1. Introduction

Control charts are vital tools in statistical quality control, offering a visual means of monitoring process parameters over time. They enable practitioners to distinguish between chance and assignable causes of variation, which signals potential issues that require intervention. By continuously monitoring process mean and variance, control charts facilitate timely decision-making and help maintain quality products. Specifically, control charts for monitoring process location are crucial for ensuring that the mean of a process remains stable over time. The early work of Shewhart (1931) laid the foundation for control charts which assume normality and utilize 3-sigma control limits to detect shifts in process mean. Page (1961) introduced cumulative sum control charts (CUSUM) which are sensitive to detect small shifts in the process mean. Roberts (1959) proposed exponentially weighted moving average (EWMA) control chart which assigns weights to previous observations, making it more sensitive to detect small shifts. Montgomery (1996) provides a comprehensive overview of control charts, discussing various types of control charts, their underlying concepts and applications in monitoring process stability in detecting shifts in process parameters.

Many control charts for process location parameter rely on the assumption that the underlying process has normal distribution. However, in practice, this assumption need not be valid. For example, in glass industry, the thickness of glass sheets used for windows, screens or optical applications is critical to monitor the process parameter. The thickness measurements are often symmetrically distributed around a target value due to production techniques like the float glass process. Some slight variations in the cooling process or other external factors can result in occasional extreme deviations which lead to a heavy-tailed distribution, where control charts based on normality may fail to detect these changes.

As a result, it is essential to propose control charts that do not rely on the assumption of normal distribution. To address this issue, it is necessary to develop control charts that are resilient to non normality. Some significant studies that do not rely on the assumption of normality include control charts based on sign statistics proposed by Amin et al. (1995), distribution-free Shewhart type control chart based on signed ranks developed by Bakir (2004), synthetic control chart developed by Pawar and Shirke (2010), Shewhart type nonparametric control chart based on the run statistic suggested by Zombade and Ghute (2019). A comprehensive overview of nonparametric control charts, detailing various methodologies for quality control is studied by Chakraborti (2019). Here, the author emphasizes the robustness and applicability of nonparametric approaches in statistical process control providing practical insights and examples for practitioners. Bhat and Patil (2024a and 2024b) propose control charts for process location based on sample median and sample midrange respectively to monitor the process location parameter, employing Shewhart's methodology.

In this paper, we propose a control chart that employs a robust statistical measure to improve monitoring accuracy and enhance shift detection in process location, especially in the presence of outliers and under non normal distributions. Section 2 deals with proposed control chart and section 3 discusses the evaluation of proposed control chart. In sections 4 and 5, we respectively deal with illustration of the proposed control chart through examples and conclusions based on our observations.

2. Control Chart Based on Gastwirth Estimator

In this section, we propose control chart based on Gastwirth estimator (G) due to Gastwirth (1966) which is a robust estimator. Suppose is a random sample of size from a distribution with cumulative density function , then G is given by

(1)

where is sample quantile.

Also, suppose represents order statistic, then

(2)

where and represent 33^rd, 50^th and 66^th percentiles of the data. The estimator is useful from practical point of view as it has better efficiency than median and is also robust. It has better computational efficiency than other robust estimators like estimators due to Harrel and Davis (1982) and Hodges and Lehmann (1963). Also, it is helpful in estimating the location parameter in the presence of outliers or when process has non-normal distribution. Its approach of using fixed weights enable its utility across various symmetric distributions.

According to DasGupta (2008), G has asymptotically normal distribution with mean

(3)

and variance , where and are respectively the population quantiles corresponding to the 33^rd, 50^th and 66^th percentiles.

(4)

where

,

for with and for .

Here, and is value of density function at the specified population quantile

Therefore,

(5)

The variance expression captures the estimator's variability as a function of the sample size and the values of the density function at the specified population quantiles and

Under the assumption that, a random sample of size is taken from a symmetric or skewed distribution with location parameter and scale parameter , following Shewhart (1931), we propose G control chart

(6)

(7)

(8)

where and are respectively upper control limit, centre line and lower control limit of G control chart. And is standard deviation of G.

3. Performance of G Control Chart

In this section, we evaluate the performance of the G control chart for various distributions under consideration. To assess the performance of the proposed G control chart, we compute mean and of G control chart under uniform (U), exponential (E), normal (N), logistic (LG), Laplace (L) and Cauchy (C) distributions. The distributions are modified in terms of and such that they have mean and variance . As G approximately follows normal distribution with mean and given by square root of (5), we have to evaluate and under various distributions. For example, to evaluate under exponential distribution,

(9)

(10)

(11)

Substituting (11) in (9), we get

(12)

On similar lines, other values of in (5) are evaluated and is obtained. The and width of G control chart for the distributions under consideration is furnished in Table 1.

Table 1.       of G control chart for various distribtions

The of the G estimator varies across distributions based on their differences in spread and tail behavior. The G statistic under Cauchy distribution has highest and under Laplace distribution has lowest . The width of G control chart is highest for Cauchy distribution followed by uniform distribution and is lowest for Laplace distribution followed by exponential distribution.

To evaluate the performance of G control chart, we consider various performance measures such as power , average run length median run length and standard deviation of run length . These measures provide a comprehensive view of the control chart’s ability to detect shifts in the process location. The refers to the probability, represents the expected number of samples required and gives the median number of samples needed to detect a shift in process location. A smaller and value indicates quicker detection of shifts. Additionally, the evaluates the consistency of shift detection, where a lower value indicates better consistency. Various measures are given by

(13)

(14)

(15)

(16)

(17)

The values of and for uniform, exponential, normal, logistic and Laplace distributions are determined by setting and for Cauchy distribution by setting due to Arnold (1965). For sample sizes and various positive shifts, the computed values of are provided in Table 2 and are provided in Table 3. and of G control chart are plotted in Figure 1 for .

Table 2.       for various distributions

Table 3.       for various distributions

Figure 1. of G Control Chart

From Table 2 and 3, for all values of we observe that, is 0.0027 for symmetric distributions under consideration at zero shift. However, since for exponential distribution, is at . The increases as and increase. For fixed and small is highest for Cauchy distribution and lowest for uniform distribution. We notice that, under symmetric distributions, and are approximately 370 and 256 respectively for zero shift. Under exponential distribution, these values are achieved at . Also, and being smaller, vary for different for zero shift. For fixed , as increases, and decrease, exhibiting effective shift detecting capability of the control chart. The same nature in and is observed for fixed and increasing . Also, decreases for increasing and , indicating stability of the control chart in detecting the shifts.

From Figure 1, it is observed that, among symmetric distributions, for specified value of , both and decrease across uniform distribution to Cauchy distribution. Hence, if the process variables are from uniform distribution, G control chart is less sensitive in detecting smaller shifts, whereas, it reflects strong sensitivity to detect shifts under heavier tailed distributions like Laplace and Cauchy distributions. Under exponential distribution, though and are decreasing for smaller shifts, the detection of shift could be illusive since these values are smaller at zero shift itself.

On comparing the G control chart with median (M) control chart due to Bhat and Patil (2024a), in terms of ARL, we observe that, is lower than under uniform, normal and logistic distributions, whereas, is higher than for Laplace distribution. These values are almost same under Cauchy distribution. This indicates that, G control chart detects shifts in the process location quickly than the M control chart when process has uniform, normal, logistic distributions and has almost equal efficiency when process has Cauchy distribution. As G curtails 29% outliers and M curtails 50% outliers, G control chart is less resilient than M control chart to outliers. Here, G control chart is better than M control chart when process variables are from light and medium tailed distributions. However, it is equally competitive when process variables have heavy tailed distributions.

4. Illustration

In this section, we illustrate G control chart with two examples given in Duncan (1955) and Ghute and Shirke (2008).

Example 1: The example due to Duncan (1955) deals with the height (inches) of fragmentation of bomb base that consists of 5 samples taken 29 times and is given in Table 4. The , control limits and width of G control chart is given in Table 5. Figure 2 is plotted using Table 4 and Table 5.

Table 4. Height (inches) of fragmentation of bomb base and computed values of G,

Table 5.       under various distributions

Figure 2. G control chart for symmetric distributions

Since and are unknown, they are estimated from the samples. Suppose samples are taken times, is estimated by taking the weighted average of 33^rd, 50^th and 66^th percentiles of the samples, with weights respectively given by 0.3, 0.4 and 0.3. That is, where and is 33^rd, 50^th and 66^th percentiles of the of sample. And is estimated by obtaining under various distributions by substituting where for λ. Here where . For .

From Table 5 and Figure 2, we observe that, is narrower for Cauchy distribution, whereas, it is wider for uniform distribution. The centre line of exponential distribution differs from that of symmetric distribution. From Figure 2, we see that, the process is out of control for Cauchy distribution, whereas, it exhibits in control status under other symmetric distributions.

Example 2: The partial data is taken from Ghute and Shirke (2008) which is due to Chen et al. (2005). The data consists of 5 samples taken 18 times from a spring manufacture process. The measurements of the inner diameter of the springs are given in Table 6 and , control limits, are given in Table 7. Figure 3 is plotted using Table 6 and Table 7. The computations are carried out on similar lines of example 1.

Table 6. Springs inner diameter and computed values of G,

Table 7.       under various distributions

Figure 3. G control chart for symmetric distributions

From Table 7 and Figure 3, we observe that, becomes narrower for heavier tailed distributions. The centre line of all symmetric distributions is the same, whereas, it is different for exponential distribution. From Figure 3, we notice that, process is out of control under Cauchy and Laplace distributions, while it remains in control under other symmetric distributions.

5. Conclusions

In this section, based on our findings, we record our conclusions on the proposed G control chart.

• A control chart based on Gastwirth estimator (G) has been proposed for monitoring process location under the assumption that, the process variables have cumulative density function, .

• The of G estimator is obtained and the control limits of the proposed control chart is developed under various symmetric and asymmetric distributions.

• The proposed control chart is evaluated using various performance measures viz. power, ARL, MRL and SDRL for different shifts and sample sizes.

• Power of the G control chart is high for Cauchy distribution as compared to other distributions.

• ARL and MRL decrease for smaller shifts and sample sizes under heavy tailed distributions like Laplace and Cauchy distributions.

• The proposed control chart has lucidity in detecting the shifts in the process location under Cauchy model.

• The G control chart outperforms M control chart due to Bhat and Patil (2024a), when process variables have uniform, normal, logistic distributions and equally performs under Cauchy distribution.

• The proposed control is deplorable under skewed distribution like exponential distribution as the detection of shifts by control chart could be misleading.

• The G control chart is useful when the distributional assumption of normality is not valid. Also, it is highly desirable when process variables are from heavy tailed distributions.

ACKNOWLEDGEMENTS

Authors thank the unknown referee whose suggestions have been useful in improvisation of the paper.