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Abstract
Calibration refers to the statistical estimation of unknown model parameters in computer experiments, such that computer experiments can match underlying physical systems. This work develops a new calibration method for imperfect computer models, Sobolev calibration, which can rule out calibration parameters that generate overfitting calibrated functions. We prove that the Sobolev calibration enjoys desired theoretical properties including fast convergence rate, asymptotic normality and semiparametric efficiency. We also demonstrate an interesting property that the Sobolev calibration can bridge the gap between two influential methods: L2 calibration and Kennedy and O’Hagan’s calibration. In addition to exploring the deterministic physical experiments, we theoretically justify that our method can transfer to the case when the physical process is indeed a Gaussian process, which follows the original idea of Kennedy and O’Hagan’s. Numerical simulations as well as a real-world example illustrate the competitive performance of the proposed method. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.
Supplementary Materials
The supplementary materials include details of the proof, additional experiment results and discussion.
Acknowledgments
The authors are grateful to the AE and reviewers for their very constructive comments and suggestions.
Disclosure Statement
The authors report there are no competing interests to declare.
Notes
1 Although a Sobolev space is typically defined on an open set, by extension and restriction theorems (DeVore and Sharpley Citation1993; Rychkov Citation1999), we can extend a Sobolev space on an open set to its closure.