Kolmogorov-Smirnov APF Test for Inhomogeneous Poisson Processes with Shift Parameter

In this article, we study the Kolmogorov-Smirnov type goodness-of-fit test for the inhomogeneous Poisson process with the unknown translation parameter as multidimensional parameter. The basic hypothesis and the alternative are composite and carry to the intensity measure of inhomogeneous Poisson process and the intensity function is regular. For this model of shift parameter, we propose test which is asymptotically partially distribution free and consistent. We show that under null hypothesis the limit distribution of this statistic does not depend on unknown parameter.


Introduction
One of the central themes of statistical theory and practice is the problem of the quality of goodness-of-fit tests. The problems of constructing the quality of goodness-of-fit tests in the case of i.i.d. are well studied in [1]. To set up a test that allows, if possible, accepting or rejecting the hypothesis to be tested against a given alternative, depending on a data set, a nonparametric study of the hypothesis tests is required, including a typical example that is the goodness-of-fit test and other important examples for applications that are the tests for symmetry, independence and homogeneity. [2] [3], and many other authors have worked in this area mainly in the mini max approach which is considered in nonparametric statistics as a good framework for determining the performance of an estimator.
In classical mathematical statistics, [4] intensely studied the Chi-square, Kolmogorov-Smirnov and Cramér-von Mises tests, and the Kolmogorov-Smirnov and Cramér-von Mises goodness-of-fit tests shown are asymptotically statistically free (i.e. have independent laws of the distribution under the null hypothesis). [5] recently studied in their paper the tests of nonparametric hypotheses for the intensity of the inhomogeneous Poisson process. The study they carried out is an extension to the Poisson processes of Ingster's work. [4] studied nonparametric tests for Gaussian white noise models with a ε noise level tending to 0. [6] presented in their article a review of several results concerning the construction of Kolmogorov-Smirnov-type and Cramér-von Mises-type fit tests for continuous-time processes. As models, they considered a small noise stochastic differential equation, an ergodic diffusion process, a Poisson process, and self-exciting or self-exciting point processes. [7] [8] consider the shift parameter model and the shift and scale parameter model, and show that the Cramér-von Mises test is asymptotically distribution free and asymptotically partially distribution free, and consistent. For each model, they proposed the tests which provide the asymptotic size α and describe the form of the power function under the local alternatives.
In applications, the hypotheses to be tested are often of a more complex nature. The first works on the problems of goodness-of-fit testing of composite hypotheses concerning classical statistics are due to [9] ( [2]) who proposed to test composite hypotheses, in the case where the distribution function under the hypothesis to be tested depends on a multidimensional unknown parameter.
The null hypothesis therefore becomes composite, i.e. it does not determine the distribution of the sample in a unique way. In the case where the parameters are estimated, the Kolmogorov-Smirnov test, as well as the Cramér-von Mises test is no longer asymptotically distribution free.
It follows that the critical values change from one null hypothesis to another. Different values of the parameter result in different critical values, often within the same parametric family. The distribution free character is therefore crucial in applications since the critical values are calculated only once for any distribution defined under the hypothesis to be tested. To work around this problem, [9] suggested the split sample method. Durbin's problem involves a martingale transformation of the parametric empirical process which was proposed by [10].
The martingale approach of [10] allows building asymptotically distribution free hypothesis tests. This approach proposed by [10] is used by various authors including [11] in the regression models, [12]. We use an approach similar to that of [10] to construct, in this article, Kolmogorov-Smirnov-type asymptotically distribution free and consistent goodness-of-fit tests.
We will consider the same model as [7]. In general, dealing with the measurement of the intensity of the Poisson process, we will consider the model de-pending on an unknown translation parameter with a composite parametric base assumption and show that the Kolmogorov-Smirnov test is asymptotically parameter free.

Statement of the Problem and Auxiliary Results
Suppose that we observe n independents inhomogeneous Poisson processes are trajectories of the Poisson processes with the mean function Here ( ) 0 λ ⋅ ≥ is the corresponding intensity function. : Then we can introduce the Kolmogorov-Smirnov (K-S) type statistic ⋅ is a known mean function of the Poisson process depending on some finite-dimensional unknown parameter ϑ ∈ Θ ⊂  . Note that under 0  there exists the true value 0 ϑ ∈ Θ such that the mean of the observed Poisson process ( ) ( ) The K-S type GoF test can be constructed by a similar way. Introduce the normalized process . The goal of this work is to show that if the unknown parameter ϑ , when ϑ ∈ Θ is the shift parameter, then it is possible to construct a test statistic ˆn Γ whose limit distribution does not depend on 0 ϑ . The test will be uniformly consistent against another class of alter- Here 0 ρ > is some given number.

The mean function under null hypothesis is
is known and therefore the solution can be calculated before the experiment using, say, numerical simulations.
We are given n independent observations ( ) ( ) Here 0 ϑ is the true value, and the intensity function is It is convenient to use two different functions and we hope that such notation will not be misleading.
Therefore, we have the parametric null hypothesis where the parametric family is Λ ⋅ is a known absolutely continuous function with properties: In this work, we denote by ( ) the derivative with respect to ϑ of any We consider the class of tests of asymptotic level ε : The test studied in this work is based on the following statistic of K-S type: As we use the asymptotic properties of the MLE ˆn ϑ , we need some regularity conditions.
Conditions  is strictly positive and three times continuously differentiable.
Note that, by these conditions, the MLE ˆn

Main Result
Let us introduce the following random variable The main result of this work is the following theorem.
Since the function Let us put ( ) ( ) ( ) It is easy to see that, if we change the variables The proof of the theorem is based on the proof of the following fundamental lemma.   Further, for the second relation, we have  Proof of the Lemma 3.4. The proof of the Lemma is based on the Central Limit theorem for stochastic integrals (see, e.g., Kutoyants [14], Theorem 1.1). We follow the proof of this theorem. In particular, we obtain the convergence when n → ∞ of the characteristic function where we put which is the characteristic function defined in (3.11). Therefore, we have the convergence of the one-dimensional distributions. In the general case, the verification of the convergence is entirely similar. In this work, we find the Kolmogorov-Smirnov GoF test based on sup-metrics in the case of the translation parameter. It is natural to ask: what if we take ( ) 2   metrics?