On asymptotic distribution of parameter free tests for ergodic diffusion processes

Abstract

We consider two problems of constructing of goodness of fit tests for ergodic diffusion processes. The first one is concerned with a composite basic hypothesis for a parametric class of diffusion processes, which includes the Ornstein-Uhlenbeck and simple switching processes. In this case we propose asymptotically parameter free tests of Cramér-von Mises type. The basic hypothesis in the second problem is simple and we propose asymptotically distribution free tests for a wider class of trend coefficients.MSC 2000 Classification: 62M02, 62G10, 62G20.Key words: Cramér-von Mises tests, ergodic diffusion process, goodness of fit test, asymptotically distribution free.In the second problem we assume that under the basic hypothesis (H 0 ) the observed process satisfieswhere S 0 (·) is a known function, i.e., (H 0 ) is simple.In both models the alternatives are nonparametric and, under the hypothesis H 0 , the diffusion processes are assumed to be ergodic with the invariant densities f (ϑ, x) and f S 0 (x) respectively. We denote the corresponding distribution functions by F (ϑ, x) and F S 0 (x).Our goal is to construct the goodness of fit tests which provide the fixed limit error ε ∈ (0, 1). Introduce the class K ε of such tests, i.e., the testsψ T satisfying the relationsand limT →∞in the first and the second problems respectively. All tests studied in the present work are of the formψ T = 1I {∆ T >cε} , where ∆ T is the Cramér-von Mises type statistic. More precisely, in the first problem ∆ T is either of the L 2 distances D F T (x) , F θ T , x and

On ADF goodness-of-fit tests for perturbed dynamical systems

“…Our objective is to realize this program. A similar result for ergodic diffusion processes is contained in Kutoyants [12] (simple basic hypothesis) and Kleptsyna and Kutoyants [7] (parametric basic hypothesis).…”

On APF test for Poisson process with shift and scale parameters

“…Here V (s) = λ 0 (s) , sλ 0 (s) − Λ 0 (s) . The convergences (12), (13) we will prove in several steps.…”

“…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.).…”

On score-functions and goodness-of-fit tests for stochastic processes

“…This work is a continuation of the study of GoF tests for diffusion processes observed in continuous time. The case of simple basic hypothesis was treated for example in the works [4], [7], [12], [1], [20], [14]. The case of parametric basic hypothesis and ADF tests was studied in the works [21], [14], [9], [15], [16].…”