A semi-empirical Bayesian chart to monitor Weibull percentiles

TL;DR

This paper introduces a Bayesian control chart for Weibull percentiles that directly monitors distribution parameters, improving small sample estimation and adapting control limits in high-reliability contexts.

Contribution

It presents a novel semi-empirical Bayesian chart that estimates Weibull percentiles without data transformation, enhancing monitoring accuracy for small samples.

Findings

Good Average Run Length performance demonstrated

Chart adapts control limits during Phase I

Provides insights into Weibull process monitoring

Abstract

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.