  This paper develops a Bayesian control chart for the percentiles of the
Weibull distribution, when both its in-control and out-of-control parameters
are unknown. The Bayesian approach enhances parameter estimates for small
sample sizes that occur when monitoring rare events as in high-reliability
applications or genetic mutations. The chart monitors the parameters of the
Weibull distribution directly, instead of transforming the data as most
Weibull-based charts do in order to comply with their normality assumption. The
chart uses the whole accumulated knowledge resulting from the likelihood of the
current sample combined with the information given by both the initial prior
knowledge and all the past samples. The chart is adapting since its control
limits change (e.g. narrow) during the Phase I. An example is presented and
good Average Run Length properties are demonstrated. In addition, the paper
gives insights into the nature of monitoring Weibull processes by highlighting
the relationship between distribution and process parameters.
