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 tp are commonly used metrics in reliability applications. Confidence intervals are used to quantify the statistical uncertainty of estimators of these two metrics. In practice, a set of pointwise confidence intervals for F(t) or the quantiles tp are often computed and plotted on one graph, which we refer to as pointwise “confidence bands.” These confidence bands for F(t) or tp can be obtained through normal approximation, likelihood, or other procedures. In this paper, we compare normal approximation and likelihood methods and introduce a new procedure to get the confidence intervals for F(t) by inverting the pointwise confidence bands of the quantile tp function. We show that 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
This preprint was published in IEEE Transactions on Reliability 57 (2008): 260–266.