Beta Binomial Confidence Bounds on the Mixed Weibull Distribution

The mixed Weibull distribution is used to model data that do not fall on a straight line on a Weibull probability plot. Data of this type, particularly if the data points follow an S-shape on the probability plot, may be indicative of more than one failure mode at work in the population of failure times. This article describes how the beta binomial method is used to estimate the confidence bounds for such analyses.

The beta binomial confidence bounds uses a procedure similar to that used in calculating median ranks (see Data and Data Types). This is a non-parametric approach to confidence interval calculations that involves the use of rank tables and is commonly known as beta binomial bounds (BB). By non-parametric, we mean that no underlying distribution is assumed. In other words, this method can be used for any distribution, without having to make adjustments in the underlying equations based on the assumed distribution.

The median ranks are calculated by solving the cumulative binomial distribution for Z (rank for the failure), at the 50% confidence interval:

where:

•  N = the sample size

•  j = the order number

• P = 0.5

The same methodology can then be repeated by changing P from 0.50 (50%) to our desired confidence level. For P = 90% one would formulate the equation as:

Keep in mind that one must be careful to select the appropriate values for P based on the type of confidence bounds one desires. For example, if 2-sided 80% confidence bounds are to be calculated, one must solve the equation twice, once with P = 0.1 and once with P = 0.9, in order to place the bounds around 80% of the population.

Using this methodology, the appropriate ranks are obtained and plotted, based on the desired confidence level. These points are then joined by a smooth curve to obtain the corresponding confidence bound. These binomial equations can again be transformed using the beta and F distributions; thus the name beta binomial confidence bounds.