variable ∆. Hence, if we take k ε to be solution of the equation

P f (∆) F (∆) + ∆ > k ε = ε,
then the BT1 ψn belongs to K ε . A similar calculation under the alternative ϑ u * = ϑ + u * ϕ n allows us to evaluate the limit power function of the BT1 as follows:

β ψn , u * = P ϑu * ϕ -1 n θn -ϑ 1 > k ε -→ P ∞ 0 vZ (v, u * ) dv ∞ 0 Z (v, u * ) dv > k ε = P f (∆ + u * ) F (∆ + u * ) + ∆ + u * > k ε .
Another possibility in Bayesian approach is to define the test as the test with the minimal mean error of the second kind. For a test ψn , let us denote α ψn , θ = 1β ψn , θ the error of the second kind and introduce the mean error of the second kind:

α ψn = b ϑ 1 α ψn , θ p (θ) dθ.
The Bayes test ψ⋆ n (X n ) is defined as the test which minimizes this mean error:

α ψ⋆ n = inf ψn∈Kε α ψn .

We can rewrite the above integral as follows

b ϑ 1 1 -E θ ψn (X n ) p (θ) dθ = b ϑ 1 1 -ψn (x n ) dP θ p (θ) dθ = 1 -ψn (x n ) dP = E 1 -ψn (X n ) ,
where we denoted P θ the distribution of the sample X n and

P (X n ∈ A) = b ϑ 1 P θ (X n ∈ A) p (θ) dθ.
The averaged power β ψn = E ψn (X n ) is the same as if we have two simple hypothesis. Under H 1 we observe a Poisson process of intensity function λ (ϑ 1 , •), and under the alternative H 2 the observed point process has random intensity and its measure is P. This process is a mixture according to the density p(θ) of inhomogeneous Poisson processes with intensities λ (θ, •), θ ∈ (ϑ 1 , b). This means that we have two simple hypotheses and the most powerful (Neyman-Pearson) test is of the form

ψ * n = 1 { L(X n )> mε} , E ϑ 1 ψ * n (X n ) = ε,
where the averaged likelihood ratio

L (X n ) = dP dP ϑ 1 (X n ) = b ϑ 1 dP θ dP ϑ 1 (X n ) p (θ) dθ.
To study this test under hypothesis we change the variables:

L (X n ) = b ϑ 1 L (θ, ϑ 1 , X n ) p (θ) dθ = ϕ n ϕ -1 n (b-ϑ 1 ) 0 Z n (v) p (ϑ 1 + vϕ n ) dv.
The limit of the last integral was already described above and this allow us to write

R n (X n ) = L (X n ) p (ϑ 1 ) ϕ n = 1 p (ϑ 1 ) ϕ -1 n (b-ϑ 1 ) 0 e v∆n-v 2 2 +rn p (ϑ 1 + vϕ n ) dv =⇒ ∞ 0 e v∆-v 2 2 dv = e ∆ 2 2 ∞ -∆ e -y 2 2 dy = √ 2π e ∆ 2 2 (1 -F (-∆)) = F (∆) f (∆) ,
where F (•) and f (•) are again the distribution function and the density of the standard Gaussian random variable ∆. Hence, if we take m ε to be solution of the equation

P F (∆) f (∆) > m ε = ε,
then the test ψ⋆ n (X n ) = 1 {Rn>mε} , which we call BT2, belongs to K ε and coincides with the test ψ * n (X n ) if we put mε = m ε p (ϑ 1 ) ϕ n . A similar calculation under the alternative ϑ u * = ϑ + u * ϕ n allows us to evaluate the limit power function of the BT2 as follows:

β ψ⋆ n , u * = P ϑu * {R n > m ε } -→ P F (∆ + u * ) f (∆ + u * ) > m ε .

Simulations

Below we present the results of numerical simulations for the power functions of the tests. We observe n independent realizations where ϕ n = (19.82 n) -1/2 . The numerical simulation of the observations allows us to obtain the power functions presented in Figures 1 and2. For example, the computation of the numerical values of the power function of the SFT was done as follows. We define an increasing sequence of u beginning at u = 0. Then, for every u, we simulate N i.i.d. observations of n-tuples of inhomogeneous Poisson processes X n,i , i = 1, ..., N, with the intensity function λ (3 + uϕ n , t) and calculate the corresponding statistics ∆ n (3, X n,i ), i = 1, ..., N. The empirical frequency of acceptation of the alternative gives us an estimate of the power function:

X j = (X j (t), t ∈ [0, 3]), j =
β (ψ ⋆ n , u) ≈ 1 N N i=1
1 {∆n(3,X n,i )>zε} .

We repeat this procedure for different values of u until the values of β (ψ ⋆ n , u) become close to 1. In the computation of the power function of the Bayes test BT1, we take as a priori law the uniform distribution, that is, ϑ ∼ U( [START_REF] Dachian | Hypotheses testing : Poisson versus stress-release[END_REF][START_REF] Yu | Asymptotically optimal algorithm of detecting Poisson signals[END_REF]). The thresholds of the BT1 are obtained by simulating M = 10 5 random variables ∆ i ∼ N (0, 1), i = 1, . . . , M, calculating for each of them the quantity f (∆ i ) F (∆ i ) + ∆ i , i = 1, . . . , M, and taking the (1ε) M-th greatest between them. Some of the thresholds are presented in Table 1.

parameter. The test statistic ∆ n = ∆ n (3, X n ) can be rewritten as follows: The last integral becomes negative for some values of u, which explains the loss of power of the SFT (for n = 10).

∆ n =

1 , 2 (( 3 +

 123 ..., n, of inhomogeneous Poisson process of intensity functionλ (ϑ, t) = 3 cos 2 (ϑt) + 1, 0 ≤ t ≤ 3, ϑ ∈ [3, 7) ,where ϑ 1 = 3. The Fisher information at the point ϑ 1 is I (ϑ 1 ) ≈ 19.82. Recall that all our tests (except Bayes tests) in regular case are LAUMP. Therefore they have the same limit power function. Our goal is to study the power functions of different tests for finite n.The normalized likelihood ratio Z n (u) is given by the expression Z n (u) = exp ϕ uϕ n ) t) + 1 3 cos 2 (3t) + 1 dX j (t)

Figure 1 :

 1 Figure 1: Power functions of SFT and BT1

  (3 t) + 1 × cos 2 (3 t)cos 2 ((3 + uϕ n ) t) dt.

	ϕ n	n j=1	0	T	λ (ϑ 1 , t) λ (ϑ 1 , t)	[dX j (t) -λ (ϑ 1 + uϕ n , t) dt]
	+	n I (ϑ 1 )	0	T	λ (ϑ 1 , t) λ (ϑ 1 , t)	[λ (ϑ 1 + uϕ n , t) -λ (ϑ 1 , t)] dt
	= -3ϕ n	n j=1	0	3	t sin(6t) 3 cos 2 (3 t) + 1	dX j (t) -3 cos 2 ((3 + uϕ n ) t)+ 1 dt
	+ 9	n I (ϑ 1 )	0	3	t sin(6t) 3 cos 2

Acknowledgements

This study was partially supported by Russian Science Foundation (research project No. 14-49-00079). The authors thank the Referee for helpful comments.

Note that for the small values of n, under alternative, the power function of SFT starts to decrease (see Figure 2). This interesting fact can be explained by the strongly non linear dependence of the likelihood ratio on the ε 0.01 0.05 0.