On The Performance of a Two-Sided Shewhart Chart for Continuous Proportions with Estimated Parameters

TL;DR

This paper evaluates the performance of a two-sided Shewhart chart for monitoring continuous proportions modeled by the Kumaraswamy distribution, focusing on parameter estimation, control limit adjustments, and practical implementation.

Contribution

It introduces a comprehensive analysis of the Shewhart chart's performance with estimated parameters for Kumaraswamy-distributed data, including empirical rules for Phase I sample size and control limit adjustments.

Findings

The chart's performance varies with Phase I sample size and out-of-control conditions.

Adjustments to control limits improve out-of-control detection.

Practitioners face a trade-off between in-control guarantee and out-of-control sensitivity.

Abstract

During the recent years there was an increased interest in studying the performance of different types of control charts, under various distributional models for continuous proportions, such as percentages and rates. In this work we consider the Kumaraswamy distribution, a popular and flexible distributional model for data in the unit interval (0,1) and investigate further the properties of a two-sided chart for individual observations for monitoring these types of processes, when the process parameters are unknown. Specifically, using Monte Carlo simulation, we evaluate the performance of the chart under a conditional perspective and provide empirical rules on how to select the appropriate size for the Phase I sample. In addition, we explore possible adjustments on the control limits of the chart, which take into account the available Phase I sample. The performance of the chart is also investigated for several out-of-control situations. The results show that for small and moderate size Phase I samples, practitioners have to choose whether they prefer a guaranteed in-control performance or an improved out-of-control performance. The implementation of the considered methods in practice is discussed via two numerical examples.