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 metries. 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 "confidence bands." 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.