Distribution-free Phase II CUSUM control chart for joint monitoring of location and scale

Mukherjee and Chakraborti1 proposed a single distribution-free (nonparametric) Shewhart-type chart based on the Lepage2 statistic for simultaneously monitoring both the location and the scale parameters of a continuous distribution when both of these parameters are unknown. In the present work, we consider a single distribution-free CUSUM chart, based on the Lepage2 statistic, referred to as the CUSUM-Lepage (denoted by CL) chart. The proposed chart is distribution-free (nonparametric) and therefore, the in control (denoted IC) properties of the chart remain invariant and known for all continuous distributions. Control limits are tabulated for implementation of the proposed chart in practice. The IC and out of control (denoted OOC) performance properties of the chart are investigated through simulation studies in terms of the average, the standard deviation, the median and some percentiles of the run length distribution. Detailed comparison with a competing Shewhart-type chart is presented. Several existing CUSUM charts are also considered in the performance comparison. The proposed CL chart is found to perform very well in the location-scale models. We also examine the effect of the choice of the reference value (k) of CUSUM chart on the performance of the CL chart. The proposed chart is illustrated with a real data set. Summary and conclusions are presented.http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1099-16382016-02-28hb201

Nonparametric (distribution-free) control charts : an updated overview and some results

Control charts that are based on assumption(s) of a specific form for the underlying process distribution are referred to as parametric control charts. There are many applications where there is insufficient information to justify such assumption(s) and, consequently, control charting techniques with a minimal set of distributional assumption requirements are in high demand. To this end, nonparametric or distribution-free control charts have been proposed in recent years. The charts have stable in-control properties, are robust against outliers and can be surprisingly efficient in comparison with their parametric counterparts. Chakraborti and some of his colleagues provided review papers on nonparametric control charts in 2001, 2007 and 2011, respectively. These papers have been received with considerable interest and attention by the community. However, the literature on nonparametric statistical process/quality control/monitoring has grown exponentially and because of this rapid growth, an update is deemed necessary. In this article, we bring these reviews forward to 2017, discussing some of the latest developments in the area. Moreover, unlike the past reviews, which did not include the multivariate charts, here we review both univariate and multivariate nonparametric control charts. We end with some concluding remarks.https://www.tandfonline.com/loi/lqen20hj2020Science, Mathematics and Technology Educatio

Some new nonparametric distribution-free control charts based on rank statistics

Ph.DDOCTOR OF PHILOSOPH

Statistical Quality Control with the qcr Package

[Abstract] The R package qcr for Statistical Quality Control (SQC) is introduced and described. It includes a comprehensive set of univariate and multivariate SQC tools that completes and increases the SQC techniques available in R. Apart from integrating different R packages devoted to SQC (qcc, MSQC), qcr provides nonparametric tools that are highly useful when Gaussian assumption is not met. This package computes standard univariate control charts for individual measurements, (Formula presented), S, R, p, np, c, u, EWMA, and CUSUM. In addition, it includes functions to perform multivariate control charts such as Hotelling T2, MEWMA and MCUSUM. As representative features, multivariate nonparametric alternatives based on data depth are implemented in this package: r, Q and S control charts. The qcr library also estimates the most complete set of capability indices from first to the fourth generation, covering the nonparametric alternatives, and performing the corresponding capability analysis graphical outputs, including the process capability plots. Moreover, Phase I and II control charts for functional data are included.The work of Salvador Naya, Javier Tarrío-Saavedra, Miguel Flores and Rubén Fernández-Casal has been supported by MINECO grant MTM2017-82724-R, and by the Xunta de Galicia (Grupos de Referencia Competitiva ED431C-2020-14 and Centro de Investigación del Sistema universitario de Galicia ED431G 2019/01), all of them through the ERDF. The research of Miguel Flores has been partially supported by Grant PII-DM-002-2016 of Escuela Politécnica Nacional of Ecuador. In addition, the research of Javier Tarrío-Saavedra has been also founded by the eCOAR project (PC18/03) of CITICXunta de Galicia; ED431C-2020-14Xunta de Galicia; ED431G 2019/01Escuela Politécnica Nacional de Ecuador; PII-DM-002-201

Affine invariant signed-rank multivariate exponentially weighted moving average control chart for process location monitoring

Multivariate statistical process control (SPC) charts for detecting possible shifts in mean vectors assume that data observation vectors follow a multivariate normal distribution. This assumption is ideal and seldom met. Nonparametric SPC charts have increasingly become viable alternatives to parametric counterparts in detecting process shifts when the underlying process output distribution is unknown, specifically when the process measurement is multivariate. This study examined a new nonparametric signed-rank multivariate exponentially weighted moving average type (SRMEWMA) control chart for monitoring location parameters. The control chart was based on adapting a multivariate spatial signed-rank test. The test was affine-invariant and the weighted version of this test was used to formulate the charting statistic by incorporating the exponentially weighted moving average (EWMA) scheme. The test\u27s in-control (IC) run length distribution was examined and the IC control limits were established for different multivariate distributions, both elliptically symmetrical and skewed. The average run length (ARL) performance of the scheme was computed using Monte Carlo simulation for select combinations of smoothing parameter, shift, and number of p-variate quality characteristics. The ARL performance was compared to the performance of the multivariate exponentially weighted moving average (MEWMA) and Hotelling T2. The control charts for observation vectors sampled the multivariate normal, multivariate t, and multivariate gamma distributions. The SRMEWMA control chart was applied to a real dataset example from aluminum smelter manufacturing that showed the SRMEWMA performed well. The newly investigated nonparametric multivariate SPC control chart for monitoring location parameters--the Signed-Rank Multivariate Exponentially Weighted Moving Average (SRMEWMA)--is a viable alternative control chart to the parametric MEWMA control chart and is sensitive to small shifts in the process location parameter. The signed-rank multivariate exponentially weighted moving average performance for data from elliptically symmetrical distributions is similar to that of the MEWMA parametric chart; however, SRMEWMA\u27s performance is superior to the performance of the MEWMA and Hotelling\u27s T2 control charts for data from skewed distributions

Monitoring for a Shift in a Process Covariance Matrix Using the Generalized Variance

The commonly recommended charts for monitoring the mean vector are affected by a shift in the covariance matrix. As in the univariate case, a chart for monitoring for a change in the covariance matrix should be examined first before examining the chart used to monitor for a change in the mean vector. One such chart is the one that plots the generalized sample variance lSl verses the sample number t. We propose to study charts based on the statistics V = l(n - 1) Σ₀-¹ Sl¹/̳p̳ and U = 1n (l (n-1) Σ₀-¹Sl¹/̳p̳), where n is the sample size and Σ₀ is the in-control value of the process covariance matrix Σ. In particular, we will study the Shewhart V and U charts supplemented with runs rules. Also, we examine the methods that are useful in studying the run length properties of the cumulative sum (CUSUM) U charts. Further, we will study the effect that estimating Σ₀ has on the performance of these charts. Guidance will be given for designing the Shewhart charts with runs rules with illustrative examples

Phase - II Lepage - type CUSUM charts for joint monitoring of location and scale

Most control charts in nonparametric statistics tend to focus on detecting changes in the location or scale parameters. Since it is ideal to monitor both, researchers have proposed to combine them into a single statistic. From there, the Lepage-type family of tests was born, and it’s been updated with many versions. Chowdhury, Mukherjee, and Chrakaborti proposed merging the Ansari-Bradley and Wilcoxon Rank Sum tests into a single plotting statistic able to track persistent changes in the location and scale parameters in Phase II in the form of a CUSUM chart, using a reference sample from Phase I. The work from Guerrero (2016) based on several tests proposed by Conover et al. (1981) shows that there may be many statistics able to perform better than the Ansari-Bradley for monitoring the scale parameter. In the present work, we propose three Control charts based on the mentioned recommendations. These are the LP-M, LP-FK, and LP-SR CUSUM charts, in which the Ansari-Bradley statistic is replaced for the Mood, the Fligner and Killeen, and the Squared ranks tests respectively. Via Monte Carlo simulations, Control limits are tabulated for practitioners, plus the In-control and Out-of-control performance of the charts are calculated and compared with the CUSUM-Lepage Chart proposed by Chowdhury. This will help to choose the most useful CUSUM chart for each scenario. An example using real data illustrates how the proposed Control Charts must be implemented