[Paper Review] On the Goodness-of-Fit Testing for Ergodic Diffusion Processes

Yury A. Kutoyants|ArXiv.org|Mar 26, 2009

Diffusion Coefficients in Liquids6 references3 citations

TL;DR

This paper proposes asymptotically distribution-free (ADF) Cramér-von Mises type goodness-of-fit tests for ergodic diffusion processes under a simple hypothesis on the drift coefficient, with known diffusion coefficient. By using the empirical distribution function and a local time estimator of the invariant density, the authors derive a transformation that renders the test statistics asymptotically distribution-free, ensuring valid critical values across different models, and establish consistency against all fixed alternatives.

ABSTRACT

We consider the goodness of fit testing problem for ergodic diffusion processes. The basic hypothesis is supposed to be simple. The diffusion coefficient is known and the alternatives are described by the different trend coefficients. We study the asymptotic distribution of the Cramer-von Mises type tests based on the empirical distribution function and local time estimator of the invariant density. At particularly, we propose a transformation which makes these tests asymptotically distribution free. We discuss the modifications of this test in the case of composite basic hypothesis.

Motivation & Objective

• To develop goodness-of-fit tests for ergodic diffusion processes under a simple hypothesis on the drift coefficient.
• To ensure the test statistics are asymptotically distribution-free (ADF) under the null hypothesis, enabling universal critical values.
• To establish consistency of the tests against all fixed alternatives, regardless of the specific alternative trend function.
• To extend the framework to composite hypotheses by incorporating parameter estimation in the test construction.

Proposed method

• Constructs a Cramér-von Mises type test statistic based on the empirical distribution function and local time estimator of the invariant density.
• Derives the asymptotic distribution of the test statistic under the null hypothesis using weak convergence theory.
• Applies a transformation to the test statistic that removes dependence on the underlying model, achieving asymptotic distribution-freeness.
• Uses the invariant density estimator based on local time to consistently estimate the true invariant density under the null.
• Establishes asymptotic normality and convergence in distribution to a functional of Brownian bridge under the null.
• Analyzes the behavior of the test under alternatives by showing divergence of the test statistic to infinity under fixed alternatives.

Experimental results

Research questions

• RQ1 Can a Cramér-von Mises type test for ergodic diffusion processes be made asymptotically distribution-free (ADF) under the null hypothesis?
• RQ2 How can the invariant density be consistently estimated from continuous-time sample paths to enable GoF testing?
• RQ3 What transformation ensures that the test statistic's limiting distribution does not depend on the model parameters or the true drift function?
• RQ4 Is the proposed test consistent against all fixed alternatives, including those with small deviations from the null?
• RQ5 How does the test perform under alternatives defined by Kullback-Leibler distance, and why might it fail to be uniformly consistent in such cases?

Key findings

• The proposed test statistic, after an appropriate transformation, converges in distribution to a functional of the Brownian bridge, making it asymptotically distribution-free.
• The test is consistent against all fixed alternatives, as the test statistic diverges to infinity under any alternative trend function.
• The asymptotic distribution of the test statistic is free of the underlying model parameters, enabling universal critical values across different diffusion models.
• The invariant density estimator based on local time achieves consistency and enables the construction of the test statistic from continuous-time observations.
• The test loses uniform consistency under alternatives defined via Kullback-Leibler distance, such as oscillatory perturbations, due to vanishing L2 distance between the true and null densities.
• For such alternatives, the paper suggests Chi-squared tests as a more suitable alternative, potentially minimax optimal.