Statistical calibration for infinite many future values in linear regression: simultaneous or pointwise tolerance intervals or what else?

Abstract

Statistical calibration using regression is a useful statistical tool with many applications. For confidence sets for x-values associated with infinitely many future y-values, there is a consensus in the statistical literature that the confidence sets constructed should guarantee a key property. While it is well known that the confidence sets based on the simultaneous tolerance intervals (STI’s) guarantee this key property conservatively, it is desirable to construct confidence sets that satisfy this property exactly. Also, there is a misconception that the confidence sets based on the pointwise tolerance intervals (PTI’s) also guarantee this property. This paper constructs the weighted simultaneous tolerance intervals (WSTI’s) so that the confidence sets based on the WSTI’s satisfy this property exactly if the future observations have the
x-values distributed according to a known specific distribution F(·). Through the lens of the WSTI’s, convincing counter examples are also provided to demonstrate that the confidence sets based on the PTI’s do not guarantee the key property in general and so should not be used. The WSTI’s have been applied to real data examples to show that the WSTI’s can produce more accurate calibration intervals than STI’s and PTI’s.

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ORCID for Wei Liu: orcid.org/0000-0002-4719-0345

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