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Calibration parameters in deterministic computer experiments are those attributes that cannot be measured or available in physical experiments. Kennedy and O'Hagan \cite{kennedy2001bayesian} suggested an approach to estimate them by using data from physical experiments and computer simulations. A theoretical framework is given which allows us to study the issues of parameter identifiability and estimation. We define the $L_2$-consistency for calibration as a justification for calibration methods. It is shown that a simplified version of the original KO method leads to asymptotically $L_2$-inconsistent calibration. This $L_2$-inconsistency can be remedied by modifying the original estimation procedure. A novel calibration method, called the $L_2$ calibration, is proposed and proven to be $L_2$-consistent and enjoys optimal convergence rate. A numerical example and some mathematical analysis are used to illustrate the source of the $L_2$-inconsistency problem.

A Theoretical Framework for Calibration in Computer Models: Parametrization, Estimation and Convergence Properties