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

The failure probability of a product F(t), and the life time quantile t _p are commonly used metrics in reliability applications. Confidence intervals are used to quantify the s-uncertainty of estimators of these two metrics. In practice, a set of pointwise confidence intervals for F(t), or the quantiles t _p are often plotted on one graph, which we refer to as pointwise ldquoconfidence bands.rdquo These confidence bands for F(t) or t _p can be obtained through s-normal approximation, maximum likelihood, or other procedures. In this paper, we compare s-normal approximation to likelihood methods, and introduce a new procedure to get the confidence intervals for F(t) by inverting the pointwise confidence bands of the quantile t _p function. 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 for F(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.