1 Introduction
The problems of the construction of goodness of fit tests in the case of i.i.d. observations are well studied [15]. Special attention is payed to the case of parametric null hypothesis. Wide class of distributions can be parametrized by the shift and scale parameters, say, . In the case of such families several authors showed that the limit distributions of the KolmogorovSmirnov and Cramervon Mises tests statistics do not depend on the unknown parameters (see [4], [6], [8], [7], [16], [17] and references therein). We call such tests asymptotically parameter free (APF).
For the continuous time stochastic processes the goodness of fit testing is not yet well developed. We can mention here several works for diffusion and Posson processes [1], [2], [3], [5], [11], [13],[14], [18]. The problem of goodness of fit testing for inhomogeneous Poisson process is interesting because there is a wide literature on the applications of inhomogeneous Poisson process models in different domains (astronomy, biology, image analysis, medicine, optical communication, physics, reliability theory, etc.). Therefore to know if the observed Poisson process corresponds to some parametric family of intensity functions is important.
We consider the problem of goodness of fit testing for inhomogeneous Poisson process which under the null hypothesis has the intensity function with shift and scale parameters. We show that as in the classical case the limit distribution of the Cramervon Mises type statistics does not depend on these unknown parameters. This allows us to construct the corresponding APF goodness of fit test of fixed asymptotic size.
2 Statement of the problem and auxiliary results
Suppose that we observe independents inhomogeneous Poisson processes , where are trajectories of the Poisson processes with the mean function Here is the corresponding intensity function.
Let us remind the construction of GoF test of Cramérvon Mises type in the case of simple null hypothesis. The class of tests of asymptotic size is
Suppose that the basic hypothesis is simple, say, where is a know continuous function satisfying . The alternative is composite (non parametric) Then we can introduce the Cramérvon Mises (CvM) type statistic
where is the empirical mean of the Poisson process. It can be verified that under this statistic converges to the following limit:
where is a standard Wiener process. Therefore the CvM type test with the threshold defined by the equation belongs to . This test is asymptotically distribution free (ADF) (see, e.g., [3]). Remind that the test is called ADF if the limit distribution of the test statistic under hypothesis does not depend on the mean function .
Let us consider the case of the parametric null hypothesis. It can be formulated as follows. We have to test the null hypothesis
against the alternative Here is a known mean function of the Poisson process depending on some finitedimensional unknown parameter . Note that under there exists the true value such that the mean of the observed Poisson process .
The CvM type GoF test can be constructed by a similar way. Introduce the normalized process Here
is some estimator of the parameter
, which is (under hypothesis ) consistent and asymptotically normal .The corresponding CvM type statistic can be
Then, under null hypothesis , we can verify the convergence
Here is the scalar product in and dot means differentiation w.r.t. . Let us denote
and introduce the vector
Then we obtain the convergencewhere
is standard Wiener process. Here the distribution of the limit random variable
depends on the true value and on the mean function .Therefore if we propose a GoF test based on this statistics, say, , then to find the threshold such that we have to solve the equation . The solution , where is the unknown true value. There are several possibilities to construct the test belonging . One is to calculate the function , verify that this function is continuous w.r.t. and then to use the consistent estimator for the threshold
. Another possibility is to use the linear transformation of the statistic
, which transforms it in the Wiener process (see, e.g., [10] or [11]). In this work we follow the third approach: we show that the limit distribution of the statistic does not depend on .In particular, the goal of this work is to show that if the unknown parameter is twodimensional , where is the shift and is the scale parameters, then it is possible to construct a test statistic whose limit distribution does not depend on . The mean function under null hypothesis is
The proposed test statistic is
Here is the maximum likelihood estimator (MLE) of the vector parameter . We show that , where , i.e., the distribution of the random variable does not depend on . Remind that the function is known and therefore the solution can be calculated before the experiment using, say, numerical simulations.
We are given independent observations of inhomogeneous Poisson processes with the mean function . We have to construct a GoF test in the hypothesis testing problem with parametric null hypothesis . More precizely, we suppose that under the mean function is absolutely continuous: . Here is the true value and the intensity function is The set and , where all constants are finite. Therefore if we denote then the mean function under null hypothesis is
It is convenient to use two different functions and and we hope that such notation will not be misleading.
Therefore, we have the parametric null hypothesis
where the parametric family is
(1) 
Here is a known absolutely continuous function
with properties:
.
We consider the class of tests of asymptotic level :
(2) 
The test studied in this work is based on the following statistic of CvM type:
(3) 
where is the MLE. Remind that the loglikelihood ratio for this model of observations is
and the MLE is defined by the equation
(4) 
Here is some fixed value.
As we use the asymptotic properties of the MLE , we need some regularity conditions, which we borrow from [12] (see the conditions B1B5 in the Section 2.1 there).
Note that the derivative (vector) of the intensity function is
(5) 
Here .
Conditions
. The intensity function is strictly positive and two times continuously differentiable.
. For any we have
(6)  
(7) 
. The function satisfies the conditions
(8) 
Of course, we suppose that the expressions under the sign of integrals are integrable in the required sense.
For the consistency of the MLE we need the identifiability condition
For any
Note that in the case of shift and scale parameters this condition is fulfilled. Indeed, suppose that for some this integral is 0. Then there exists () such that . Recall that the functions are continuous. Therefore or after the change of variables we have
Of course, such function . Hence, the condition of identifiability is fulfilled.
To construct the test statistics we need the following property of the mean function
For all
(9) 
This condition can be expressed in terms of the function like (6)(7). Indeed we have
As the function is bounded, it is sufficient to suppose (8) and we obtain (9).
Let us introduce the Fisher information matrix
where the matrix does not depend on . Note that the matrix is non degenerate. Indeed, the determinant is
Remind that by CauchySchwartz inequality
The equality in CauchySchwartz inequality () we obtain if and only if Of course such equality is impossible, if or . As the function is positive and differentiable, we have
We suppose that the intensity function is strictly positive because if we have a set of positive Lebesgue measure, where and the unknown parameters are shift and scale, then the measures induced by the observations will be not equivalent. The properties of the MLE will be different.
3 Main result
Introduce the following random variable:
(11) 
where and is a Wiener process. The main result of this work is the following theorem.
Theorem 1
Let the conditions be fulfilled then the test
belongs to the class .
Proof. We can write
Here the vector and we used the Taylor formula.
We have to show that under the null hypothesis
(12)  
(13) 
Here .
The convergences (12), (13) we will prove in several steps.
 A

. We show that we have the convergence of finite dimensional distributions
(14) where we put and
 B

. We verify the estimate: for and any
(15) where the constant does not depend on .
 C

. We show that for any there exists such that for all
 D

. We check (13) by direct calculations.
To prove A
we recall that by the central limit theorem
(16) 
where is a Wiener process. Moreover, the vector for any and is asymptotically normal
We know as well that the MLE is asymptotically normal. The Wiener process and the Gaussian vector are correlated. To clarify this dependence and to prove the joint asymptotic normality of the MLE and of this vector we recall how the asymptotic normality of the MLE can be proved. We follow below the approach developed by Ibragimov and Khasminskii [9].
Introduce the normalized likelihood ratio Here . Under the presented here conditions the random field admits the representation (LAN)
(17) 
where and the vector
By the central limit theorem
(18) 
Let us denote the limit random field
Recall that we have the representation
with the same Wiener process as in (16). Moreover, for the MLE we have the limit
where the vector (see (5)). This representation, which we prove below, allows us to say what is the correlation between and :
Let us return to the proof of the asymptotic normality of the MLE. The random field we extend on the whole plane continuously decreasing to zero outside of . Denote the measurable space of the continuous random surfaces tending to zero at infinity with the uniform metrics and Borelian algebra. Introduce the measures and induced by the realizations of and in the space respectively. Suppose that we already proved the weak convergence
(19) 
Then we have the convergence of the distributions of the continuous functionals to the distribution of . Consider a convex set . We can write
Note that is a continuous functional on the space . The random function takes its
maximum at the point . To prove the joint convergence in distribution of the vector
and we
denote
introduce the product space
with the corresponding Borelian algebra . To verify the
weak convergence , where we
a) prove the convergence of the
finitedimensional distributions
b) prove the tightness of the corresponding family of measures.
The convergence a) follows from the LAN (17), (18). The prove of b) is a part of the Theorem 1.10.1 in [9]. The conditions are sufficient for the verification of the conditions B1B5 of the Theorem 1.10.1 in [9]. Therefore we obtain the joint asymptotic normality of the vector
Hence we obtain the convergence of the finitedimensional distributions (14). Let us check B. We have
Hence ()
For the second term we have
The inequality C follows from the similar estimates.
because
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