ON THE ASYMPTOTIC STABILITY OF THE INTRINSIC AND FRACTIONAL BAYES FACTORS FOR TESTING SOME DIFFUSION MODELS

S. SIVAGANESAN^{1} AND RAMA T. LINGHAM^{2}

1Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, U.S.A. 2Division of Statistics, Northern Illinois University, DeKalb, IL 60115, U.S.A.

(Received May 17, 1999; revised May 14, 2001)

Abstract. Random processes, from which a single sample path data are available on a fine time scale, abound in many areas including finance and genetics. An effective way to model such data is to consider a suitable continuous-time-scale analog, X_{t} say, for the underlying process. We consider three diffusion models for the process X_{t} and address model selection under improper priors. Specifically, fractional and intrinsic Bayes factors (FBF and IBF) for model selection are considered. Here, we focus on the asymptotic stability of the IBF’s and FBF’s for comparing these models. Specifically, we propose to employ certain novel transformations of the data in order to ensure the asymptotic stability of the IBF’s. While we use different transformations for pairwise comparisons of the models, we also show that a single common transformation can be used when simultaneously comparing all three models. We then demonstrate that, when FBF’s are used to compare these models, we may have to employ different, model-specific training fractions in order to achieve asymptotic stability of the FBF’s.

Key words and phrases: Fractional Bayes factor, Girsanov formula, intrinsic Bayes factor, Jeffreys prior, local asymptotic normality, mean-reversion, Wiener process.