The Relationship Between Confidence Intervals for Failure Probabilities and Life Time Quantiles

The failure probability of a productF(t), and the life time quantiletpare commonly used metrics in reliability applications. Confidence intervals are used to quantify thes-uncertainty of estimators of these two metrics. In practice, a set of pointwise confidence intervals forF(t), or the quantilestpare often plotted on one graph, which we refer to as pointwise ldquoconfidence bands.rdquo These confidence bands forF(t) ortpcan be obtained throughs-normal approximation, maximum likelihood, or other procedures. In this paper, we compares-normal approximation to likelihood methods, and introduce a new procedure to get the confidence intervals forF(t) by inverting the pointwise confidence bands of the quantiletpfunction. We show why it is valid to interpret the set of pointwise confidence intervals for the quantile function as a set of pointwise confidence intervals forF(t), and vice-versa. Our results also indicate that the likelihood-based pointwise confidence bands have desirable statistical properties, beyond those that were known previously.