Univariate parametric and nonparametric statistical quality control techniques with estimated process parameters

Chapter 1 gives a brief introduction to statistical quality control (SQC) and provides background information regarding the research conducted in this thesis. We begin Chapter 2 with the design of Shewhart-type Phase I S2, S and R control charts for the situation when the mean and the variance are both unknown and are estimated on the basis of m independent rational subgroups each of size n available from a normally distributed process. The derivations recognize that in Phase I (with unknown parameters) the signaling events are dependent and that more than one comparison is made against the same estimated limits simultaneously; this leads to working with the joint distribution of a set of dependent random variables. Using intensive computer simulations, tables are provided with the charting constants for each chart for a given false alarm probability. Second an overview of the literature on Phase I parametric control charts for univariate variables data is given assuming that the form of the underlying continuous distribution is known. The overview presents the current state of the art and what challenges still remain. It is pointed out that, because the Phase I signaling events are dependent and multiple signaling events are to be dealt with simultaneously (in making an in-control or not-in-control decision), the joint distribution of the charting statistics needs to be used and the recommendation is to control the probability of at least one false alarm while setting up the charts. In Chapter 3 we derive and evaluate expressions for the run-length distributions of the Phase II Shewhart-type p-chart and the Phase II Shewhart-type c-chart when the parameters are estimated. We then examine the effect of estimating and on the performance of the p-chart and the c-chart via their run-length distributions and associated characteristics such as the average run-length, the false alarm rate and the probability of a “no-signal”. An exact approach based on the binomial and the Poisson distributions is used to derive expressions for the Phase II run-length distributions and the related Phase II characteristics using expectation by conditioning (see e.g. Chakraborti, (2000)). We first obtain the characteristics of the run-length distributions conditioned on point estimates from Phase I and then find the unconditional characteristics by averaging over the distributions of the point estimators. The in-control and the out-of-control properties of the charts are looked at. The results are used to discuss the appropriateness of the widely followed empirical rules for choosing the size of the Phase I sample used to estimate the unknown parameters; this includes the number of reference samples m and the sample size n. Chapter 4 focuses on distribution-free control charts and considers a new class of nonparametric charts with runs-type signaling rules (i.e. runs of the charting statistics above and below the control limits) for both the scenarios where the percentile of interest of the distribution is known and unknown. In the former situation (or Case K) the charts are based on the sign test statistic and enhance the sign chart proposed by Amin et al. (1995); in the latter scenario (or Case U) the charts are based on the two-sample median test statistic and improve the precedence charts by Chakraborti et al. (2004). A Markov chain approach (see e.g. Fu and Lou, (2003)) is used to derive the run-length distributions, the average run-lengths, the standard deviation of the run-lengths etc. for our runs rule enhanced charts. In some cases, we also draw on the results of the geometric distribution of order k (see e.g. Chapter 2 of Balakrishnan and Koutras, (2002)) to obtain closed form and explicit expressions for the run-length distributions and/or their associated performance characteristics. Tables are provided for implementation of the charts and examples are given to illustrate the application and usefulness of the charts. The in-control and the out-of-control performance of the charts are studied and compared to the existing nonparametric charts using criteria such as the average run-length, the standard deviation of the run-length, the false alarm rate and some percentiles of the run-length, including the median run-length. It is shown that the proposed “runs rules enhanced” sign charts offer more practically desirable in-control average run-lengths and false alarm rates and perform better for some distributions. 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.Thesis (PhD)--University of Pretoria, 2009.Mathematics and Applied Mathematicsunrestricte

On the bivariate Kummer-beta type IV distribution

In this paper the non-central bivariate Kummer-beta type IV distribution is introduced and derived via the Laplace transform of the non-central bivariate beta distribution by Gupta et al. (2009). We focus on and discuss the central bivariate Kummer-beta type IV distribution; this distribution is a special case of the non-central bivariate Kummer-beta type IV distribution and extends the popular Jones’ bivariate beta distribution. The probability density functions of the product and the ratio of the components of the central bivariate Kummer-beta type IV distribution are also derived and we provide tabulations of the associated lower percentage points as well as some upper percentage points that are useful in reliability.National Research Foundation, South Africahttp://www.tandfonline.com/loi/lsta20Statistic

A generally weighted moving average chart for time between events

Shewhart-type attribute charts are known to be inefficient for small changes in monitoring nonconformities. An alternative way is to use a time-weighted chart to monitor the time between events (TBE). We propose a one-sided Generally Weighted Moving Average control chart to monitor the time between events (TBE); regarded as the GWMA-TBE chart. To aid the implementation of the chart, the necessary design parameters are provided. An extensive performance analysis shows that the GWMA-TBE chart is better than the well-known EWMA and Shewhart charts at detecting very small to moderate changes. Finally, a summary and some conclusions are provided.STATOMET (Grant number: SMB2015A), University of Pretoria, South Africa.http://www.tandfonline.com/loi/lssp202018-05-09hj2018Statistic

Robustness of the EWMA control chart for individual observations

The traditional exponentially weighted moving average (EWMA) chart is one of the most popular control charts used in practice today. The in-control robustness is the key to the proper design and implementation of any control chart, lack of which can render its out-of-control shift detection capability almost meaningless. To this end, Borror et al. [5] studied the performance of the traditional EWMA chart for the mean for i.i.d. data. We use a more extensive simulation study to further investigate the in-control robustness (to non-normality) of the three different EWMA designs studied by Borror et al. [5]. Our study includes a much wider collection of non-normal distributions including light- and heavy-tailed and symmetric and asymmetric bi-modal as well as the contaminated normal, which is particularly useful to study the effects of outliers. Also, we consider two separate cases: (i) when the process mean and standard deviation are both known and (ii) when they are both unknown and estimated from an in-control Phase I sample. In addition, unlike in the study done by Borror et al. [5], the average run-length (ARL) is not used as the sole performance measure in our study, we consider the standard deviation of the run-length (SDRL), the median run-length (MDRL), and the first and the third quartiles as well as the first and the 99th percentiles of the in-control run-length distribution for a better overall assessment of the traditional EWMA chart’s in-control performance. Our findings sound a cautionary note to the (over) use of the EWMA chart in practice, at least with some types of non-normal data. A summary and recommendations are provided.STATOMET and the Department of Statistics at the University of Pretoria.http://www.tandfonline.com/loi/cjas20nf201

Improved Shewhart-type runs-rules nonparametric sign charts

Runs-rules are typically incorporated in control charts to increase their sensitivity to detect small process shifts. However, a drawback of this approach is that runs-rules charts are unable to detect large shifts quickly. In this paper improved runs-rules are introduced to the nonparametric sign chart, to address this limitation. Improved runs-rules are incorporated to maintain sensitivity to small process shifts, while having the added ability to detect large shifts in the process more efficiently. Performance comparisons between sign charts with runsrules and sign charts with improved runs-rules illustrate that the improved runs-rules are superior in performance for large shifts in the process, while maintaining the same sensitivity in the detection of small shifts.http://www.tandfonline.com/loi/lsta20hb201

A new development in the bivariate beta field

In this paper, the bivariate Kummer-beta type IV distribution, which extends the Jones’ bivariate beta distribution, is discussed. The probability density functions of the product and ratio of the components of this distribution are derived. Also, a shape analysis is done to investigate the effect of the new parameter.http://www.satnt.ac.z

Generalized multivariate beta distribution : control charting when the measurements are from an exponential distribution

In Statistical Process Control (SPC) there exists a need to model the runlength distribution of a Q-chart that monitors the process mean when measurements are from an exponential distribution with an unknown parameter. To develop exact expressions for the probabilities of run-lengths the joint distribution of the charting statistics is needed. This gives rise to a new distribution that can be regarded as a generalized multivariate beta distribution. An overview of the problem statement as identified in the field of SPC is given and the newly developed generalized multivariate beta distribution is proposed. Statistical properties of this distribution are studied and the effect of the parameters of this generalized multivariate beta distribution on the correlation between two variables is also discussed.The National Research Foundation, South Africa (Grant: FA2007043000003 and the Thuthuka programme: TTK20100707000011868).http://www.springerlink.com/content/0932-5026/nf201

A nonparametric exponentially weighted moving average signed-rank chart for monitoring location

Nonparametric control charts can provide a robust alternative in practice to the data analyst when there is a lack of knowledge about the underlying distribution. A nonparametric exponentially weighted moving average (NPEWMA) control chart combines the advantages of a nonparametric control chart with the better shift detection properties of a traditional EWMA chart. A NPEWMA chart for the median of a symmetric continuous distribution was introduced by Amin and Searcy (1991) using the Wilcoxon signed-rank statistic (see Gibbons and Chakraborti, 2003). This is called the nonparametric exponentially weighted moving average Signed-Rank (NPEWMA-SR) chart. However, important questions remained unanswered regarding the practical implementation as well as the performance of this chart. In this paper we address these issues with a more indepth study of the two-sided 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, including the traditional EWMA chart for subgroup averages and some nonparametric charts i.e. runs-rules enhanced Shewhart-type SR charts and the NPEWMA chart based on signs. Results show that the NPEWMA-SR chart performs just as well as and in some cases better than the competitors. A summary and some concluding remarks are given.http://www.elsevier.com/locate/csdanf201

A generally weighted moving average signed-rank control chart

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.Research of the first author was supported in part by STATOMET at the University of Pretoria, South Africa and National Research Foundation through the SARChI Chair at the University of Pretoria, South Africa.http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1099-16382017-12-31hb2017Statistic

Noncentral generalized multivariate Beta Type II distribution

The distribution of the variables that originates from monitoring the variance when the mean encountered a sustained shift is considered — specifically for the case when measurements from each sample are independent and identically distributed normal random variables. It is shown that the solution to this problem involves a sequence of dependent random variables that are constructed from independent noncentral chi- squared random variables. This sequence of dependent random variables are the key to understanding the performance of the process used to monitor the variance and are the focus of this article. For simplicity, the marginal (i.e. the univariate and bivariate) distributions and the joint (i.e. the trivariate) distribution of only the first three random variables following a change in the variance is considered. A multivariate generalization is proposed which can be used to calculate the entire run-length (i.e. the waiting time until the first signal) distribution.The National Research Foundation, South Africa (Grant: FA2007043000003 and the Thuthuka programme : TTK20100707000011868)http://www.ine.pt/revstat/inicio.htmlam201