Note on a paradox in decision-theoretic interval estimation

TL;DR

This paper addresses a paradox in decision-theoretic interval estimation for the normal mean with unknown variance, proposing a simple modification to the loss function that removes the paradoxical behavior of the resulting confidence interval.

Contribution

It introduces a modified loss function that, when combined with a standard prior, yields confidence intervals without paradoxical behavior in the normal mean case.

Findings

Modified loss function removes paradoxical behavior

Results in confidence intervals consistent with classical methods

Addresses decision-theoretic approach limitations

Abstract

Confidence intervals are assessed according to two criteria, namely expected length and coverage probability. In an attempt to apply the decision-theoretic method to finding a good confidence interval, a loss function that is a linear combination of the interval length and the indicator function that the interval includes the parameter of interest has been proposed. We consider the particular case that the parameter of interest is the normal mean, when the variance is unknown. Casella, Hwang and Robert, Statistica Sinica, 1993, have shown that this loss function, combined with the standard noninformative prior, leads to a generalized Bayes rule that is a confidence interval for this parameter which has "paradoxical behaviour". We show that a simple modification of this loss function, combined with the same prior, leads to a generalized Bayes rule that is the usual confidence interval i.e. the "paradoxical behaviour" is removed.