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

• Published in: IEEE transactions on reliability Vol. 57; no. 2; pp. 260 - 266
• Main Authors: Yili Hong, Meeker, W.Q., Escobar, L.A.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2008; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Approximation; Asymptotic approximation; Ball bearings; Bands; Confidence; confidence bands; Confidence intervals; Data analysis; Distribution functions; Estimators; Failure; life data analysis; Life estimation; likelihood confidence interval; Mathematical analysis; maximum likelihood; Maximum likelihood estimation; Probability; Quantiles; Random variables; s -normal approximation; Statistics; Studies; Warranties
• ISSN: 0018-9529; 1558-1721
• DOI: 10.1109/TR.2008.920352

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.