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.