Calibration Intervals in Linear Regression Models

Abstract

Many of the existing methods of finding calibration intervals in simple linear regression rely on the inversion of prediction limits. In this article, we propose an alternative procedure which involves two stages. In the first stage, we find a confidence interval for the value of the explanatory variable which corresponds to the given future value of the response. In the second stage, we enlarge the confidence interval found in the first stage to form a confidence interval called, calibration interval, for the value of the explanatory variable which corresponds to the theoretical mean value of the future observation. In finding the confidence interval in the first stage, we have used the method based on hypothesis testing and percentile bootstrap. When the errors are normally distributed, the coverage probability of resulting calibration interval based on hypothesis testing is comparable to that of the classical calibration interval. In the case of non normal errors, the coverage probability of the calibration interval based on hypothesis testing is much closer to the target value than that of the calibration interval based on percentile bootstrap.

Keywords:

• Bootstrap
• Calibration intervals
• Linear regression
• Quadratic-normal distribution

Mathematics Subject Classification:

• Primary 62J05
• Secondary 62F40

Acknowledgment

The authors are very grateful to the referee for making various suggestions which have considerably improved the article.

Notes

*CL: Classical calibration interval; HT: Calibration interval based on hypothesis testing; EL.CL: Expected length for classical calibration interval; EL.HT: Expected length for calibration interval based on hypothesis testing.

*BT: Percentile bootstrap calibration interval; HT: Calibration interval based on hypothesis testing; EL.BT: Expected length for percentile bootstrap calibration interval; EL.HT: Expected length for calibration interval based on hypothesis testing.