For model-comparison purpose, we study asymptotic behavior of the marginal quasi-log likelihood associated with a family of locally asymptotically quadratic (LAQ) statistical experiments. Our result entails a far-reaching extension of applicable scope of the classical approximate Bayesian model comparison due to Schwarz, with frequentist-view theoretical foundation. In particular, the proposed statistics can deal with both ergodic and non-ergodic stochastic process models, where the corresponding $M$-estimator may of multi-scaling type and the asymptotic quasi-information matrix may be random. We also deduce the consistency of the multistage optimal-model selection where we select an optimal sub-model structure step by step, so that computational cost can be much reduced. Focusing on some diffusion type models, we illustrate the proposed method by the Gaussian quasi-likelihood for diffusion-type models in details, together with several numerical experiments.
Shoichi Eguchi. Hiroki Masuda. "Schwarz type model comparison for LAQ models." Bernoulli 24 (3) 2278 - 2327, August 2018. https://doi.org/10.3150/17-BEJ928