Simultaneous Tolerance Intervals for the Linear Regression Model

Statistical tolerance intervals are developed for the normal regression model. These intervals are constructed to guarantee at least P content for all possible values of the predictor variates. The confidence-set approach suggested by Wilson (1967) is used. Wilson used an ellipsoidal confidence set for the regression coefficients, β, and standard deviation of the residuals, σ, which imposes an unnecessary lower bound on σ. By modifying the ellipsoidal confidence set to remove the lower bound imposed on σ, we obtain narrower tolerance intervals. Another confidence set formed from the product set of confidence sets for β and σ is used to construct tolerance intervals. The tolerance intervals of the two new procedures are compared with those of the Wilson method for a simple linear regression example. The tolerance intervals based on the product confidence set are found to be efficient and easy to compute compared with those constructed from the ellipsoidal and the modified ellipsoidal confidence sets.