On goodness-of-fit testing for ergodic diffusion process with shift parameter

Stat Inference Stoch Process

2014

We consider the problem of the construction of the asymptotically distribution free test by the observations of ergodic diffusion process. It is supposedd that under the basic hypothesis the trend coefficient depends on the finite dimensional parameter and we study the Cramér-von Mises type statistics. The underlying statistics depends on the deviation of the local time estimator from the invariant density with parameter replaced by the maximum likelihood estimator. We propose a linear transformation which yields the convergence of the test statistics to the integral of Wiener process. Therefore the test based on this statistics is asymptotically distribution free.MSC 2000 Classification: 62M02, 62G10, 62G20.

Section: Introductionmentioning

confidence: 99%

On asymptotically distribution free tests with parametric hypothesis for ergodic diffusion processes

Kleptsyna

Stat Inference Stoch Process

2014

“…The general case of ergodic diffusion processes with one-dimensional shift parameter was studied by Negri and Zhou [16]. They showed that the limit distribution of the Cramér-von Mises statistic does not depend on the unknown parameter.…”

Section: Introductionmentioning

confidence: 99%

On asymptotic distribution of parameter free tests for ergodic diffusion processes

Stat Inference Stoch Process

2014

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

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

2016

Math. Meth. Stat.

The problems of the construction of the asymptotically distribution free goodness-of-fit tests for three models of stochastic processes are considered. The null hypothesis for all models is composite parametric. All tests are based on the score-function processes, where the unknown parameter is replaced by the MLE. We show that a special change of time transforms the limit score-function processes into the Brownian bridge. This property allows us to construct the asymptotically distribution free tests for the following three models of stochastic processes : dynamical systems with small noise, ergodic diffusion processes, inhomogeneous Poisson processes and nonlinear AR time series.MSC 2000 Classification: 62M02, 62G10, 62G20.