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Journal ArticleDOI

A generally weighted moving average exceedance chart

25 Mar 2018-Journal of Statistical Computation and Simulation (Taylor & Francis)-Vol. 88, Iss: 9, pp 1759-1781

AbstractDistribution-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. H...

Topics: Chart (55%), Control chart (53%), Moving average (52%)

Summary (3 min read)

Introduction

  • 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 [2], 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 [12]).
  • 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 [2] 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 [2], Lu [6] and Sheu and Yang [13]).
  • 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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1
A Generally Weighted Moving Average Exceedance Chart
Niladri Chakraborty
a
, Schalk W. Human
a
and Narayanaswamy Balakrishnan
a,b
a
Department of Statistics, University of Pretoria, Pretoria 0002, South Africa;
b
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario,
Canada L8S 4K1;
Niladri Chakraborty contact:
Email: niladriorama@gmail.com
Schalk W. Human contact:
Email: schalk.human@up.ac.za; Phone: (+27) 83 759 1604
Narayanaswamy Balakrishnan contact:
Email: bala@mcmaster.ca; Phone: (905) 525-9140 Ext: 23420
N. Chakraborty has received his PhD in Mathematical Statistics from the Department of
Statistics, University of Pretoria, South Africa. Presently his main area of research is
statistical quality control.
S. W. Human has obtained PhD in Mathematical Statistics from the Department of Statistics,
University of Pretoria, South Africa. In a worldwide collaboration, he has authored/co-
authored numerous accredited peer-reviewed journal articles and has presented his research
work at several national and international conferences. He also serves as a reviewer for a
number of reputed international statistical journals. Based on his academic achievements, he
was awarded several bursaries from University of Pretoria and the National Research
Foundation, South Africa. He also has worked as a supervisor for a number of Doctoral and
Master’s candidates at the University of Pretoria, South Africa. His main area of research
interests includes statistical quality control.
N. Balakrishnan is a distinguished university professor in the Department of Mathematics
and Statistics at McMaster University, Hamilton, Ontario, Canada. Prof. Balakrishnan is also
an extra-ordinary professor at the Department of Statistics, University of Pretoria. He is a
fellow of the American Statistical Association, a fellow of the Institute of Mathematical
Statistics, and an elected member of the International Statistical Institute. His research
interests are quite wide covering a range of topics including ordered data analysis,
distribution theory, quality control, reliability, survival analysis and robust inference. He is
currently the editor-in-chief of Communications in Statistics and the Encyclopaedia of
Statistical Sciences.

2
Abstract:
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. However, a distribution-free control chart for monitoring the process location
often requires information on the in-control process median. 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. In view of this, a time-
weighted control chart, labelled as the Generally Weighted Moving Average (GWMA)
exceedance (EX) chart (in short GWMA-EX chart), is proposed for detection of a shift in the
unknown process location; this chart is based on an exceedance statistic when there is no
information available on the process distribution. An extensive performance analysis shows
that the proposed GWMA-EX control chart is, in many cases, better than its contenders.
Keywords: nonparametric control chart; GWMA chart; exceedance statistic; precedence
statistic; average run-length; Monte Carlo simulation.
1. Introduction
Control charts are efficient tools in statistical process control (SPC) that aim at efficient
monitoring of streaming process and detecting changes, if any, in process performance as
early as possible so that a corrective measure can be taken to ensure minimal loss due to a
downfall in quality. Shewhart-type charts, proposed by Walter A. Shewhart in 1920s, might
be appealing in practice for its simplicity, but time-weighted control charts such as
Cumulative Sum (CUSUM) or Exponentially Weighted Moving Average (EWMA) charts
have proven to be more efficient than Shewhart-type charts in detecting small persistent shift
(see Montgomery [1]). Generalizing the EWMA charting procedure, Sheu and Lin [2]
proposed a GWMA control chart for the normal distribution (denoted by GWMA-
chart)
that has been shown to be more effective than EWMA, CUSUM and Shewhart-type charts
(see Hsu et al. [3]) in detecting small shift in process. Chakraborty et al. [4] proposed a
parametric GWMA chart to monitor time-between-failures when the process distribution is

3
not normal. However, the underlying process distribution may not be known or satisfy the
distributional assumption always. It has been observed that performance of a parametric
GWMA chart deviates in departure from distributional assumption (see Chakraborty et al. [5]
) even when the process parameters are not shifted from its in-control standard. When the
performance of a control chart is independent (or nearly independent) of the underlying
process distribution, it is said to be a robust control chart and, for parametric GWMA chart,
the in-control robustness is often adversely affected. It should be mentioned that Lu [6] and
Chakraborty et al. [5] proposed GWMA control charts based on the sign statistic (denoted by
GWMA-SN chart) and Wilcoxon signed-rank statistic (denoted by GWMA-SR chart),
respectively, for the case when the true process median is known (Case K).
In many practical situations, however, the true process median may not be known (Case U)
that to some extent limits the applicability of the distribution-free GWMA charts based on
sign and Wilcoxon signed-rank statistics. Exceedance (or precedence) tests are well known
nonparametric two-sample tests based on the number of observations from one of the samples
that exceed (or precede) a specified (say, the -th) order statistic of the other sample for
distributional shift. Precedence/exceedance test statistics are linearly related, and the tests
based on these statistics are found to be useful in a number of applications including quality
control and reliability studies with lifetime data. Balakrishnan and Ng [7] have provided a
detail overview of the precedence/exceedance tests and their properties and applications.
Balakrishnan and Ng [7] (see page 51) have stated that, ‘Wilcoxon rank-sum test performs
better than precedence tests if the underlying distribution is close to symmetry, such as the
normal distribution, gamma distribution with large values of shape parameter and lognormal
distribution with small values of the shape parameter. However, 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

4
power than the Wilcoxon’s rank-sum test for small values of . It is evident that the more
right skewed the underlying distribution is, the more powerful the precedence test is.’ Here,
corresponds to the

order statistic that is being used as a reference for the precedence
chart. Chakraborti et al. [8] studied a class of nonparametric Shewhart-type charts based on
precedence statistics, referred to as the Shewhart-type precedence charts. Graham et al. [9],
Graham et al. [10], and Mukherjee et al. [11], respectively, studied EWMA and CUSUM
charts based on exceedance statistics for small process shift. In this article, we construct a
distribution-free GWMA chart based on what is known as exceedance statistic for monitoring
unknown median of a streaming process. This chart is referred to as the GWMA exceedance
(or GWMA-EX) chart.
In Section 2, a GWMA control chart based on exceedance statistic is designed and the
necessary theoretical framework is developed. Design and implementation issues of the
proposed chart are addressed in Section 3. Next, an illustrative example is provided in
Section 4. Finally, some concluding remarks are made in Section 5.
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. Now, suppose



,
, is the

test sample of size 1, that follow an unknown continuous
distribution
󰇛󰇜
󰇛 󰇜. Note that a location model for the distribution of the test
sample is assumed as we 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. The

5
process is said to be in-control (IC) when , i.e., when . Let
󰇛󰇜
be the

order
statistic obtained from the Phase I sample of size .
We define the exceedance statistic as

the number of

󰇛󰇜
in the

sample, for
1, 2, …, . For the sake of notational simplicity, we use
hereafter to denote the
exceedance statistic for the

sample in Phase II.
2.1. GWMA-EX plotting statistic
GWMA-EX chart is constructed by taking a weighted average of a sequence of the
. Let
denote the number of samples until the next occurrence of an event since its last
occurrence. Then, by summing over all values of , we 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. The probability 
󰇟
󰇠
is considered as the weight
for the starting value, denoted by
, which is taken as the unconditional IC expectation of
given by
󰇛

󰇜
 󰇡

󰇢 (see Appendix A4). Therefore, the plotting statistic
for the GWMA-TBE chart is defined as
󰇟󰇠



󰇟
󰇠
for  (2)
As in Sheu and Lin [2], 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 [12]). So, the weights are given by
󰇛

󰇜
. By substituting the
probability mass function (p.m.f.) of the two-parameter discrete Weibull distribution in equation (2),
GWMA-EX chart plotting statistic is defined as
󰇛
󰇛

󰇜
󰇜


, for  , (3)

Citations
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Journal ArticleDOI
Abstract: An overview of monitoring schemes from a class called generally weighted moving average (GWMA) is provided. A GWMA scheme is an extended version of the exponentially weighted moving average (EWMA) scheme with an additional adjustment parameter that introduces more flexibility in the GWMA model as it adjusts the kurtosis of the weighting function so that the GWMA scheme can be designed such that it has an advantage over the corresponding EWMA scheme in the detection of certain shift values efficiently. The parametric and distribution-free GWMA schemes to monitor various quality characteristics and its existing enhanced versions (i.e. double GWMA, composite Shewhart-GWMA, mixed GWMA-CUSUM and mixed CUSUM-GWMA) have better performance than their corresponding EWMA counterparts in many situations; hence, all such existing research works discussing GWMA-related schemes (i.e. 61 publications in total) are documented and categorized in such a manner that it is easy to identify research gaps. Finally, a number of possible future research ideas are provided.

10 citations



Journal ArticleDOI
TL;DR: The D GWMA-EX chart combines the better shift detection properties of a DGWMA chart with the robust in-control performance of a nonparametric chart, by using all the information from the start until the most recent sample to decide if a process is in- control (IC) or out-of-control (OOC).
Abstract: Since the inception of control charts by W. A. Shewhart in the 1920s they have been increasingly applied in various fields. The recent literature witnessed the development of a number of nonparametric (distribution-free) charts as they provide a robust and efficient alternative when there is a lack of knowledge about the underlying process distribution. In order to monitor the process location, information regarding the in-control process median is typically required. However, in practice this information might not be available due to various reasons. To this end, a generalized type of nonparametric time-weighted control chart labelled as the Double Generally Weighted Moving Average (DGWMA) based on the exceedance statistic (EX) is proposed. The DGWMA-EX chart includes many of the wellknown existing time-weighted control charts as special or limiting cases for detecting a shift in the unknown location parameter of a continuous distribution. The DGWMA-EX chart combines the better shift detection properties of a DGWMA chart with the robust in-control performance of a nonparametric chart, by using all the information from the start until the most recent sample to decide if a process is in-control (IC) or out-of-control (OOC). An extensive simulation study reveals that the proposed DGWMA-EX chart, in many cases, outperforms its counterparts.

9 citations


Journal ArticleDOI
Abstract: Article history: Received July 15 2019 Received in Revised Format September 1 2019 Accepted September 1 2019 Available online September 2 2019 Distribution-free (or nonparametric) monitoring schemes are needed in industrial, chemical and biochemical processes or any other analytical non-industrial process when the assumption of normality fails to hold. The Mann-Whitney (MW) test is one of the most powerful tests used in the design of these types of monitoring schemes. This test is equivalent to the Wilcoxon ranksum (WRS) test. In this paper, we propose a new distribution-free generally weighted moving average (GWMA) monitoring scheme based on the WRS statistic. The performance of the proposed scheme is investigated using the average run-length, the standard deviation of the runlength, percentile of the run-length and some characteristics of the quality loss function through extensive simulation. The proposed scheme is compared with the existing parametric and nonparametric GWMA monitoring schemes and other well-known control schemes. The effect of the estimated design parameters as well as the effect of the Phase I sample size on the Phase II performance of the new monitoring scheme are also investigated. The results show that the proposed scheme presents better and attractive mean shifts detection properties, and therefore outperforms the existing monitoring schemes in many situations. Moreover, it requires a reasonable number of Phase I observations to guarantee stability and accuracy in the Phase II performance. © 2020 by the authors; licensee Growing Science, Canada

6 citations


Cites background from "A generally weighted moving average..."

  • ...Chakraborty et al. (2018) proposed a robust GWMA exceedance chart (EX-GWMA) for monitoring the location parameter....

    [...]

  • ...(2007), and for other more recent improvements or enhancements, see for example, Haridy et al. (2019), Haq (2019), Adegoke et al....

    [...]


References
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Book
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TL;DR: Part I: Introduction Chapter 1: Quality Improvement in the Modern Business Environment Chapter 2: The DMAIC Process Chapter 3: Statistical Methods Useful in Quality Control and Improvement Chapter 4: Inferences about Process Quality
Abstract: Part I: Introduction Chapter 1: Quality Improvement in the Modern Business Environment Chapter 2: The DMAIC Process Part II: Statistical Methods Useful in Quality Control and Improvement Chapter 3: Modeling Process Quality Chapter 4: Inferences about Process Quality Part III: Basic Methods of Statistical Process Control and Capability Analysis Chapter 5: Methods and Philosophy of Statistical Process Control Chapter 6: Control Charts for Variables Chapter 7: Control Charts for Attributes Chapter 8: Process and Measurement System Capability Analysis Part IV: Other Statistical Process-Monitoring and Control Techniques Chapter 9: Cumulative Sum and Exponentially Weighted Moving Average Control Charts Chapter 10: Other Univariate Statistical Process Monitoring and Control Techniques Chapter 11: Multivariate Process Monitoring and Control Chapter 12: Engineering Process Control and SPC Part V: Process Design and Improvement with Designed Experiments Chapter 13: Factorial and Fractional Experiments for Process Design and Improvements Chapter 14: Process Optimization and Designed Experiments Part VI: Acceptance Sampling Chapter 15: Lot-by-Lot Acceptance Sampling for Attributes Chapter 16: Other Acceptance Sampling Techniques Appendix

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Journal ArticleDOI
Abstract: The discrete Weibull distribution is defined to correspond with the Weibull distribution in continuous time. A few properties of the discrete Weibull distribution are discussed.

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Journal ArticleDOI
Abstract: A generalization of the exponentially weighted moving average (EWMA) control chart is proposed and analyzed. The generalized control chart we have proposed is called the generally weighted moving average (GWMA) control chart. The GWMA control chart, with time‐varying control limits to detect start‐up shifts more sensitively, performs better in detecting small shifts of the process mean. We use simulation to evaluate the average run length (ARL) properties of the EWMA control chart and GWMA control chart. An extensive comparison reveals that the GWMA control chart is more sensitive than the EWMA control chart for detecting small shifts in the mean of a process. To enhance the detection ability of the GWMA control chart, we submit the composite Shewhart‐GWMA scheme to monitor process mean. The composite Shewhart‐GWMA control chart with/without runs rules is more sensitive than the GWMA control chart in detecting small shifts of the process mean. The resulting ARLs obtained by the GWMA control chart...

132 citations


"A generally weighted moving average..." refers background or methods in this paper

  • ...However, Sheu and Lin [2], Sheu and Yang [13], Lu [6] and Chakraborty et al....

    [...]

  • ...For a detail study on GWMA-X̄ chart for normal distribution, the reader is referred to Sheu and Lin [2]....

    [...]

  • ...As in Sheu and Lin [2], 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 [12])....

    [...]

  • ...Generalizing the EWMA charting procedure, Sheu and Lin [2] 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 [2] noted that (q,α) combinations in the intervals 0....

    [...]


01 Jan 2000
Abstract: Distribution-free Shewhart-type control charts are proposed for future sample percentiles based on a reference sample. These charts have a key advantage that their in-control run length distribution do not depend on the underlying continuous process distribution. Tables are given to help implement the charts for given sample sizes and false alarm rates. Expressions for the exact run length distribution and the average run length (ARL) are obtained using expectation by conditioning. Properties of the charts are studied, via evaluations of the run length distribution and the ARL. These computations show that in certain cases the proposed charts have attractive ARL properties over standard parametric charts such as the CUSUM and the EWMA. Calculations are illustrated with several short examples. Also included is a numerical example, using data from Montgomery (1997), where an application of the precedence chart produced slightly different results.

104 citations


Journal ArticleDOI
TL;DR: The charts proposed are preferable from a robustness point of view, have attractive ARL properties and would be particularly useful in situations where one uses a classical Shewhart "X" -chart.
Abstract: A class of Shewhart-type distribution-free control charts is considered. A key advantage of these charts is that the in-control run length distribution is the same for all continuous process distributions. Exact expressions for the run length distribution and the average run length (ARL) are derived and properties of the charts are studied via evaluations of the run length distribution probabilities and the ARL. Tables are provided for implementation for some typical ARL values and false alarm rates. The charts proposed are preferable from a robustness point of view, have attractive ARL properties and would be particularly useful in situations where one uses a classical Shewhart X-chart. A numerical illustration is given.

95 citations


"A generally weighted moving average..." refers background in this paper

  • ...[8] studied a class of nonparametric Shewhart-type charts based on precedence statistics, referred to as the Shewhart-type precedence charts....

    [...]