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Multimodel inference based on smoothed information criteria

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Abstract

The multimodel inference makes statistical inferences from a set of plausible models rather than from a single model. In this paper, we focus on the multimodel inference based on smoothed information criteria proposed by seminal monographs (see Buckland et al. (1997) and Burnham and Anderson (2003)), which are termed as smoothed Akaike information criterion (SAIC) and smoothed Bayesian information criterion (SBIC) methods. Due to their simplicity and applicability, these methods are very widely used in many fields. By using an illustrative example and deriving limiting properties for the weights in the linear regression, we find that the existing variance estimation for SAIC is not applicable because of a restrictive condition, but for SBIC it is applicable. Especially, we propose a simulation-based inference for SAIC based on the limiting properties. Both the simulation study and real data example show the promising performance of the proposed simulation-based inference.

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References

  1. Ando T, Li K C. A weight-relaxed model averaging approach for high-dimensional generalized linear models. Ann Statist, 2017, 45: 2654–2679

    Article  MathSciNet  Google Scholar 

  2. Aspinall R. Modelling land use change with generalized linear models—a multi-model analysis of change between 1860 and 2000 in Gallatin Valley, Montana. J Environ Manag, 2004, 72: 91–103

    Article  Google Scholar 

  3. Buckland S T, Burnham K P, Augustin N H. Model selection: An integral part of inference. Biometrics, 1997, 53: 603–618

    Article  Google Scholar 

  4. Burnham K P, Anderson D R. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. New York: Springer-Verlag, 2003

    MATH  Google Scholar 

  5. Burnham K P, Anderson D R, Huyvaert K P. AIC model selection and multimodel inference in behavioral ecology: Some background, observations, and comparisons. Behav Ecology Sociobiology, 2011, 65: 23–35

    Article  Google Scholar 

  6. Cade B S. Model averaging and muddled multimodel inferences. Ecology, 2015, 96: 2370–2382

    Article  Google Scholar 

  7. Campbell D, Mørkbak M R, Olsen S B. The link between response time and preference, variance and processing heterogeneity in stated choice experiments. J Environ Econom Manag, 2018, 88: 18–34

    Article  Google Scholar 

  8. Charkhi A, Claeskens G, Hansen B E. Minimum mean squared error model averaging in likelihood models. Statist Sinica, 2016, 26: 809–840

    MathSciNet  MATH  Google Scholar 

  9. Chen J, Li D G, Linton O, et al. Semiparametric ultra-high dimensional model averaging of nonlinear dynamic time series. J Amer Statist Assoc, 2018, 113: 919–932

    Article  MathSciNet  Google Scholar 

  10. Diks C G H, Vrugt J A. Comparison of point forecast accuracy of model averaging methods in hydrologic applications. Stoch Environ Res Risk Assess, 2010, 24: 809–820

    Article  Google Scholar 

  11. Fletcher D, Dillingham P W. Model-averaged confidence intervals for factorial experiments. Comput Statist Data Anal, 2011, 55: 3041–3048

    Article  MathSciNet  Google Scholar 

  12. Fragoso T M, Bertoli W, Louzada F. Bayesian model averaging: A systematic review and conceptual classification. Internat Statist Rev, 2018, 86: 1–28

    Article  MathSciNet  Google Scholar 

  13. Hansen B E. Least squares model averaging. Econometrica, 2007, 75: 1175–1189

    Article  MathSciNet  Google Scholar 

  14. Hansen B E, Racine J S. Jackknife model averaging. J Econometrics, 2012, 167: 38–46

    Article  MathSciNet  Google Scholar 

  15. Hjort N L, Claeskens G. Frequentist model average estimators. J Amer Statist Assoc, 2003, 98: 879–899

    Article  MathSciNet  Google Scholar 

  16. Hocking R R. Methods and Applications of Linear Models, 2nd ed. Hoboken: John Wiley & Sons, 2003

    Book  Google Scholar 

  17. Hoeting J A, Madigan D, Raftery A E, et al. Bayesian model averaging: A tutorial. Statist Sci, 1999, 14: 382–417

    Article  MathSciNet  Google Scholar 

  18. Hong H, Preston B. Bayesian averaging, prediction and nonnested model selection. J Econometrics, 2012, 167: 358–369

    Article  MathSciNet  Google Scholar 

  19. Hsiao C, Wan S K. Is there an optimal forecast combination? J Econometrics, 2014, 178: 294–309

    Article  MathSciNet  Google Scholar 

  20. Leamer E E. Specification searches: Ad hoc inference with nonexperimental data. Technometrics, 1981, 23: 112–113

    Google Scholar 

  21. Li C, Li Q, Racine J, et al. Optimal model averaging of varying coefficient models. Statist Sinica, 2018, 28: 2795–2809

    MathSciNet  MATH  Google Scholar 

  22. Li D G, Linton O, Lu Z D. A flexible semiparametric forecasting model for time series. J Econometrics, 2015, 187: 345–357

    Article  MathSciNet  Google Scholar 

  23. Link W A, Barker R J. Model weights and the foundations of multimodel inference. Ecology, 2006, 87: 2626–2635

    Article  Google Scholar 

  24. Liu C A. Distribution theory of the least squares averaging estimator. J Econometrics, 2015, 186: 142–159

    Article  MathSciNet  Google Scholar 

  25. Liu Q F, Okui R. Heteroskedasticity-robust Cp model averaging. Econom J, 2013, 16: 463–472

    Article  MathSciNet  Google Scholar 

  26. Lu X, Su L J. Jackknife model averaging for quantile regressions. J Econometrics, 2015, 188: 40–58

    Article  MathSciNet  Google Scholar 

  27. Lukacs P M, Burnham K P, Anderson D R. Model selection bias and Freedman’s paradox. Ann Inst Statist Math, 2010, 62: 117–125

    Article  MathSciNet  Google Scholar 

  28. Mitra P, Lian H, Mitra R, et al. A general framework for frequentist model averaging. Sci China Math, 2019, 62: 205–226

    Article  MathSciNet  Google Scholar 

  29. Rao C R. Linear statistical Inference and Its Applications. Hoboken: Wiley, 1973

    Book  Google Scholar 

  30. Symonds M R E, Moussalli A. A brief guide to model selection, multimodel inference and model averaging in behavioural ecology using Akaike’s information criterion. Behav Ecology Sociobiology, 2011, 65: 13–21

    Article  Google Scholar 

  31. Turek D, Fletcher D. Model-averaged Wald confidence intervals. Comput Statist Data Anal, 2012, 56: 2809–2815

    Article  MathSciNet  Google Scholar 

  32. Wan A T K, Zhang X Y. On the use of model averaging in tourism research. Ann Tourism Res, 2009, 36: 525–532

    Article  Google Scholar 

  33. Xie T. Prediction model averaging estimator. Econom Lett, 2015, 131: 5–8

    Article  MathSciNet  Google Scholar 

  34. Yang Y H. Adaptive regression by mixing. J Amer Statist Assoc, 2001, 96: 574–588

    Article  MathSciNet  Google Scholar 

  35. Yuan Z, Yang Y H. Combining linear regression models: When and how? J Amer Statist Assoc, 2005, 100: 1202–1214

    Article  MathSciNet  Google Scholar 

  36. Zhang X Y, Liu C A. Inference after model averaging in linear regression models. Econometric Theory, 2019, 35: 816–841

    Article  MathSciNet  Google Scholar 

  37. Zhang X Y, Lu Z D, Zou G H. Adaptively combined forecasting for discrete response time series. J Econometrics, 2013, 176: 80–91

    Article  MathSciNet  Google Scholar 

  38. Zhang X Y, Yu D L, Zou G H, et al. Optimal model averaging estimation for generalized linear models and generalized linear mixed-effects models. J Amer Statist Assoc, 2016, 111: 1775–1790

    Article  MathSciNet  Google Scholar 

  39. Zhang X Y, Zou G H, Carroll R J. Model averaging based on Kullback-Leibler distance. Statist Sinica, 2015, 25: 1583–1598

    MathSciNet  MATH  Google Scholar 

  40. Zhang X Y, Zou G H, Liang H. Choice of weights in FMA estimators under general parametric models. Sci China Math, 2013, 56: 443–457

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by National Key R&D Program of China (Grant No. 2020AAA 0105200), National Natural Science Foundation of China (Grant Nos. 12001559, 71925007, 71988101 and 72042019), Ministry of Education of China (Grant No. 17YJC910011), the Youth Innovation Promotion Association of the Chinese Academy of Sciences, the Beijing Academy of Artificial Intelligence, and Academy for Multidisciplinary Studies, Capital Normal University. The authors thank two referees for helpful comments.

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Correspondence to Xinyu Zhang.

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Zhao, S., Zhang, X. Multimodel inference based on smoothed information criteria. Sci. China Math. 64, 2563–2578 (2021). https://doi.org/10.1007/s11425-020-1798-y

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