An Explanatory Study on the Non-Parametric Multivariate T2 Control Chart

Most control charts require the assumption of normal distribution for observations. When distribution is not normal, one can use non-parametric control charts such as sign control chart. A deficiency of such control charts could be the loss of information due to replacing an observation with its sign or rank. Furthermore, because the chart statistics of T2 are correlated, the T2 chart is not a desire performance. Non-parametric bootstrap algorithm could help to calculate control chart parameters using the original observations while no assumption regarding the distribution is needed. In this paper, first, a bootstrap multivariate control chart is presented based on Hotelling’s T2 statistic then the performance of the bootstrap multivariate control chart is compared to a Hotelling’s T2 parametric multivariate control chart, a multivariate sign control chart, and a multivariate Wilcoxon control chart using a simulation study. Ultimately, the bootstrap multivariate control chart is used in an empirical example to study the process of sugar production

Abordagens Paramétricas e Não Paramétricas para Monitoramento de Parâmetro de Locação – Caso Univariado

The Shewhart control chart is a powerful statistical tool in process control. The operation of these control charts is the periodic sampling off items produced. They are analyzed according to some characteristic of interest. The quality characteristic can be an attribute or a variable. The chart contains two horizontal lines, called upper and lower control limits. The width of the range between these limits is chosen so that, when the sampling point is within the control limits, it is considered that the process is operating under control. However, when a point occurs outside these limits, it is considered that the process is out of control, requiring management intervention for the process to operating again in statistical control conditions. The in-control performance of non-parametric individuals control charts based on kernel estimators are studied by simulation. Three different procedures are adopted for kernel estimator bandwidth selection. It turns out that the alternative control charts are robust against deviations from symmetry and perform reasonably well under normality of the observations.O gráfico de controle de Shewhart é uma poderosa ferramenta em controle estatístico de processos. A operação desses gráficos de controle consiste na coleta periódica de itens produzidos, analisando-os de acordo com alguma característica de interesse. A característica de qualidade pode ser um atributo ou uma variável. O gráfico contém duas linhas horizontais, denominadas limites superior e inferior de controle. A amplitude do intervalo entre esses limites é escolhida de maneira que, quando o ponto amostral estiver dentro dos limites de controle, considera-se que o processo esteja operando sob controle. Entretanto, quando um ponto ocorrer fora desses limites, considera-se que o processo está fora de controle, exigindo intervenção gerencial para que o processo opere novamente em condições de controle estatístico. No presente trabalho são estudadas as consequências das várias estimativas paramétricas efetuadas para a construção de gráficos de controle de média e de medidas individuais. Em particular, são verificados os efeitos dessas estimativas no comprimento médio de sequência (CMS), que é bastante utilizado para medir o desempenho desses gráficos. São apresentadas também duas abordagens não paramétricas para determinação dos limites de gráficos de controle de média amostral e de medidas individuais: reamostragem por bootstrap e núcleo estimador. É analisado o desempenho de gráficos de controle por média, cujos limites são construídos por intermédio de metodologia de reamostragem por bootstrap e o desempenho de gráficos de controle de medidas individuais, construído por intermédio das metodologias de núcleos estimadores da função de distribuição. A determinação dos limites de controle baseia-se em observações obtidas na denominada Fase I, na qual são coletados os dados da característica de qualidade de interesse. São apresentados resultados de análise de sensibilidade de um conjunto de misturas de normais que simulam situações de não normalidade, em especial quanto à assimetria e a curtose da função de densidade de probabilidade da característica de qualidade de interesse

Robust control charts via winsorized and trimmed estimators

In process control, cumulative sum (CUSUM), exponentially weighted moving average (EWMA), and synthetic control charts are developed to detect small and moderate shifts. Small shifts which are hard to detect can be costly to the process control if left undetected for a long period. These control charts are not reliable under non-normality as the design structure of the charts is based on the sample mean. Sample mean is sensitive to outliers, a common cause of non-normality. In circumventing the problem, this study applied robust location estimators in the design structure of the control charts, instead of the sample mean. For such purpose, four robust estimators namely 20%-trimmed mean, median, modified one-step M-estimator (MOM), and winsorized MOM (WMOM) were chosen. The proposed charts were tested on several conditions which include sample sizes, shift sizes, and different types of non-normal distributions represented by the g-and-h distribution. Random variates for each distribution were obtained using SAS RANNOR before transforming them to the desired type of distribution. Robustness and detection ability of the charts were gauged through average run length (ARL) via simulation study. Validation of the charts’ performance which was done through real data study, specifically on potential diabetic patients at Universiti Utara Malaysia shows that robust EWMA chart and robust CUSUM chart outperform the standard charts. The findings concur with the results of simulation study. Even though robust synthetic chart is not among the best choice as it cannot detect small shifts as quickly as CUSUM or EWMA, its performance is much better than the standard chart under non-normality. This study reveals that all the proposed robust charts fare better than the standard charts under non-normality, and comparable with the latter under normality. The most robust among the investigated charts are EWMA control charts based on MOM and WMOM. These robust charts can fast detect small shifts regardless of distributional shapes and work well under small sample sizes. These characteristics suit the industrial needs in process monitoring