Monitoring the shape parameter of a Weibull renewal process

Abstract

This research arose from a challenge faced in real practice—monitoring changes to the Weibull shape parameter. From first-hand experience, we understand that a mechanism for such a purpose is very useful. This article is primarily focused on monitoring the shape parameter of a Weibull renewal process. We derive a novel statistic on the Weibull shape parameter making use of maximum likelihood theory, which is demonstrated to follow an approximately normal distribution. This desirable normality property makes the statistic well suited for use in monitoring the Weibull shape parameter. It also allows for a simple approach to constructing a Shewhart-type control chart, named the Beta chart. The parameter values required to design a Beta chart are provided. A self-starting procedure is also proposed for setting up the Phase I Beta chart. The Average Run Length (ARL) performance of the Beta chart is evaluated through Monte Carlo simulation. A comparison with a moving range exponentially weighted moving average (EWMA) chart from the literature shows that the Beta chart has much better ARL performance when properly designed. Application examples, using both simulated and real data, demonstrate that the Beta chart is effective and makes good sense in real practice.

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