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Two requirements for pivoting a cumulative distribution function (CDF) inorder to construct exact confidence intervals or bounds for a real-valuedparameter $\theta$ are the monotonicity of this CDF with respect to $\theta$and the existence of solutions of some pertinent equations for $\theta$. Thesecond requirement is not fulfilled by the CDF of the maximum likelihoodestimator of the exponential scale parameter when the data come from somelife-testing scenarios such as type-I censoring, hybrid type-I censoring, andprogressive type-I censoring that are subject to time constraints. However, themethod has been used in these cases probably because the non-existence of thesolution usually happens only with small probability. Here, we illustrate theproblem by giving formal details in the case of type-I censoring and byproviding some further examples. We also present a suitable extension of thebasic pivoting method which is applicable in situations wherein the consideredequations have no solution.

On the method of pivoting the CDF for exact confidence intervals with illustration for exponential mean under life-test with time constraints