Exceedance probabilities for parametric control charts

Common control charts assume normality and known parameters. Quite often, these assumptions are not valid and large relative errors result in the usual performance characteristics such as the false alarm rate or the average run length. A fully nonparametric approach can form an attractive alternative but requires more Phase I observations than usually available. Sufficiently general parametric families then provide realistic intermediate models. In this article, the performance of charts based on such families is considered. Exceedance probabilities of the resulting stochastic performance characteristics during in-control are studied. Corrections are derived to ensure that such probabilities stay within prescribed bounds. Attention is also devoted to the impact of the corrections for an out-of-control process. Simulations are presented both to illustrate and to demonstrate that the approximations obtained are sufficiently accurate for practical usage.     

Using Control Charts to Corroborate Bribery in Jai Alai

Statistical process control charts were used in the State of Florida District Court to help establish the guilt of an individual who was alleged to have affected the outcome of jai alai contests by bribing some of the contestants to lose. By placing wagers on the nonbribed contestants the briber gains an increased chance of winning, which is to the detriment of the other bettors. This paper gives an example of how statistical process control techniques can be employed to detect the unusually high bets that generally accompany bribery of the contestants. If the management of the jai alai gaming facility had been using control charts on a regular basis, the game fixing might have been detected much sooner.     

A Comparison of the Performance of Phase II Simple Linear Profile Control Charts when Parameters are Estimated

The standard deviation of the average run length (SDARL) is an important performance metric in studying the performance of control charts with estimated in-control parameters. Only a few studies in the literature, however, have considered this measure when evaluating control chart performance. The current study aims at comparing the in-control performance of three phase II simple linear profile monitoring approaches; namely, those of Kang and Albin (2000), Kim et al. (2003), and Mahmoud et al. (2010). The comparison is performed under the assumption of estimated parameters using the SDARL metric. In general, the simulation results of the current study show that the method of Kim et al. (2003) has better overall statistical performance than the competing methods in terms of SDARL values. Some of the recommended approaches based solely on the usual average run length properties can have poor SDARL performance.     

One-Class Classification-Based Control Charts for Monitoring Autocorrelated Multivariate Processes

In recent years, statistical process control (SPC) of multivariate and autocorrelated processes has received a great deal of attention. Modern manufacturing/service systems with more advanced technology and higher production rates can generate complex processes in which consecutive observations are dependent and each variable is correlated. These processes obviously violate the assumption of the independence of each observation that underlies traditional SPC and thus deteriorate the performance of its traditional tools. The popular way to address this issue is to monitor the residuals—the difference between the actual value and the fitted value—with the traditional SPC approach. However, this residuals-based approach requires two steps: (1) finding the residuals; and (2) monitoring the process. Also, an accurate prediction model is necessary to obtain the uncorrelated residuals. Furthermore, these residuals are not the original values of the observations and consequently may have lost some useful information about the targeted process. The main purpose of this article is to examine the feasibility of using one-class classification-based control charts to handle multivariate and autocorrelated processes. The article uses simulated data to present an analysis and comparison of one-class classification-based control charts and the traditional Hotelling's T ² chart.     

Progressive Variance Control Charts for Monitoring Process Dispersion

In a process, the deviation from location or scale parameters affects the quality of the process and waste resources. So it is essential to monitor such processes for possible changes due to any assignable causes. Control charts are the most famous tool used to meet this intention. It is useless to monitor process location until the assurance that process dispersion is in-control. This study proposes some new two-sided memory control charts named as progressive variance (PV) control charts which are based on sample variance to monitor changes in process dispersion assuming normality of quality characteristic to be monitored. Simulation studies are made, and an example is discussed to evaluate the performance of the proposed charts. The comparison of the proposed chart is made with exponentially weighted moving average- and cumulative sum-type charts for process dispersion. The study shows that performance of the proposed charts are uniformly better than its competitors for detecting positive shifts while for detecting negative shift in the variance their performance is better for small shifts and reasonably good for moderated shifts.     

EWMA Control Charts for Monitoring High Yield Processes

The main objective of this article is to scrutinize the efficiency and verify the performance superiority of the one-sided EWMA control chart on high-yield processes. The proposed control chart is designed to detect both upward and downward shifts of the fraction of non conforming products and is developed based on non transformed geometric counts. Its algorithmic function is theoretically established and numerous performance measures are extracted using analytical methods based on the Markov modeling of the chart. Comparisons with traditional high yield control charts are conducted. Optimality tables and nomograms are included to help graphical determination of the optimal chart parameters.     

<Emphasis Type="Italic">R</Emphasis>-charts for the exponential,Laplace and logistic processes

This paper (i) discusses theR-chart with asymmetric probability control limits under the assumption that the distribution of the quality characteristic under study is either exponential, Laplace, or logistic, (ii) examines the effect of the estimated probability limits on the performance of theR-chart, and (iii) obtains the desired probability limits of theR-chart that has a specified false alarm rate when probability limits must be estimated from preliminary samples taken from either the exponential, Laplace, or logistic processes.     

Comments on the Mixture Detection Rule Used in SPC Control Charts

When calculating independently the false alarm rate of the eight usual runs rules used in SPC control chart, it appears that the proposed rule designed to detect mixture patterns corresponds to a Type-I error strongly lower than the seven other rules. This discrepancy is underlined and the mixture rule is showed to be useless both for in-control and out-of-control processes. Thus a modification of the mixture detection rule is proposed and the impact of this new mixture rule is then illustrated and discussed using Monte Carlo calculations.     

On the Monitoring of Autocorrelated Linear Profiles

Control charts are commonly used to monitor quality of a process or product characterized by a quality characteristic or a vector of quality characteristics. However, in many practical situations the quality of a process or product can be characterized by a function or profile. Here we consider a linear function and investigate the violation of common independence assumption implicitly considered in most control charting applications. We specifically consider the case when profiles are not independent from each other over time. In this article, the effect of autocorrelation between profiles is investigated using average run length (ARL) criterion. Simulation results indicate significant impact on the ARL values when autocorrelation is overlooked. In addition, three methods based on time series approach are used to eliminate the effect of autocorrelation. Their performances are compared using ARL criterion.     

Phase I Monitoring of Polynomial Profiles

In some statistical process control applications, quality of a process or product is characterized by a relationship between two or more variables which is referred to as profile. In many practical situations, a profile can be modeled as a polynomial regression. In this article, three methods are developed for monitoring polynomial profiles in Phase I. Their performance is evaluated using power criterion. Furthermore, a method based on likelihood ratio test is developed to identify the location of shifts. Numerical simulation is used to evaluate the performance of the developed method.     

ARL-unbiased control charts with alarm and warning lines for monitoring Weibull percentiles using the first-order statistic

This article discusses methodology for constructing control charts to monitor the percentiles of a Weibull process with known shape parameter. Periodic samples are censored at the smallest observed value. Charts with alarm and warning limits are studied, and these limits are derived using theoretical results based on the first-order statistic. The performance of the proposed charts is evaluated and compared using average run lengths. A numerical application concerning life tests of an electronic product is presented to illustrate the methods.     

Monitoring two-stage processes with a profile at the second stage

In the multistage processes, quality of a process or a product at each stage is related to the previous stage(s). This property is referred to as a cascade property. Sometimes, quality of a process is characterized by a profile. In this paper, we consider a two-stage process with a normal quality characteristic in the first stage and a simple linear regression profile in the second stage. Then we propose two methods to monitor quality characteristics in both stages. The performance of the proposed two methods is evaluated through a numerical example in terms of average run length criterion.     

An Examination of the Robustness to Non Normality of the EWMA Control Charts for the Dispersion

ABSTRACT

The EWMA control chart is used to detect small shifts in a process. It has been shown that, for certain values of the smoothing parameter, the EWMA chart for the mean is robust to non normality. In this article, we examine the case of non normality in the EWMA charts for the dispersion. It is shown that we can have an EWMA chart for dispersion robust to non normality when non normality is not extreme.     

An Empirical-Likelihood-Based Multivariate EWMA Control Scheme

Nonparametric control charts are useful in statistical process control (SPC) when there is a lack of or limited knowledge about the underlying process distribution, especially when the process measurement is multivariate. This article develops a new multivariate SPC methodology for monitoring location parameter based on adapting a well-known nonparametric method, empirical likelihood (EL), to on-line sequential monitoring. The weighted version of EL ratio test is used to formulate the charting statistic by incorporating the exponentially weighted moving average control (EWMA) scheme, which results in a nonparametric counterpart of the classical multivariate EWMA (MEWMA). Some theoretical and numerical studies show that benefiting from using EL, the proposed chart possesses some favorable features. First, it is a data-driven scheme and thus is more robust to various multivariate non-normal data than the MEWMA chart under the in-control (IC) situation. Second, it is transformation-invariant and avoids the estimation of covariance matrix from the historical data by studentizing internally, and hence its IC performance is less deteriorated when the number of reference sample is small. Third, in comparison with the existing approaches, it is more efficient in detecting small and moderate shifts for multivariate non-normal process.     

Robust Estimation of Parameters in Simple Linear Profiles Using M-Estimators

In many situations, the quality of a process or product may be better characterized and summarized by a relationship between the response variable and one or more explanatory variables. Parameter estimation is the first step in constructing control charts. Outliers may hamper proper classical estimators and lead to incorrect conclusions. To remedy the problem of outliers, robust methods have been developed recently. In this article, a robust method is introduced for estimating the parameters of simple linear profiles. Two weight functions, Huber and Bisquare, are applied in the estimation algorithm. In addition, a method for robust estimation of the error terms variance is proposed. Simulation studies are done to investigate and evaluate the performance of the proposed estimator, as well as the classical one, in the presence and absence of outliers under different scenarios by the means of MSE criterion. The results reveal that the robust estimators proposed in this research perform as well as classical estimators in the absence of outliers and even considerably better when outliers exist. The maximum value of variance estimate in one scenario obtained from classical estimator is 10.9, while this value is 1.66 and 1.27 from proposed robust estimators when its actual value is 1.     

The Performance of Phase II Simple Linear Profile Approaches when Parameters Are Estimated

Previous studies of statistical performance of Phase II simple linear profile approaches were reported only for the case of known profile parameters assumption. The main objective of this article is to evaluate and compare the performance of these approaches when the profile parameters are estimated from an in-control Phase I profile data set. Simulations establish that the performance of these approaches is strongly affected when the parameters are estimated compared to the known parameters case. The in-control performance of the competing approaches significantly deteriorates if estimated parameters are used with control limits intended for known parameters, especially when only a few Phase I samples are used to estimate the parameters. The results show also that some profile monitoring approaches need much larger number of Phase I profiles than other approaches to achieve the expected statistical performance. They also show that the profile monitoring approach proposed by Mahmoud et al. (2010 Mahmoud , M. A. , Morgan , J. P. , Woodall , W. H. ( 2010 ). The monitoring of simple linear regression profiles with two observations per sample . Journal of Applied Statistics 37 : 1249 – 1263 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) has generally better out-of-control run length performance than the competing approaches when the estimated parameters are used in the charts design.     

Bayes Estimation of Shift Point in Normal Sequence and Its Application to Statistical Process Control

A sequence of independent observations X ₁, X ₂, …, X _m , X _m+1, …, X _n was observed on some measurable characteristic X in statistical process control. The shift in process mean is reflected in the sequence after X _m . The Bayes estimators of shift point m, and past and future process means, μ₁ and μ₂, are derived using various priors and loss functions. An application in statistical process control is given and a simulation study of the estimators is carried out.     

Diagnosis Aids in Multivariate Multiple Linear Regression Profiles Monitoring

Diagnosis aids in addition to detecting the out-of-control state is an important issue in multivariate multiple linear regression profiles monitoring; because a large number of parameters and profiles in this structure are involved. In this paper, we specifically concentrate on identification of profile(s) and parameter(s) which have changed during the process in multivariate multiple linear regression profiles structure in Phase II. We demonstrate the effectiveness of our proposed approaches through Monte Carlo simulations and a real case study in terms of accuracy percent.