1
A Double Generally Weighted Moving Average Exceedance Control Chart
H. Masoumi Karakani
1
Department of Statistics
University of Pretoria
Pretoria, 0002
Lynnwood Road, Hillcrest
South Africa
hosseinstatistics@gmail.com
http://orcid.org/0000-0002-2505- 6532
S.W. Human
Department of Statistics
University of Pretoria
Pretoria, 0002
Lynnwood Road, Hillcrest
South Africa
schalk.human@up.ac.za
J. van Niekerk
Department of Statistics
University of Pretoria
Pretoria, 0002
Lynnwood Road, Hillcrest
South Africa
janet.vanniekerk@up.ac.za
http://orcid.org/0000-0002-4334-2057
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 well-
known 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.
Keywords: Average run length; Control chart; DGWMA; Exceedance statistic;
Nonparametric.
1. Introduction
Statistical process control (SPC) refers to the collection of statistical procedures and problem
solving tools used to control and monitor the quality of the output of some production process
1
Corresponding author: e-mail: hosseinstatistics@gmail.com
2
(Balakrishnan et al.
1
). It is often of interest to detect any changes in location and/or dispersion
as early as possible and SPC possesses some of the extensively used tools to detect the
presence of causes of variation and to maintain stability. One of these tools is the control
chart and designed to detect changes in a process from an in-control to an out-of-control
state. Control charts are widely used to analyse and understand process variables, monitor
effects of the variables on the difference between target and actual performance, and
determine if a process is under statistical control. If a charting statistic plots within the upper
and lower control limits , it is considered to be in-control (IC) and if a charting statistic plots
on or outside either of the limits, it is declared to be out-of-control (OOC). Control charts
usually assume a known (normal) distribution for the process, however in many applications,
the underlying process distribution is unknown and/or not normal and hence the statistical
properties of commonly used charts, designed to perform best under the normal distribution
assumption, could be highly affected. Nonparametric control charts provide a robust
alternative when there is a lack of knowledge about the underlying process distribution. A
chart is called distribution-free or nonparametric if its IC run length distribution remains
invariant for all continuous process distributions. However, in some cases, symmetry of the
underlying distribution is required for the chart to be nonparametric. The number of plotting
statistics to be plotted until the first out-of-control signal occurs, is a discrete random variable
and is called the run length.
Walter A. Shewhart (1891-1967) proposed Shewhart-type charts, laying the foundation of
SPC. The interested reader is referred to Shewhart
2,3
. Shewhart-type control charts are the
most widely known charts in practice because of their global performance. The charting
statistic for the Shewhart-type charts is typically the value of the corresponding sample
statistic. As an example, assume that the observations from the process being monitored are
mutually independent and from a normal distribution with known mean and known
variance
. Then the symmetrically placed control limits for a Shewhart
chart are given
by
and
, where denotes the sample size, and are
the upper and lower control limits, respectively, and is the distance of the control
limits from the centerline. Because Shewhart-type charts only use the most recent sample to
decide if the process is IC or OOC, they are inefficient in detecting small and minor shifts in
the process. To overcome the difficulties of Shewhart-type charts in detecting process shifts,
it is recommended to use time-weighted or memory-type charts such as the Cumulative Sum
(CUSUM) proposed by Page
4
, the Exponentially Weighted Moving Average (EWMA)
3
proposed by Roberts
5
, the Double Exponentially Weighted Moving Average (DEWMA)
proposed by Shamma and Shamma
6
and the Generally Weighted Moving Average (GWMA)
proposed by Sheu and Lin
7
; these charts sequentially accumulate information over time to
determine the state of statistical control. The interested reader is referred to Montgomery
8
for
more details. Sheu and Hsieh
9
proposed a Double Generally Weighted Moving Average
(DGWMA) chart for the normal distribution (denoted by DGWMA-
) by combining the
DEWMA-
chart proposed by Zhang and Chen
10
and the GWMA-
chart proposed by Sheu
and Lin
7
. They have shown that the DGWMA-
chart is more sensitive in detecting minor
shifts in the process. The interested reader is referred to the works by Tai et al.
11
and Huang
et al.
12
. In typical applications Shewhart-type and time-weighted charts are based on the fact
that the observations of the underlying process are assumed to follow a normal or specified
probability distribution. However, in many situations, the assumption of normality may not
be justified or valid when the observations are from a non-normal or unknown distribution.
The CUSUM signed-rank charts were developed by Bakir and Reynolds
13
and the Shewhart-
type signed-rank chart by Bakir
14
. For more details the interested reader is referred to Amin
et al.
15
, Chakraborti et al.
16
and Bakir
17
. More recently, Lu
18
and Chakraborty et al.
19
proposed nonparametric GWMA charts based on the sign statistic (denoted by GWMA-SN)
and Wilcoxon signed-rank statistic (denoted by GWMA-SR), respectively, for the case when
the true process median is known; this is referred to as Case K. The parametric DGWMA
scheme has been shown to improve the detection ability of the GWMA chart. To this end,
Lu
20
proposed a nonparametric DGWMA chart (denoted by DGWMA-SN) when the true
process proportion is known. However, the true process median may not be known (referred
to as Case U) which limits the applicability of the distribution-free DGWMA charts based on
well-known nonparametric statistics, e.g. the sign and Wilcoxon signed-rank statistics.
Precedence or exceedance tests, based on precedence or exceedance statistics, are well
known nonparametric two-sample tests which do not suffer from the limits of the
aforementioned. Precedence statistics are defined as the number of observations from one of
the samples that exceeds a specified (
) order statistic of the other sample. A class of
nonparametric Shewhart-type charts, referred to as Shewhart-type precedence charts were
studied by Graham et al.
21
. For more information in terms of nonparametric control charts
please refer to Chakraborti et al.
22
. More recently, Chakraborty et al.
23
proposed a
nonparametric GWMA exceedance chart, referred to as the GWMA-EX chart, which
outperforms the EWMA-EX chart. Relatively little work has been done on nonparametric
4
schemes in the context of a DGWMA chart. Motivated by these findings, we construct a
distribution-free DGWMA chart based on an exceedance statistics for monitoring the
unknown median of a process. This chart is referred to as the DGWMA exceedance (or
DGWMA-EX) chart and integrates the virtues of both the GWMA and DEWMA charts to
achieve improved detection ability, when compared with the nonparametric GWMA-EX
chart. The proposed chart can be viewed as a generalized nonparametric time-weighted
control chart which includes other nonparametric time-weighted charts such as the GWMA-
EX, EWMA-EX and Shewhart-EX charts as limiting cases. Furthermore, the nonparametric
DEWMA chart based on exceedance statistics, labelled as the DEWMA-EX chart, which is a
special case of the DGWMA-EX chart, will be proposed and discussed as well. To the best of
our knowledge there is no research published on the DEWMA-EX chart, a special case of the
proposed DGWMA-EX chart, in the SPC literature, hence this paper also introduces this
chart and discusses some of its properties. The structure of the rest of the paper is as follows:
Section 2 provides the necessary theoretical framework for the DGWMA-EX chart. In
Section 3, the run length distribution and design of the proposed chart are studied. An
illustrative example is provided in Section 4, while some conclusions are provided in Section
5.
2. Preliminaries and Statistical Framework of the DGWMA exceedance chart
Let
iid
denote a Phase I reference sample from an in-control (IC)
process with an unknown continuous cumulative distribution function (c.d.f.)
where
denotes the unknown location parameter. Let
,
denote the
test sample in Phase II of size , with an unknown continuous c.d.f.
. The main intention is to design a control chart for monitoring the
unknown process location. The unknown/true value of the location parameter is denoted by
and the shifted location parameter is denoted by
; where is the
location shift. The process is declared to be IC when the unknown continuous c.d.f.’s and
are equal (i.e. or ) and out-of-control (OOC) when or .
Let
denote the number of observations in the
Phase II sample that exceeds
,
1,2,…, , i.e. the
order statistic from the Phase I sample of size . The statistic
is called the exceedance statistic and the probability
, is the
exceedance probability. For inference purposes, the exceedance and precedence tests are
equivalent in the sense that the two statistics are linearly related. Hereafter,
will be used to
denote the exceedance statistic for the
sample in Phase II.
5
2.1. Charting statistic
The DGWMA-EX chart is an extension of the GWMA-EX chart by invoking the DEWMA
technique i.e. performing “smoothing” twice. The GWMA-EX chart proposed by
Chakraborty et al.
23
is constructed by taking a weighted average of a sequence of the
exceedance statistic
’s. Let
and
be two discrete random variables denoting 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:
for and
(1)
A GWMA is a weighted moving average of a sequence of
’s, where the probability
is known as the weightr the
most recent statistic
among the last of
the
’s. The probability
is considered the weight for the starting value, denoted
by
, and is typically taken as the unconditional in-control (IC) expected value of the
exceedance statistic under consideration, i.e.,
. Hence, the
charting statistic for the GWMA-EX chart is as follows:
for
(2)
The distribution of
can be written as (Sheu and Hsieh
9
):
(3)
where
and
are the parameters and . Equation (3) is the
probability mass function (p.m.f.) of the two-parameter discrete Weibull distribution
introduced by Nakagawa and Osaki
24
. By substituting the p.m.f. of the two-parameter
discrete Weibull distribution in Equation (2), the charting statistic for the GWMA-EX is:
for
(4)
where
.
Now, to propose the DGWMA-EX chart as an extension of the GWMA-EX chart, the
DGWMA-EX charting statistic is defined as:
(5)
where
is the starting value, and
(6)
where
and
are the parameters and , similar to Equation (3).
Note that the superscripts that are used to denote the charting statistics for the GWMA-EX
and the DGWMA-EX charts (i.e.
and
, respectively) also denote the order in which we