A Generally Weighted Moving Average Signed-Rank Control Chart
N. Chakraborty
a
, S. Chakraborti
a,b
, S.W. Human
1
a
, N. Balakrishnan
c
a
Department of Statistics, University of Pretoria, Pretoria, Lynnwood Road, Hillcrest, South Africa, 0002
b
Department of Information Systems, Statistics and Management Science, University of Alabama, Tuscaloosa, AL 35487, U.S.A.
c
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Abstract: The idea of process monitoring emerged so as to preserve and improve the quality of a
manufacturing process. In this regard, control charts are widely accepted tools in the manufacturing sector for
monitoring the quality of a process. However, a specific distributional assumption for any process is
restrictive and often criticised. Distribution-free control charts are efficient alternatives when information on
the process distribution is partially or completely unavailable. In this article, we propose a distribution-free
generally weighted moving average (GWMA) control chart based on the Wilcoxon signed-rank (SR) statistic.
Extensive simulation is done to study the performance of the proposed chart. The performance of the proposed
chart is then compared to a number of existing control charts including the parametric GWMA chart for
subgroup averages, a recently proposed GWMA chart based on the sign statistic and an exponentially
weighted moving average (EWMA) chart based on the signed-rank statistic. The simulation results reveal that
the proposed chart performs just as well and in many cases better than the existing charts, and therefore can
serve as a useful alternative in practice.
Keywords: Control chart; Distribution-free; Average run-length; Generally weighted moving average;
Signed-rank statistic,
1. Introduction
Control charts are known to be efficient tools for monitoring quality of products. In the present era of
highly sophisticated technology, it is of increasing importance to design control charts that are efficient in
detecting small shifts in the characteristics of interest in a production process. While the Shewhart-type charts
are the best known and most widely used in practice due to their inherent simplicity and global performance,
other classes of charts, such as the exponentially weighted moving average (EWMA) and the generally
weighted moving average (GWMA) charts are useful and sometimes more naturally appropriate.
The traditional EWMA chart for the mean was introduced by Roberts
1
and includes the Shewhart-type
chart as a special case. The literature on EWMA charts is vast and still continues to grow at a considerable
pace. The reader is referred to the overview on EWMA charts by Ruggeri et al.
2
and the references therein. A
generalization of the EWMA chart, referred to as the Generally Weighted Moving Average (GWMA) chart,
was proposed by Sheu and Lin
3
and they showed that it does perform better in detecting small shifts in the
process mean.
In typical applications of the GWMA chart, it is usually assumed that the underlying process distribution
is normal. If normality is in doubt or cannot be justified, a distribution-free control chart is more desirable. For
an overview on distribution-free control charts and their advantages, the reader is referred to Chakraborti et
al.
4
.
1
Corresponding author: e-mail: schalk.human@up.ac.za
1
Amin and Searcy
5
proposed an EWMA chart based on the Wilcoxon signed-rank statistic (EWMA-SR
chart) for monitoring the known value of the median of a process. Graham et al.
6
further studied the practical
implementation and performance of the EWMA-SR chart. Graham et al.
7
proposed an EWMA chart based on
the sign statistic (EWMA-SN chart). Lu
8
proposed the GWMA chart based on the sign statistic, called the
GWMA sign (GWMA-SN) chart, and showed that it outperforms the EWMA-SN chart for small shifts and
performs similarly for large shifts. Lu
8
also showed that the GWMA-SN chart is more efficient than the
parametric GWMA chart for the mean (GWMA- chart) for underlying normal and many non-normal
distributions. It is therefore only natural to further investigate and develop the GWMA control chart based on
some other (and more efficient) nonparametric or distribution-free statistics.
The Wilcoxon signed-rank (SR) test is a popular nonparametric alternative to the paired two-sample t-
test, but has the advantage that normality is not needed; only symmetry of the underlying continuous process
distribution is needed which is easy to verify (see Konijin
9
for a discussion on some tests of symmetry).
Furthermore, it is well known that the SR test is more efficient than the sign (SN) test for a number of non-
normal symmetric continuous distributions (see Gibbons and Chakraborti
10
. In addition, it can be shown that
the Asymptotic Relative Efficiency (ARE) of the SR test relative to the Student t-test is at least 0.864 for any
symmetric continuous distribution (see Gibbons and Chakraborti
10
, page 508).
In this article, we propose a GWMA chart based on the Wilcoxon signed-rank statistic (hereafter referred
to as the GWMA-SR chart) to monitor the known value of the median of a process with a continuous
distribution; the objective is to gain better sensitivity for small sustained upward or downward step shifts. The
median is taken as the location parameter of interest as it is more robust and therefore a better choice than the
mean for measuring the central value. Moreover, the Wilcoxon signed-rank test considers the median as the
location parameter of interest.
The rest of this article is organised as follows: The GWMA-SR chart is defined in Section 2. In Section 3,
the design and implementation of the proposed control chart is provided. A detailed empirical study
comparing the performance of the GWMA-SR chart with a number of existing control charts is provided in
Section 4. An illustrative example is given in Section 5. Finally, a concluding summary and some
recommendations are presented in Section 6.
2. GWMA signed-rank (GWMA-SR) control chart
Let be the quality characteristic of interest and assume its underlying distribution to be continuous and
symmetric around . We take to be the median as it is a more robust measure and a better representative of
the central value of a distribution than the mean. Let
denote the known value of .
Suppose
, where and , denotes the
observation in the
random sample
(or rational subgroup) of size > 1. Let
denote the ranks of the absolute differences
, for
within the
subgroup. Define the statistic
, , (1)
2
where
if > 0 or < 0, respectively. Therefore, it is easily verified that
is the difference between the sum of the ranks of the
’s corresponding to positive and negative
differences, respectively, within the
subgroup. Also,
can be re-written as
because
the sum of all the ranks within a sample
; this relationship between the statistics will be used
later to show that the proposed GWMA control chart is non-parametric or distribution-free. Note that, as the
random sample is assumed to be drawn from a continuous distribution, the probability of tied observations, i.e.
, is theoretically zero and therefore ignored.
Let be the discrete random variable denoting the number of samples until the next occurrence of an
event since its last occurrence. Then, by summing over all possible time periods, we can write
. (2)
A generally weighted moving average (GWMA) is a weighted moving average (WMA) of a sequence of
statistics with the probability being regarded as the weight of the
most recent statistic
. In other words, the probability is the weight of the latest or most recent observation
and the probability is the weight of the most out-dated or oldest observation
. The probability
is taken to be the weight of the starting value, denoted by
, which is typically taken as the in-
control (IC) expected value of the statistic under consideration, i.e.
. Therefore, the
charting statistic for the GWMA-SR chart can be defined as
for , (3)
where
is the starting value of the chart. As in Sheu and Lin
3
, the distribution of is
taken to be
, where and are two parameters; this is the discrete
two-parameter Weibull distribution (Nakagawa and Osaki
11
). By substituting
and the probability mass
function (p.m.f.) of the two-parameter Weibull distribution in equation (3), the charting statistic for the
GWMA-SR chart simplifies to
for . (4)
Note that the GWMA-SR chart reduces to the EWMA-SR chart when and , where
is the smoothing parameter of the EWMA chart. The EWMA-SR chart further reduces to the Shewhart-SR
chart when and . The GWMA chart can therefore be viewed as a generalisation of both EWMA
and Shewhart-type charts with an additional parameter that provides more flexibility in designing the chart.
The in-control (IC) expected value and variance of the charting statistic
are given by
= 0 (5)
and
, (6)
respectively, where
is the sum of squares of the weights. Equations (5) and (6) are
derived using the results in Gibbons and Chakraborti
10
(page 198) for the mean and variance of the well-
3
known Wilcoxon signed-rank test statistic coupled with the properties of the EWMA charting statistic (see
Montgomery
12
, page 419).
The exact time-varying (and symmetrically placed) upper control limit (
), lower control limit
(L
) and centerline (
) of the GWMA-SR chart are given by
and
, (7)
where 0 is the distance of the control limits from the centerline.
The asymptotic variance of the charting statistic
is given by
,
where
, which is an increasing function of and converges as (see the Appendix for
more detail).
The steady-state control limits are used when the process has been running for several time periods and are
based on the asymptotic variance of the charting statistic (see Lucas and Saccucci
13
). The steady-state control
limits and the centerline are given by
and
, (8)
where the subscript “” denotes the steady-state values.
The following points are worth noting:
i. The distribution of the
statistic is discrete, symmetric about zero and its extreme values, i.e. the
minimum and maximum, are
and
, respectively; these extremes occur when all the
observations within a subgroup are less than or greater than
, respectively;
ii. We study two-sided GWMA-SR charts with symmetrically placed control limits, i.e. equidistant from the
centerline. The methodology can be easily modified wherein a one-sided chart is more meaningful or when
two-sided control charts with asymmetric control limits are necessary;
iii. Steady-state control limits are used in order to simplify the application/implementation of the chart. For the
sake of notational simplicity, we will use and hereafter to denote the steady-state control limits;
iv. If any charting statistic
plots on or outside either of the control limits given by equation (8), a signal is
given and the process is declared to be out-of-control (OOC). Otherwise, the process is considered to be in-
control (IC), which implies that no location shift has occurred, and the charting procedure continues on.
In the next section, we discuss the design of the proposed GWMA-SR chart in more detail.
3. The design and implementation of GWMA-SR chart
Performance measures are needed to design and compare the performance of control charts. The
traditional approach of evaluating a control chart is to obtain the run-length distribution and its associated
characteristics. The run-length is a discrete random variable that represents the number of samples which must
be collected (or, equivalently, the number of charting statistics that must be plotted) in order for the chart to
4
detect a shift or give a signal. An intuitively appealing and popular measure of a chart’s performance is the
average run-length (), which is the expected number of charting statistics that must be plotted in order for
the chart to signal (see Human and Graham
14
). Clearly, for an efficient control chart, one would like to have
the in-control (denoted
) to be “large” and the out-of-control (denoted
) to be “small”.
Although other measures such as the standard deviation of the run-length and various upper and
lower percentiles could be and have been used to supplement the evaluation of control charts, the is the
most widely used measure due to its intuitive appealing interpretation. Therefore, we use here the to
design and compare the performance of the proposed GWMA-SR chart to other charts.
The design of a control chart typically involves solving for the combination of the chart’s parameters, i.e.
, and , so as to obtain a pre-specified in-control average run-length denoted by
. The computational
aspects of the run-length distribution for the GWMA-SR chart are discussed next, followed by the design of
the GWMA-SR control chart.
3.1 The run-length distribution of the GWMA-SR chart
Suppose the run-length random variable is denoted by and that
denotes the signalling event at the
sample. The complimentary event is the non-signalling event and is the event that there is no signal at the
sample, i.e.
for 1,2,3,.... Then, in general, the run-length distribution can be
written as
, for 1,2,3,...; i.e. there is a signal for the first time at the
sample. For any , we can re-write the event that a charting statistic is between the control limits, i.e.
], as
, where
and
, respectively, and
and
, for 2,3,4,… (9)
The benefit of re-writing the non-signalling event is as follows: Instead of working with the joint
distribution of a sequence of dependent charting statistics, i.e. the
’s (which is compared to the steady-state
control limits in equation (8)), one works with the joint distribution of a sequence of independent sample
statistics
(which is compared to the varying limits in (9)). The run-length distribution can therefore be
written as
and
, (10)
where
for ; see the Appendix for more detail.
As the samples are assumed to be independent, the
’s are independent and an expression for
can be
obtained using the relationship
; see Gibbons and Chakraborti
10
(page 198) for more
detail on the statistic
. An expression for
is given by
(11)
5