## Bernoulli

• Bernoulli
• Volume 24, Number 3 (2018), 2278-2327.

### Schwarz type model comparison for LAQ models

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 24, Number 3 (2018), 2278-2327.

Dates
Revised: November 2016
First available in Project Euclid: 2 February 2018

https://projecteuclid.org/euclid.bj/1517540475

Digital Object Identifier
doi:10.3150/17-BEJ928

Mathematical Reviews number (MathSciNet)
MR3757530

Zentralblatt MATH identifier
06839267

#### Citation

Eguchi, Shoichi; Masuda, Hiroki. Schwarz type model comparison for LAQ models. Bernoulli 24 (2018), no. 3, 2278--2327. doi:10.3150/17-BEJ928. https://projecteuclid.org/euclid.bj/1517540475

#### References

• [1] Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory (Tsahkadsor, 1971) 267–281. Budapest: Akadémiai Kiadó.
• [2] Akaike, H. (1974). A new look at the statistical model identification. IEEE Trans. Automat. Control AC-19 716–723.
• [3] Bickel, P.J., Li, B., Tsybakov, A.B., van de Geer, S.A., Yu, B., Valdés, T., Rivelo, C., Fan, J. and van der Vaart, A. (2006). Regularization in statistics. TEST 15 271–344.
• [4] Boswijk, H.P. (2010). Mixed normal inference on multicointegration. Econometric Theory 26 1565–1576.
• [5] Bozdogan, H. (1987). Model selection and Akaike’s information criterion (AIC): The general theory and its analytical extensions. Psychometrika 52 345–370.
• [6] Brouste, A., Fukasawa, M., Hino, H., Iacus, S.M., Kamatani, K., Koike, Y., Masuda, H., Nomura, R., Ogihara, T., Shimizu, Y., Uchida, M. and Yoshida, N. (2014). The yuima project: A computational framework for simulation and inference of stochastic differential equations. J. Stat. Softw. 57 1–51.
• [7] Burnham, K.P. and Anderson, D.R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. New York: Springer.
• [8] Casella, G., Girón, F.J., Martínez, M.L. and Moreno, E. (2009). Consistency of Bayesian procedures for variable selection. Ann. Statist. 37 1207–1228.
• [9] Cavanaugh, J.E. and Neath, A.A. (1999). Generalizing the derivation of the Schwarz information criterion. Comm. Statist. Theory Methods 28 49–66.
• [10] Chan, N.H., Huang, S.-F. and Ing, C.-K. (2013). Moment bounds and mean squared prediction errors of long-memory time series. Ann. Statist. 41 1268–1298.
• [11] Chan, N.H. and Ing, C.-K. (2011). Uniform moment bounds of Fisher’s information with applications to time series. Ann. Statist. 39 1526–1550.
• [12] Chen, J. and Chen, Z. (2008). Extended Bayesian information criteria for model selection with large model spaces. Biometrika 95 759–771.
• [13] Claeskens, G. and Hjort, N.L. (2008). Model Selection and Model Averaging. Cambridge: Cambridge Univ. Press.
• [14] Dziak, J.J., Coffman, D.L., Lanze, S.T. and Li, R. (2012). Sensitivity and specificity of information criteria. PeerJ PrePrints 3.
• [15] Eguchi, S. and Masuda, H. (2015). Quasi-Bayesian model comparison for LAQ models. Technical report, MI Preprint Series 2015-7, Kyushu University.
• [16] Fasen, K. and Kimmig, S. (2015). Information criteria for multivariate CARMA processes. Bernoulli. To appear. Available at arXiv:1505.00901.
• [17] Foster, D.P. and George, E.I. (1994). The risk inflation criterion for multiple regression. Ann. Statist. 22 1947–1975.
• [18] Fujii, T. and Uchida, M. (2014). AIC type statistics for discretely observed ergodic diffusion processes. Stat. Inference Stoch. Process. 17 267–282.
• [19] Genon-Catalot, V. and Jacod, J. (1993). On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann. Inst. Henri Poincaré Probab. Stat. 29 119–151.
• [20] Gobet, E. (2002). LAN property for ergodic diffusions with discrete observations. Ann. Inst. Henri Poincaré Probab. Stat. 38 711–737.
• [21] Goutis, C. and Robert, C.P. (1998). Model choice in generalised linear models: A Bayesian approach via Kullback–Leibler projections. Biometrika 85 29–37.
• [22] Kamatani, K. and Uchida, M. (2015). Hybrid multi-step estimators for stochastic differential equations based on sampled data. Stat. Inference Stoch. Process. 18 177–204.
• [23] Kashyap, R.L. (1982). Optimal choice of AR and MA parts in autoregressive moving average models. IEEE Trans. Pattern Anal. Mach. Intell. 4 99–104.
• [24] Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scand. J. Stat. 24 211–229.
• [25] Kim, J.Y. (1998). Large sample properties of posterior densities, Bayesian information criterion and the likelihood principle in nonstationary time series models. Econometrica 66 359–380.
• [26] Konishi, S., Ando, T. and Imoto, S. (2004). Bayesian information criteria and smoothing parameter selection in radial basis function networks. Biometrika 91 27–43.
• [27] Konishi, S. and Kitagawa, G. (1996). Generalised information criteria in model selection. Biometrika 83 875–890.
• [28] Konishi, S. and Kitagawa, G. (2008). Information Criteria and Statistical Modeling. Springer Series in Statistics. New York: Springer.
• [29] Lavine, M. and Schervish, M.J. (1999). Bayes factors: What they are and what they are not. Amer. Statist. 53 119–122.
• [30] Liu, W. and Yang, Y. (2011). Parametric or nonparametric? A parametricness index for model selection. Ann. Statist. 39 2074–2102.
• [31] Lv, J. and Liu, J.S. (2014). Model selection principles in misspecified models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 141–167.
• [32] Masuda, H. (2013). Convergence of Gaussian quasi-likelihood random fields for ergodic Lévy driven SDE observed at high frequency. Ann. Statist. 41 1593–1641.
• [33] Masuda, H. and Uehara, Y. (2016). On stepwise estimation of Lévy driven stochastic differential equation (Japanese). Proc. Inst. Statist. Math.
• [34] Nishii, R. (1984). Asymptotic properties of criteria for selection of variables in multiple regression. Ann. Statist. 12 758–765.
• [35] Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461–464.
• [36] Sclove, S.L. (1987). Application of model-selection criteria to some problems in multivariate analysis. Psychometrika 52 333–343.
• [37] Sei, T. and Komaki, F. (2007). Bayesian prediction and model selection for locally asymptotically mixed normal models. J. Statist. Plann. Inference 137 2523–2534.
• [38] Uchida, M. (2010). Contrast-based information criterion for ergodic diffusion processes from discrete observations. Ann. Inst. Statist. Math. 62 161–187.
• [39] Uchida, M. and Yoshida, N. (2001). Information criteria in model selection for mixing processes. Stat. Inference Stoch. Process. 4 73–98.
• [40] Uchida, M. and Yoshida, N. (2006). Asymptotic expansion and information criteria. SUT J. Math. 42 31–58.
• [41] Uchida, M. and Yoshida, N. (2011). Estimation for misspecified ergodic diffusion processes from discrete observations. ESAIM Probab. Stat. 15 270–290.
• [42] Uchida, M. and Yoshida, N. (2012). Adaptive estimation of an ergodic diffusion process based on sampled data. Stochastic Process. Appl. 122 2885–2924.
• [43] Uchida, M. and Yoshida, N. (2013). Quasi likelihood analysis of volatility and nondegeneracy of statistical random field. Stochastic Process. Appl. 123 2851–2876.
• [44] Uchida, M. and Yoshida, N. (2016). Model selection for volatility prediction. In The Fascination of Probability, Statistics and Their Applications 343–360. Cham: Springer.
• [45] van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge: Cambridge Univ. Press.
• [46] Vehtari, A. and Ojanen, J. (2012). A survey of Bayesian predictive methods for model assessment, selection and comparison. Stat. Surv. 6 142–228.
• [47] Yoshida, N. (2011). Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations. Ann. Inst. Statist. Math. 63 431–479.