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