One-sided simultaneous tolerance limits for regression

Mee, Eberhardt, and Reeve (1989) recently produced tables of factors for simultaneous two-sided tolerance intervals for linear regression. These factors, obtained using numerical quadrature, provide narrower intervals than were previously available. Using identical notation, this article presents simultaneous one-sided tolerance limits for regression models. Since one-sided tolerance limits are equivalent to one-sided confidence limits on percentiles, the bounds proposed here provide simultaneous one-sided confidence limits for a specified percentile of the conditional distributions of the dependent variable. Although this specific problem had not been addressed previously in the literature, several authors have proposed simultaneous two-sided confidence intervals for a specified percentile of the dependent variable in regression (Steinhorst and Bowden 1971 ; Thomas and Thomas 1986; Turner and Bowden 1977). The limits proposed here have several applications to calibration (or discrimination). For example, one use is the construction of one-sided confidence limits for future unobserved values of the independent variable. The analogous two-sided inference in calibration has been discussed by Lieberman, Miller and Hamilton (1967) and Scheffe' (1973).