Kriging models are widely employed due to their simplicity and flexibility in a variety of fields. To gain more distributional information about the unknown parameters, we focus on constructing the fiducial distribution of Kriging model parameters. To solve the challenge of constructing the fiducial marginal distribution for the spatially related parameter, we substitute the Bayesian posterior distribution for the fiducial distribution of this spatially related parameter and present a quasi-fiducial distribution for Kriging model parameters. A Gibbs sampling algorithm is given to get the samples of the quasi-fiducial distribution. Then a model selection criterion based on the quasi-fiducial distribution is proposed. Numerical studies demonstrate that the proposed method is superior to the Lasso and Elastic Net.

References

• Chen, H., Huang, H., Lin, D. K., & Liu, M. Q. (2016). Uniform sliced Latin hypercube designs. Applied Stochastic Models in Business and Industry, 32(5), 574–584. https://doi.org/10.1002/asmb.2192
• Cui, Y., Hannig, J., & Edlefsen, P. (2024). Semiparametric fiducial inference. https://arxiv.org/abs/2404.18779(open in a new window)
• Hannig, J. (2009). On generalized fiducial inference. Statistica Sinica, 19(2), 491–544. https://doi.org/10.5705/ss.2011.102
• Hannig, J. (2013). Generalized fiducial inference via discretization. Statistica Sinica, 23(2), 489–514.
• Hodges, J. S., & Reich, B. J. (2010). Adding spatially-correlated errors can mess up the fixed effect you love. The American Statistician, 64(4), 325–334. https://doi.org/10.1198/tast.2010.10052
• Huang, H., Lin, D. K., Liu, M. Q., & Zhang, Q. (2020). Variable selection for Kriging in computer experiments. Journal of Quality Technology, 52(1), 40–53. https://doi.org/10.1080/00224065.2019.1569959
• Hung, Y. (2011). Penalized blind Kriging in computer experiments. Statistica Sinica, 21(3), 1171–1190. https://doi.org/10.5705/ss.2011.v21n3a
• Joseph, V. R., Hung, Y., & Sudjianto, A. (2008). Blind Kriging: A new method for developing metamodels. Journal of Mechanical Design, 130(3), 350–353. https://doi.org/10.1115/1.2829873
• Morris, M. D., Mitchell, T. J., & Ylvisaker, D. (1993). Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction. Technometrics, 35(3), 243–255. https://doi.org/10.1080/00401706.1993.10485320
• Neal, R. M. (2003). Slice sampling. Annals of Statistics, 31(3), 705–767. https://doi.org/10.1214/aos/1056562461
• Paciorek, C. J. (2010). The importance of scale for spatial-confounding bias and precision of spatial regression estimators. Statistical Science, 25(1), 107–125. https://doi.org/10.1214/10-STS326
• Park, I. (2021). Lasso Kriging for efficiently selecting a global trend model. Structural and Multidisciplinary Optimization, 64(3), 1527–1543. https://doi.org/10.1007/s00158-021-02939-7
• Sacks, J., Schiller, S. B., & Welch, W. J. (1989). Designs for computer experiments. Technometrics, 31(1), 41–47. https://doi.org/10.1080/00401706.1989.10488474
• Shao, J. (2010). Mathematical Statistics (2nd ed.). Springer.
• Simpson, T. W., Mauery, T. M., Korte, J. J., & Mistree, F. (2001). Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA Journal, 39(12), 2233–2241. https://doi.org/10.2514/2.1234
• Tuo, R., & Wu, C. J. (2015). Efficient calibration for imperfect computer models. Annals of Statistics, 43(6), 2331–2352. https://doi.org/10.1214/15-AOS1314
• Welch, W. J., Buck, R. J., Sacks, J., Wynn, H. P., Mitchell, T. J., & Morris, M. D. (1992). Screening, predicting, and computer experiments. Technometrics, 34(1), 15–25. https://doi.org/10.2307/1269548
• Williams, J. P., & Hannig, J. (2019). Nonpenalized variable selection in high-dimensional linear model settings via generalized fiducial inference. Annals of Statistics, 47(3), 1723–1753. https://doi.org/10.1214/18-AOS1733
• Zhao, Y., Feng, Z., Li, M., & Li, X. (2023). Modified penalized blind Kriging for efficiently selecting a global trend model. Journal of Statistical Computation and Simulation, 93(17), 3052–3066. https://doi.org/10.1080/00949655.2023.2216335
• Zhao, Y., Liang, H., & Li, X. (2023). Fiducial variable selection for the high-dimensional regression model. Scientia Sinica Mathematica, 53(6), 839–858. https://doi.org/10.1360/SCM-2022-0245

To cite this article: Chen Fan, Shuqin Zhang & Xinmin Li (2025) The quasi-fiducial model selection for Kriging model, Statistical Theory and Related Fields, 9:3, 285-296, DOI: 10.1080/24754269.2025.2537484

To link to this article: https://doi.org/10.1080/24754269.2025.2537484