An ARL-unbiased thinning-based EWMA chart to monitor counts

Abstract

Shewhart control charts are known to be somewhat insensitive to shifts of small and moderate size. Expectedly, alternative control schemes such as the exponentially weighted moving average (EWMA) charts have been proposed to speed up the detection of such shifts. Unfortunately, applying the ordinary EWMA recursion to count data leads to a control statistic no longer with a fixed discrete range. Therefore, we propose a novel chart which relies on a EWMA control statistic where the usual scalar product is replaced by a thinning operation. Actually, we use the new fractional binomial thinning to avoid the typical over-smoothing ascribable to ceiling, rounding, and flooring operations. The properties of this discrete statistic are similar to the ones of its continuous EWMA counterpart and the run length (RL) performance of the associated chart can be computed exactly using the Markov chain approach for independent and identically distributed (i.i.d.) counts. Moreover, this chart is set in such way that: the average run length (ARL) curve attains a maximum in the in-control situation, i.e., the chart is ARL-unbiased; and the in-control ARL is equal to a pre-specified value. We use the R statistical software to provide compelling illustrations of this unconventional EWMA chart and to compare its RL performance with the ones of a few competing control charts for the mean of i.i.d. Poisson counts.

KEYWORDS:

• Average run length unbiased design
• exponentially weighted moving average
• fractional binomial thinning
• randomised signals
• statistical process control

Subject Classifications:

• 62F03
• 62P30

Funding

This work was partially supported by FCT (Fundação para a Ciência e a Tecnologia) through projects UID/Multi/04621/2013, PEst-OE/MAT/UI0822/2014 and PEst-OE/MAT/UI4080/2014.

Acknowledgements

We are greatly indebted to the Referee and the Associate Editor who selflessly devoted their time to review our work.