A Double Moving Average Control Chart

The double exponentially weighted moving average (DEWMA) technique has been investigated in recent years for detecting shifts in the process mean and has been shown to be more efficient than the corresponding exponentially weighted moving average (EWMA) technique. In this article, we extend the DEWMA technique of performing exponential smoothing twice to the double moving average (DMA) technique by computing the moving average twice. Using simulation, we show that our proposed DMA chart improves upon the ARL performance of the moving average (MA) chart in detecting mean shifts of small to moderate magnitudes. It is also shown through simulation that, generally, the DMA charts with spans, w = 10 and 15 provide comparable average run length (ARL) performances to the EWMA and cumulative sum (CUSUM) charts, designed for detecting small shifts.     

The Design of the Multivariate Synthetic Exponentially Weighted Moving Average Control Chart

A multivariate synthetic exponentially weighted moving average (MSEWMA) control chart is presented in this study. The MSEWMA control chart consists of a multivariate exponentially weighted moving average (MEWMA) control chart and a conforming run length control chart. The average run length of the MSEWMA control chart is obtained using a Markov chain approach. From the numerical comparisons, it is shown that the MSEWMA control chart is more efficient than the multivariate synthetic T ² control chart and the MEWMA control chart for detecting shifts in the process mean vector.     

THE CONTROL CHART FOR INDIVIDUAL OBSERVATIONS FROM A MULTIVARIATE NON-NORMAL DISTRIBUTION

The Hotelling's T²statistic has been used in constructing a multivariate control chart for individual observations. In Phase II operations, the distribution of the T²statistic is related to the F distribution provided the underlying population is multivariate normal. Thus, the upper control limit (UCL) is proportional to a percentile of the F distribution. However, if the process data show sufficient evidence of a marked departure from multivariate normality, the UCL based on the F distribution may be very inaccurate. In such situations, it will usually be helpful to determine the UCL based on the percentile of the estimated distribution for T². In this paper, we use a kernel smoothing technique to estimate the distribution of the T²statistic as well as of the UCL of the T²chart, when the process data are taken from a multivariate non-normal distribution. Through simulations, we examine the sample size requirement and the in-control average run length of the T²control chart for sample observations taken from a multivariate exponential distribution. The paper focuses on the Phase II situation with individual observations.     

Monitoring Nonlinear Profiles with Random Effects by Nonparametric Regression

The monitoring of process/product profiles is presently a growing and promising area of research in statistical process control. This study is aimed at developing monitoring schemes for nonlinear profiles with random effects. We utilize the technique of principal components analysis to analyze the covariance structure of the profiles and propose monitoring schemes based on principal component (PC) scores. The number of the PC scores used in constructing control charts is crucial to the detecting power. In the Phase I analysis of historical data, due to the dependency of the PC-scores, we adopt the usual Hotelling T ² chart to check the stability. For Phase II monitoring, we study individual PC-score control charts, a combined chart scheme that combines all the PC-score charts, and a T ² chart. Although an individual PC-score chart may be perfect for monitoring a particular mode of variation, a chart that can detect general shifts, such as the T ² chart and the combined chart scheme, is more feasible in practice. The performances of the schemes under study are evaluated in terms of the average run length.     

Note on the Cramér-Rao inequality in the nonregular case: the family of uniform distributions

We consider the family of uniform distributions with range of unit length. The main result of this note asserts that the average variance of any unbiased estimator of the midpoint of the range is not less than (2(n+1))(n+2))^-1 and this lower bound is sharp. The proof is based upon a nonregular version of the Cramér-Rao inequality.     

Accounting for phase I sampling variability in the performance of the MEWMA control chart with estimated parameters

In this article, we assess the performance of the multivariate exponentially weighted moving average (MEWMA) control chart with estimated parameters while considering the practitioner-to-practitioner variability. We evaluate the chart performance in terms of the in-control average run length (ARL) distributional properties; mainly the average (AARL), the standard deviation (SDARL), and some percentiles. We show through simulations that using estimates in place of the in-control parameters may result in an in-control ARL distribution that almost completely lies below the desired value. We also show that even with the use of larger amounts of historical data, there is still a problem with the excessive false alarm rates. We recommend the use of a recently proposed bootstrap-based design technique for adjusting the control limits. The technique is quite effective in controlling the percentage of short in-control ARLs resulting from the estimation error.     

Economic-statistical design of synthetic double sampling T2 chart

Abstract

An economic-statistical design of the synthetic double sampling (synDS) T² chart is presented in this study. The cost function is minimized to obtain the optimal design parameters of the synDS T² chart by incorporating the statistical constraints or the constraints on the average number of samples. An example is provided and a sensitivity analysis is conducted to study the effect of model parameters on the optimal solution of the design. The numerical comparison shows that the synDS T² chart performs better than the synthetic T² chart and the multivariate exponentially weighted moving average chart, in terms of the cost.     

A new confidence interval for the difference between two binomial proportions of paired data

Motivated by a study on comparing sensitivities and specificities of two diagnostic tests in a paired design when the sample size is small, we first derived an Edgeworth expansion for the studentized difference between two binomial proportions of paired data. The Edgeworth expansion can help us understand why the usual Wald interval for the difference has poor coverage performance in the small sample size. Based on the Edgeworth expansion, we then derived a transformation based confidence interval for the difference. The new interval removes the skewness in the Edgeworth expansion; the new interval is easy to compute, and its coverage probability converges to the nominal level at a rate of O(n^−1/2). Numerical results indicate that the new interval has the average coverage probability that is very close to the nominal level on average even for sample sizes as small as 10. Numerical results also indicate this new interval has better average coverage accuracy than the best existing intervals in finite sample sizes.     

Independence and sphericity tests for the residuals of space-time arma models

The mechanics of the procedure for building space-time autoregressive moving average (STARMA) models is dependent upon the form of G, the variance-covariance matrix of the underlying errors.This paper presents large sample tests of the hypotheses that G is diagonal and that G equals o² I. Tables of the critical values for these tests are constructed     

A Markov-chain-based regression model with random effects for the analysis of 18O-labelled mass spectra

The enzymatic ¹⁸O-labelling is a useful technique for reducing the influence of the between-spectra variability on the results of mass-spectrometry experiments. A difficulty in applying the technique lies in the quantification of the corresponding peptides due to the possibility of an incomplete labelling, which may result in biased estimates of the relative peptide abundance. To address the problem, Zhu et al. [A Markov-chain-based heteroscedastic regression model for the analysis of high-resolution enzymatically ¹⁸O-labeled mass spectra, J. Proteome Res. 9(5) (2010), pp. 2669–2677] proposed a Markov-chain-based regression model with heteroscedastic residual variance, which corrects for the possible bias. In this paper, we extend the model by allowing for the estimation of the technical and/or biological variability for the mass spectra data. To this aim, we use a mixed-effects version of the model. The performance of the model is evaluated based on results of an application to real-life mass spectra data and a simulation study.     

Estimation of quadratic functions of the Bernoulli parameter in inverse sampling method

The problem of estimating quadratic functions p² + βp (for different values of β), with reference to the estimation of genetic parameter and its functions, is discussed for the inverse sampling method. Then the inverse sampling method is compared with fixed size sampling method adopting the inverse of the precision per unit average sampling size as a criterion of comparison.     

Group sequential x2 statistic for testing the difference of two mean vectors in clinical trials

In this study we discuss the group sequential procedures for comparing two treatments based on multivariate observations in clinical trials. Also we suppose that a response vector on each of two treatments has a multivariate normal distribution with unknown covariance matrix. Then we propose a group sequential x² statistic in order to carry out repeated significance test for hypothesis of no difference between two population mean vectors. In order to realize the group sequential test where average sample number is reduced, we propose another modified group sequential x² statistic by extension of Jennison and Turnbull ( 1991 ). After construction of repeated confidence boundaries for making the repeated significance test, we compare two group sequential procedures based on two statistics regarding the average sample number and the power of the test in the simulations.     

The existence of the strong combined-optimal design

The technique of fold-over is useful for conducting follow-up experiments. Based on the minimum aberration criterion, Li and Lin (2003) developed an algorithm and used computer to search the corresponding optimal foldover designs for 16 and 32 runs in the 2 ^k-p design. In their study, they found that the 2¹⁰⁻⁶ design is the only one that is not a strong combined-optimal design among all the designs. However, they did not interpret the reason causing the phenomenon. This article will explore under what kind of conditions, that the strong combined-optimal design will exist, and the solutions of the related problems.     

A new S2 control chart using repetitive sampling

A new S² control chart is presented for monitoring the process variance by utilizing a repetitive sampling scheme. The double control limits called inner and outer control limits are proposed, whose coefficients are determined by considering the average run length (ARL) and the average sample number when the process is in control. The proposed control chart is compared with the existing Shewhart S² control chart in terms of the ARLs. The result shows that the proposed control chart is more efficient than the existing control chart in detecting the process shift.     

Integration of support vector machines and control charts for multivariate process monitoring

Statistical process control tools have been used routinely to improve process capabilities through reliable on-line monitoring and diagnostic processes. In the present paper, we propose a novel multivariate control chart that integrates a support vector machine (SVM) algorithm, a bootstrap method, and a control chart technique to improve multivariate process monitoring. The proposed chart uses as the monitoring statistic the predicted probability of class (PoC) values from an SVM algorithm. The control limits of SVM-PoC charts are obtained by a bootstrap approach. A simulation study was conducted to evaluate the performance of the proposed SVM–PoC chart and to compare it with other data mining-based control charts and Hotelling's T ² control charts under various scenarios. The results showed that the proposed SVM–PoC charts outperformed other multivariate control charts in nonnormal situations. Further, we developed an exponential weighed moving average version of the SVM–PoC charts for increasing sensitivity to small shifts.     

An economic statistical design of the Poisson EWMA control charts for monitoring nonconformities

An economic statistical model of the exponentially weighted moving average (EWMA) control chart for the average number of nonconformities in the sample is proposed. The statistical and economic performance of proposed design are evaluated using the average run length (ARL) and the hourly expected cost, respectively. A Markov chain approach is applied to derive expressions for ARL. The cost model is established based on the general cost function given in Lorenzen and Vance [The economic design of control charts: a unified approach. Technometrics. 1986;28:3–11]. An example is provided to illustrate the application of the proposed model. A sensitivity analysis is also carried out to investigate the effects of model parameters on the solution of the economic statistical design by using the design of experiments (DOE) technique.     

Towards the implementation of a universal control chart and estimation of its average run length using a spreadsheet: An artificial neural network is employed to model the parameters in a special case

A control chart procedure has previously been proposed (Champ et al., 1991) for which the Shewhart X ¥ -chart, the cumulative sum chart, and the exponentially weighted moving average chart are special cases. The rapid and easy production of these charts, plus many others, is proposed using spreadsheets. In addition, for all these novel charts, the average run lengths are generated as a guide to their likely behaviour. The cumulative sum chart is widely employed in quality control and is considered in greater detail. Charts are designed to exhibit acceptable average run lengths both when the process is in and out of control. A functional technique for parameter selection for such a chart is introduced that results in target average run lengths. It employs the method of artificial neural networks to derive appropriate coefficients. This approach may be extended to any of the charts previously introduced.     

A Bayes-type estimator for the Bradley-Terry model for paired comparison

A Bayes-type estimator is proposed for the worth parameter π_i and for the treatment effect parameter ln π_i in the Bradley-Terry Model for paired comparison. In contrast to current Bayes estimators which require iterative numberical calculations, this estimator has a closed form expression. This estimation technique is also extended to obtain estimators for the Luce Multiple Comparison Model. An application of this technique to a 2³ factorial experiment with paired comparisons is presented.     

Towards the implementation of a universal control chart and estimation of its average run length using a spreadsheet: an artificial neural network is employed to model the parameters in a special case

A control chart procedure has previously been proposed (Champ et al., 1991) for which the Shewhart X ¯-chart, the cumulative sum chart, and the exponentially weighted moving average chart are special cases. The rapid and easy production of these charts, plus many others, is proposed using spreadsheets. In addition, for all these novel charts, the average run lengths are generated as a guide to their likely behaviour. The cumulative sum chart is widely employed in quality control and is considered in greater detail. Charts are designed to exhibit acceptable average run lengths both when the process is in and out of control. A functional technique for parameter selection for such a chart is introduced that results in target average run lengths. It employs the method of artificial neural networks to derive appropriate coefficients. This approach may be extended to any of the charts previously introduced.     

A note on the asymptotic generalised variance for a moving average process

Relationships between the asymptotic generalised variance, of an r^th order moving average process, and the ordinary variance, of an associated r^th order autoregressive process, are established for r=1 and 2. This is extended to the case r=3, and a generalisation is suggested for all r>O, though this has not been checked beyond r=4.