JavaScript is disabled for your browser. Some features of this site may not work without it.

Graham, Marien Alet

Date:
2013

Distribution-free (nonparametric) control charts can be useful to the quality practitioner
when the underlying distribution is not known. The term nonparametric is not intended to imply
that there are no parameters involved, in fact, quite the contrary. While the term distribution-free
seems to be a better description of what we expect from these charts, that is, they remain valid for a
large class of distributions, nonparametric is perhaps the term more often used. In the statistics
literature there is now a rather vast collection of nonparametric tests and confidence intervals and
these methods have been shown to perform well compared to their normal theory counterparts.
Remarkably, even when the underlying distribution is normal, the efficiency of some nonparametric
tests relative to the corresponding (optimal) normal theory methods can be as high as 0.955 (see e.g.
Gibbons and Chakraborti (2010) page 218). For some other heavy-tailed and skewed distributions,
the efficiency can be 1.0 or even higher. It may be argued that nonparametric methods will be ‘less
efficient’ than their parametric counterparts when one has a complete knowledge of the process
distribution for which that parametric method was specifically designed. However, the reality is that
such information is seldom, if ever, available in practice. Thus it seems natural to develop and use
nonparametric methods in statistical process control (SPC) and the quality practitioners will be well
advised to have these techniques in their toolkits. In this thesis we only propose univariate
nonparametric control charts designed to track the location of a continuous process since very few
charts are available for monitoring the scale and simultaneously monitoring the location and scale
of a process.
Chapter 1 gives a brief introduction to SPC and provides background information regarding
the research conducted in this thesis. This will aid in familiarizing the reader with concepts and
terminology that are helpful to the following chapters. Details are given regarding the three main
classes of control charts, namely the Shewhart chart, the cumulative sum (CUSUM) chart and the
exponentially weighted moving average (EWMA) chart.
We begin Chapter 2 with a literature overview of Shewhart-type Phase I control charts
followed by the design and implementation of these charts. A nonparametric Shewhart-type Phase I
control chart for monitoring the location of a continuous variable is proposed. The chart is based on
the pooled median of the available Phase I samples and the charting statistics are the counts
(number of observations) in each sample that are less than the pooled median. The derivations
recognize that in Phase I the signalling events are dependent and that more than one comparison is
© University of Pretoria
v
made against the same estimated limits simultaneously; this leads to working with the joint
distribution of a set of dependant random variables. An exact expression for the false alarm
probability is given in terms of the multivariate hypergeometric distribution and this is used to
provide tables for the control limits. Some approximations are discussed in terms of the univariate
hypergeometric and the normal distributions.
In Chapter 3 Phase II control charts are introduced and considered for the case when the
underlying parameters of the process distribution are known or specified. This is referred to as the
‘standard(s) known’ case and is denoted Case K. Two nonparametric Phase II control charts are
considered in this chapter, with the first one being a nonparametric exponentially weighted moving
average (NPEWMA)-type control chart based on the sign (SN) statistic. A Markov chain approach
(see e.g. Fu and Lou (2003)) is used to determine the run-length distribution of the chart and some
associated performance characteristics (such as the average, standard deviation, median and other
percentiles). In order to aid practical implementation, tables are provided for the chart’s design
parameters. An extensive simulation study shows that on the basis of minimal required
assumptions, robustness of the in-control run-length distribution and out-of-control performance,
the proposed NPEWMA-SN chart can be a strong contender in many applications where traditional
parametric charts are currently used. Secondly, we consider the NPEWMA chart that was
introduced by Amin and Searcy (1991) using the Wilcoxon signed-rank statistic (see e.g. Gibbons
and Chakraborti (2010) page 195). This is called the nonparametric exponentially weighted moving
average signed-rank (NPEWMA-SR) chart. In their article important questions remained
unanswered regarding the practical implementation as well as the performance of this chart. In this
thesis we address these issues with a more in-depth study of the NPEWMA-SR chart. A Markov
chain approach is used to compute the run-length distribution and the associated performance
characteristics. Detailed guidelines and recommendations for selecting the chart’s design
parameters for practical implementation are provided along with illustrative examples. An extensive
simulation study is done on the performance of the chart including a detailed comparison with a
number of existing control charts. Results show that the NPEWMA-SR chart performs just as well
as and in some cases better than the competitors.
In Chapter 4 Phase II control charts are introduced and considered for the case when the
underlying parameters of the process distribution are unknown and need to be estimated. This is
referred to as the ‘standard(s) unknown’ case and is denoted Case U. Two nonparametric Phase II
control charts are proposed in this chapter. They are a Phase II NPEWMA-type control chart and a
nonparametric cumulative sum (NPCUSUM)-type control chart, based on the exceedance statistics,
© University of Pretoria
vi
respectively, for detecting a shift in the location parameter of a continuous distribution. The
exceedance statistics can be more efficient than rank-based methods when the underlying
distribution is heavy-tailed and / or right-skewed, which may be the case in some applications,
particularly with certain lifetime data. Moreover, exceedance statistics can save testing time and
resources as they can be applied as soon as a certain order statistic of the reference sample is
available. We also investigate the choice of the order statistics (percentile), from the reference
(Phase I) sample that defines the exceedance statistic. It is observed that other choices, such as the
third quartile, can play an important role in improving the performance of these exceedance charts.
It is seen that these exceedance charts perform as well as and, in many cases, better than its
competitors and thus can be a useful alternative chart in practice.
Chapter 5 wraps up this thesis with a summary of the research carried out and offers
concluding remarks concerning unanswered questions and / or future research opportunities.
© University

Thesis (PhD)--University of Pretoria, 2013.