Statistical calibration and exact one-sided simultaneous tolerance intervals for polynomial regression

Statistical calibration using linear regression is a useful statistical tool having many applications. Calibration for infinitely many future $y$-values requires the construction of simultaneous tolerance intervals (STI's). As calibration often involves only two variables $x$ and $y$ and polynomial regression is probably the most frequently used model for relating $y$ with $x$, construction of STI's for polynomial regression plays a key role in statistical calibration for infinitely many future $y$values. The only exact STI's published in the statistical literature are provided by Mee {\it et al.} (1991) and Odeh and Mee (1990). But they are for a multiple linear regression model, in which the covariates are assumed to have no functional relationships. When applied to polynomial regression, the resultant STI's are conservative. In this paper, exact one-sided STI's have been constructed for a polynomial regression model over any given interval. The available computer program allows the exact methods developed in this paper to be implemented easily. Real examples are given for illustration. Statistical calibration and exact one-sided simultaneous tolerance intervals for polynomial regression Yang Han , Wei Liua,∗, Frank Bretz , Fang Wan , Ping Yang aS3RI and School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK bUniversity of Exeter Medical School, Exeter, EX1 2LU, UK cNovartis Pharma AG, Basel, 4002, Switzerland dShanghai University of Finance and Economics, Shanghai, 200433, China eDepartment of Statistics, The Chinese University of Hong Kong, Hong Kong