Monitoring the ratio of population means of a bivariate normal distribution using CUSUM type control charts

Monitoring the Ratio of Two Normal Variables Based on Triple Exponentially Weighted Moving Average Control Charts with Fixed and Variable Sampling Intervals

In statistical process control (SPC), the ratio of two normal random variables (RZ) is a valuable statistical indicator to be taken as the charting statistic. In this work, we propose a triple exponentially weighted moving average (TEWMA) chart for monitoring the RZ. Additionally, the variable sampling interval (VSI) strategy has been adopted to different control charts by researchers. With the application of this strategy, the VSI-TEWMA-RZ chart is then developed to further improve the performance of the proposed TEWMA-RZ chart. The run length (RL) properties of the proposed TEWMA-RZ and VSI-TEWMA-RZ charts are obtained by the widely used Monte-Carlo (MC) simulations. Through the comparisons with the VSI-EWMA-RZ and the VSI-DEWMA-RZ charts, the VSI-TEWMA-RZ chart is statistically more sensitive than the VSI-EWMA-RZ and the VSI-DEWMA-RZ charts in detecting small and moderate shifts. Moreover, it turned out that the VSI-TEWMA-RZ chart has better performance than the TEWMA-RZ chart on the whole. Furthermore, this paper illustrates the implementation of the proposed charts with an example from the food industry.

Modeling the Characteristics of Unhealthy Air Pollution Events: A Copula Approach

This study proposes the concept of duration (D) and severity (S) measures, which were derived from unhealthy air pollution events. In parallel with that, the application of a copula model is proposed to evaluate unhealthy air pollution events with respect to their duration and severity characteristics. The bivariate criteria represented by duration and severity indicate their structural dependency, long-tail, and non-identically marginal distributions. A copula approach can provide a good statistical tool to deal with these issues and enable the extraction of valuable information from air pollution data. Based on the copula model, several statistical measurements are proposed for describing the characteristics of unhealthy air pollution events, including the Kendall’s τ correlation of the copula, the conditional probability of air pollution severity based on a given duration, the joint OR/AND return period, and the conditional D|S and conditional S|D return periods. A case study based on air pollution data indices was conducted in Klang, Malaysia. The results indicate that a copula approach is beneficial for deriving valuable information for planning and mitigating the risks of unhealthy air pollution events.