A generally weighted moving average exceedance chart
Summary (3 min read)
- Distribution-free control charts gained momentum in recent years as they are more efficient in detecting a shift when there is a lack of information regarding the underlying process distribution.
- This is somewhat challenging because, in practice, any information on the location parameter might not be known in advance and estimation of the parameter is therefore required.
- Under some right-skewed distributions such as the exponential distribution, gamma distribution with shape parameter 2.0 and lognormal distribution with shape parameter 0.5, the precedence tests possess higher power than the Wilcoxon’s rank-sum test for small values of .
- Design and implementation issues of the proposed chart are addressed in Section 3.
2. GWMA exceedance chart: Theoretical framework
- It is assumed that the in-control reference sample iid , where is the cumulative distribution function (c.d.f.) of an unknown continuous distribution and is the unknown location parameter of interest.
- Note that a location model for the distribution of the test sample is assumed as the authors intend to design a control chart for monitoring the process location.
- Let be the unknown true value of the parameter and be the shifted parameter when the process goes out-of-control (OOC); here, is the location shift.
- For the sake of notational simplicity, the authors use hereafter to denote the exceedance statistic for the sample in Phase II.
2.1. GWMA-EX plotting statistic
- Let denote the number of samples until the next occurrence of an event since its last occurrence.
- Then, by summing over all values of , the authors can write ∑ ∑ . (1) A generally weighted moving average (GWMA) is a weighted moving average (WMA) of a sequence of statistics, where the probability is regarded as the weight for the most recent statistic among the last of statistics.
- Therefore, the plotting statistic for the GWMA-TBE chart is defined as ∑ for (2) As in Sheu and Lin , the distribution of is taken to be , where and are the two parameters; this is the discrete two-parameter Weibull distribution (see Nakagawa and Osaki ).
- For this reason, the authors determine the control limits by using unconditional IC expectation and variance of .
2.2. Control limits
- So, the exact control limits are given by √ .
- For the sake of notational simplicity, the authors use , hereafter to denote the steady-state control limits;.
- The authors study two-sided GWMA-EX charts with symmetrically placed control limits, i.e., equidistant from the centreline .
- The methodology can be easily modified wherein an one-sided chart is more meaningful or when a two-sided control chart with asymmetric control limits is necessary;.
- Otherwise, the process is said to be IC, which implies that no location shift has occurred, and so the charting procedure continues.
3. The design and implementation of GWMA-EX chart
- Run-length distribution and its characteristics are commonly used measures to design and study the performance of a control chart.
- The average run-length (ARL) is a popular measure for a chart’s performance.
- Computational aspects of the run-length distribution for GWMA-EX chart are described next.
3.1. Computation of the run-length distribution
- Computation of the run-length distribution for GWMA-EX chart requires some effort.
- This is because of the involvement of , which is the order statistic from the reference sample, is itself a random variable.
- There are a number of methods available to calculate the run- length distribution of a time-weighted control chart.
- Three methods are discussed below in the context of the proposed chart along with their pros and cons.
3.1.1. Exact approach
- Suppose the run-length random variable is denoted by , and denotes the signalling event at the sample.
- ARL value computation by (10) is quite involved even with the use of a computer.
- In a Case U GWMA chart, computation of unconditional is even heftier with the Markov chain approach.
- 6) The number of test samples until a falls on or outside either of the limits is taken as an observation from the run-length distribution;.
3.2. The in-control design and implementation
- The typical industry standards for are 370 or 500 and the authors consider the former in their study.
- To this end, Sheu and Lin  noted that ( ) combinations in the intervals 0.5 0.9 and 0.5 1 enhanced the sensitivity of the GWMA- ̅ chart and outperformed the EWMA- ̅ chart for small shifts (i.e., less than 1.5 standard deviations in the location).
- The authors limit ourselves mostly to the choice of median since it is a robust measure of the central tendency of distributions of any shape and a popular choice of percentiles in practice.
- Considering practical limitations, the authors only select a few parameter combinations, as stated above, but this study can easily be extended for a larger range of parameter values following the same guidelines as they have provided here.
3.3. The out-of-control performance
- The GWMA charts are generally more sensitive than the EWMA charts in detecting small shifts (see the results in Sheu and Lin , Lu  and Sheu and Yang ).
- To study the performance of the GWMA-EX chart, the authors use the combinations of the parameters from Table 1; these combinations ensure that the attained is close to 370.
- The authors also compare the performance of GWMA-EX chart with GWMA- ̅ chart for normal distribution designed under Case U.
- The small variation in the obtained results in Table 5 is due to variability arising from Monte Carlo simulation.
- Note that the authors define the shift differently for the gamma distribution than for symmetric distributions.
4. Illustrative example
- To illustrate the application of the proposed GWMA-EX chart, the authors draw 49 samples of size 5 from normal(0,1) distribution as Phase I dataset to estimate the process median.
- The authors also draw 200 Phase II random samples from normal(0.25,1) distribution which can be regarded as OOC observations from a process with location shift 0.25.
- The attained for these two charts are close to 370 which put them at a similar IC performance level and are therefore comparable.
- The centreline is equal to the unconditional IC expectation of the plotting statistic;.
5. Concluding remarks
- The rigid assumption of normality for the process distribution may not hold in practice while designing a control chart for monitoring a streaming process.
- The performance of the classical GWMA- ̅ chart becomes worse under skewed distributions when the process distribution is unknown.
- A distribution-free GWMA control chart based on exceedance statistic, referred to as GWMA-EX chart, has been constructed in this paper for a process for which information on the underlying process distribution as well as the process median are not available.
- Design and relative performance study of the proposed GWMA-EX chart has been carried out.
- It has been observed that the proposed chart, with no information on the true IC process median or process distribution, is robust to non-normality when the process is IC and it performs just as well and in many cases better than the existing EWMA chart based on exceedance statistic when the shift is small.
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Cites background from "A generally weighted moving average..."
...Several authors, like Khoo et al. (2012), Graham, Chakraborti, and Mukherjee (2014), Teoh et al. (2014) and Chakraborty, Human, and Balakrishnan (2018), recommended using median run length (MRL) in place of the ARL, as a robust alternative with various advantages....
"A generally weighted moving average..." refers background or methods in this paper
...However, Sheu and Lin , Sheu and Yang , Lu  and Chakraborty et al....
...For a detail study on GWMA-X̄ chart for normal distribution, the reader is referred to Sheu and Lin ....
...As in Sheu and Lin , the distribution of N is taken to be Pr[N = i] = q(i−1) − qiα , where 0 ≤ q < 1 and α > 0 are the two parameters; this is the discrete two-parameter Weibull distribution (see Nakagawa and Osaki )....
...Generalizing the EWMA charting procedure, Sheu and Lin  proposed a Generally Weighted Moving Average (GWMA) control chart for the normal distribution (denoted by GWMA-X̄ chart) that has been shown to be more effective than EWMA, CUSUM and Shewhart-type charts (see Hsu et al....
...To this end, Sheu and Lin  noted that (q,α) combinations in the intervals 0....
"A generally weighted moving average..." refers background in this paper
... studied a class of nonparametric Shewhart-type charts based on precedence statistics, referred to as the Shewhart-type precedence charts....