Adaptive Nonparametric CUSUM Control Chart with Variable Sampling Interval Strategy

Statistical process control (SPC) is an important quality control tool that uses statistical methods to monitor and control processes. One major concern of SPC is to analyze the stability of a production process, and to give a warning signal when the process has some changes, that is, the process has shifts happened. The control performances of a control chart are generally evaluated with the average run length (ARL), which denotes the average number of observations from the process beginning to giving a warning signal. A control chart with a smaller ARL when the process has a shift is better, under the condition of a same ARL when the process is steady. This technology has currently been adopted in many fields, such as health care, finance, biotechnology and chemical engineering. More introductions about SPC technology and its applications can be found in, for example, Lucas and Saccucci (1990) [1], Thomas and William (1991) [2], Woodall (2000) [3], Montgomery (2009) [4], Zou et al. (2009) [5], Shardt et al. (2012) [6]. 

CUSUM (Page 1954) [7] is an effective control chart for detecting small and moderate shifts; it is proposed based on the sequential probability ratio test. The conventional CUSUM control chart is generally designed by assuming known the size of shifts. However, to predetermine the exact size of shifts is a difficult thing in most practical process settings. In order to detect both small and large shifts, Lucas (1982) [8] proposed a combined Shewhart-CUSUM chart which is designed by way of adding the Shewhart control limit to a conventional CUSUM control chart. Where after, the emergence of the adaptive control chart resolves this problem significantly. The main design idea of adaptive charts is that the parameters value setting is decided by sample observations, so that the chart can dynamically monitor various sizes of shifts. Sparks (2000) [9] proposed an adaptive CUSUM chart by an one-step-ahead formula to dynamically forecast the parameters values of the CUSUM chart. Shu and Jiang (2006) [10] calculated the chart ARLs of Sparks (2000) by Markov-chain. With that Shu et al. (2008) [11] proposed another adaptive CUSUM chart by improving the dynamically forecasting tool that Sparks (2000) used. Other adaptive charts are discussed by Wu et al. (2009) [12], Li and Wang (2010) [13], Capizzi and Masarotto (2012) [14], among others.

However, these control charts mentioned above often assume that sample observations follow a known parametric distribution, most commonly follow normal distribution. But in most practical processes, the distribution form of sample observations is unknown or not normal. Through the literature we known some nonparametric CUSUM control charts can monitor processes with unknown sample distribution form. For example, Bakir and Reynolds (1979) [15] proposed a nonparametric CUSUM chart based on the Wilcoxon signed-ranks statistic. McDonald (1990) [16] proposed another nonparametric CUSUM chart based on sequential ranks of observations. Qiu and Hawkins (2001) [17] discussed a multivariate nonparametric CUSUM chart based on the sequential ranks method that McDonald used. More recently, based on ranks, Liu et al. (2014) [18] proposed an adaptive nonparametric CUSUM (ANC) chart for detecting variables which shift size and distribution form are both unknown.