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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.
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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
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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
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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
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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)