A Sum of Squares Generally Weighted Moving Average Control Chart

In this article, we propose a new control chart called the sum of squares generally weighted moving average (SS-GWMA) control chart to simultaneously detect both the increase and decrease in the mean and/or variability. This new scheme is compared with the sum of squares exponentially weighted moving average (SS-EWMA) control chart. A simulation study is conducted to show that SS-GWMA control charts outperform SS-EWMA charts, in terms of the average run length (ARL), standard deviation of run length (SDRL), and diagnostic abilities. The design of SS-GWMA control charts is also discussed.     

ARL Performance of the Tukey's Control Chart

A Tukey's control chart was designed to monitor single observation data. Its easy control-limits setting and simple statistical concept are the most important advantages. In this study, we setup the Tukey's control chart based on known probability distribution and construct its average run length calculator. ARL performance of the Tukey's control chart is evaluated under a normal distribution and non normal distributions, respectively. The comparison of ARL performance reveals that Tukey's control chart is not sensitive to shift detection when the process heavily violates the normal assumption. The number of observations needed for Tukey's control chart setup is less than Shewhart's control chart. Tukey's control chart is a better choice for the process mean monitoring.     

Economic-Statistical Design of EWMA Control Charts Based on Taguchi's Loss Function

This paper proposes an economic-statistical design of the EWMA chart with time-varying control limits in which the Taguchi's quadratic loss function is incorporated into the economic-statistical design based on Lorenzen and Vance's economical model. A nonlinear programming with statistical performance constraints is developed and solved to minimize the expected total quality cost per unit time. This model, which is divided into three parts, depends on whether production continues during the period when the assignable cause is being searched for and/or repaired. Through a computational procedure, the optimal decision variables, including the sample size, the sampling interval, the control limit width, and the smoothing constant, can be solved for by each model. It is showed that the optimal economic-statistical design solution can be found from the set of optimal solutions obtained from the statistical design, and both the optimal sample size and sampling interval always decrease as the magnitude of shift increases.     

A Note on “Modified Tukey's Control Chart”

Tercero-Gomez et al. [Communications in Statistics - Simulation and Computation, 2012, 41(9), pp. 1566–1579] modified the Tukey's chart to improve the insensitivity of signaling mean shifts when the population is skewly distributed. They examine the efficiency for several skewed populations with the modified Tukey's chart (MTCC). The parameters of the MTCC were intuitively calibrated to fit several gamma distributions without following a specific mathematical procedure. Moreover, they performed no optimization for specific distributions. This study optimizes the MTCC with a statistical model for each skewed population and reexamines its efficiency. A numerical comparison shows that the optimal MTCC is sensitive to detect the shift of the means.     

Hayter and Tsui's test with double sampling for the vector mean of multivariate normal population

Abstract

In this paper, we introduce a version of Hayter and Tsui's statistical test with double sampling for the vector mean of a population under multivariate normal assumption. A study showed that this new test was more or as efficient than the well-known Hotelling's T² with double sampling. Some nice features of Hayter and Tsui's test are its simplicity of implementation and its capability of identifying the errant variables when the null hypothesis is rejected. Taking that into consideration, a new control chart called HTDS is also introduced as a tool to monitor multivariate process vector mean when using double sampling.     

Robust extreme ranked set sampling

In this paper, a robust extreme ranked set sampling (RERSS) procedure for estimating the population mean is introduced. It is shown that the proposed method gives an unbiased estimator with smaller variance, provided the underlying distribution is symmetric. However, for asymmetric distributions a weighted mean is given, where the optimal weights are computed by using Shannon's entropy. The performance of the population mean estimator is discussed along with its properties. Monte Carlo simulations are used to demonstrate the performance of the RERSS estimator relative to the simple random sample (SRS), ranked set sampling (RSS) and extreme ranked set sampling (ERSS) estimators. The results indicate that the proposed estimator is more efficient than the estimators based on the traditional sampling methods.     

A new multivariate CUSUM chart using principal components with a revision of Crosier's chart

In this article, we introduce a new multivariate cumulative sum chart, where the target mean shift is assumed to be a weighted sum of principal directions of the population covariance matrix. This chart provides an attractive performance in terms of average run length (ARL) for large-dimensional data and it also compares favorably to existing multivariate charts including Crosier's benchmark chart with updated values of the upper control limit and the associated ARL function. In addition, Monte Carlo simulations are conducted to assess the accuracy of the well-known Siegmund's approximation of the average ARL function when observations are normal distributed. As a byproduct of the article, we provide updated values of upper control limits and the associated ARL function for Crosier's multivariate CUSUM chart.     

On the distribution of the correlation coefficient when sampling from a mixture of two bivariate normal densities: Robustness and the effect of outliers

The distribution of the sample correlation coefficient is derived when the population is a mixture of two bivariate normal distributions with zero mean but different covariances and mixing proportions 1 - λ and λ respectively; λ will be called the proportion of contamination. The test of ρ = 0 based on Student's t, Fisher's z, arcsine, or Ruben's transformation is shown numerically to be nonrobust when λ, the proportion of contamination, lies between 0.05 and 0.50 and the contaminated population has 9 times the variance of the standard (bivariate normal) population. These tests are also sensitive to the presence of outliers.     

A Multivariate Synthetic Control Chart for Monitoring Process Mean Vector

In this article, a multivariate synthetic control chart is developed for monitoring the mean vector of a normally distributed process. The proposed chart is a combination of the Hotelling's T ² chart and Conforming Run Length chart. The operation, design, and performance of the chart are described. Average run length comparisons between some other existing control charts and the synthetic T ² chart are presented. They indicate that the synthetic T ² chart outperforms Hotelling's T ² chart and T ² chart with supplementary runs rules.     

Optimal acceptance sampling policy considering Bayesian risks

In this paper, we propose a sampling policy considering Bayesian risks. Various definitions of producer's risk and consumer's risk have been made. Bayesian risks for both producer and consumer are proven to give better information to decision-makers than classical definitions of the risks. So considering the Bayesian risk constraints, we seek to find optimal acceptance sampling policy by minimizing total cost, including the cost of rejecting the batch, the cost of inspection, and the cost of defective items detected during the operation. Proper distributions to construct the objective function of the model are specified. In order to demonstrate the application of the proposed model, we illustrate a numerical example. Furthermore, the results of the sensitivity analysis show that lot size, the cost of inspection, and the cost of one defective item are key factors in sampling policies. The acceptable quality level, the lot tolerance proportion defective, and Bayesian risks also affect the sampling policy, but variations of acceptable quality level and producer Bayesian risks, for values more than a specified value, cause no changes in sampling policy.     

Testing for mean and variance differences with samples from distributions that may be non-normal with unequal variances

The problem of comparing several samples to decide whether the means and/or variances are significantly different is considered. It is shown that with very non-normal distributions even a very robust test to compare the means has poor properties when the distributions have different variances, and therefore a new testing scheme is proposed. This starts by using an exact randomization test for any significant difference (in means or variances) between the samples. If a non-significant result is obtained then testing stops. Otherwise, an approximate randomization test for mean differences (but allowing for variance differences) is carried out, together with a bootstrap procedure to assess whether this test is reliable. A randomization version of Levene's test is also carried out for differences in variation between samples. The five possible conclusions are then that (i) there is no evidence of any differences, (ii) evidence for mean differences only, (iii) evidence for variance differences only, (iv) evidence for mean and variance differences, or (v) evidence for some indeterminate differences. A simulation experiment to assess the properties of the proposed scheme is described. From this it is concluded that the scheme is useful as a robust, conservative method for comparing samples in cases where they may be from very non-normal distributions.     

A powerful test for Balaam's design

The crossover trial design (AB/BA design) is often used to compare the effects of two treatments in medical science because it performs within‐subject comparisons, which increase the precision of a treatment effect (i.e., a between‐treatment difference). However, the AB/BA design cannot be applied in the presence of carryover effects and/or treatments‐by‐period interaction. In such cases, Balaam's design is a more suitable choice. Unlike the AB/BA design, Balaam's design inflates the variance of an estimate of the treatment effect, thereby reducing the statistical power of tests. This is a serious drawback of the design. Although the variance of parameter estimators in Balaam's design has been extensively studied, the estimators of the treatment effect to improve the inference have received little attention. If the estimate of the treatment effect is obtained by solving the mixed model equations, the AA and BB sequences are excluded from the estimation process. In this study, we develop a new estimator of the treatment effect and a new test statistic using the estimator. The aim is to improve the statistical inference in Balaam's design. Simulation studies indicate that the type I error of the proposed test is well controlled, and that the test is more powerful and has more suitable characteristics than other existing tests when interactions are substantial. The proposed test is also applied to analyze a real dataset. Copyright © 2015 John Wiley & Sons, Ltd.     

New robust estimators for detecting non-random patterns in multivariate control charts: a simulation approach

In the past decade, different robust estimators have been proposed by several researchers to improve the ability to detect non-random patterns such as trend, process mean shift, and outliers in multivariate control charts. However, the use of the sample mean vector and the mean square successive difference matrix in the T ² control chart is sensitive in detecting process mean shift or trend but less sensitive in detecting outliers. On the other hand, the minimum volume ellipsoid (MVE) estimators in the T ² control chart are sensitive in detecting multiple outliers but less sensitive in detecting trend or process mean shift. Therefore, new robust estimators using both merits of the mean square successive difference matrix and the MVE estimators are developed to modify Hotelling's T ² control chart. To compare the detection performance among various control charts, a simulation approach for establishing control limits and calculating signal probabilities is provided as well. Our simulation results show that a multivariate control chart using the new robust estimators can achieve a well-balanced sensitivity in detecting the above-mentioned non-random patterns. Finally, three numerical examples further demonstrate the usefulness of our new robust estimators.     

A nonparametric hypothesis test for heteroscedasticity

In this paper, a hypothesis test for heteroscedasticity is proposed in a nonparametric regression model. The test statistic, which uses the residuals from a nonparametric fit of the mean function, is based on an adaptation of the well-known Levene's test. Using the recent theory for analysis of variance when the number of factor levels goes to infinity, the asymptotic distribution of the test statistic is established under the null hypothesis of homocedasticity and under local alternatives. Simulations suggest that the proposed test performs well in several situations, especially when the variance is a nonlinear function of the predictor.     

Economic-statistical design of X ¥ charts for non-normal data by considering quality loss

When the X ¥ control chart is used to monitor a process, three parameters should be determined: the sample size, the sampling interval between successive samples, and the control limits of the chart. Duncan presented a cost model to determine the three parameters for an X ¥ chart. Alexander et al. combined Duncan's cost model with the Taguchi loss function to present a loss model for determining the three parameters. In this paper, the Burr distribution is employed to conduct the economic-statistical design of X ¥ charts for non-normal data. Alexander's loss model is used as the objective function, and the cumulative function of the Burr distribution is applied to derive the statistical constraints of the design. An example is presented to illustrate the solution procedure. From the results of the sensitivity analyses, we find that small values of the skewness coefficient have no significant effect on the optimal design; however, a larger value of skewness coefficient leads to a slightly larger sample size and sampling interval, as well as wider control limits. Meanwhile, an increase on the kurtosis coefficient results in an increase on the sample size and wider control limits.     

Kendall's tau for serial dependence

The authors show how Kendall's tau can be adapted to test against serial dependence in a univariate time series context. They provide formulas for the mean and variance of circular and noncircular versions of this statistic, and they prove its asymptotic normality under the hypothesis of independence. They present also a Monte Carlo study comparing the power and size of a test based on Kendall's tau with the power and size of competing procedures based on alternative parametric and nonparametric measures of serial dependence. In particular, their simulations indicate that Kendall's tau outperforms Spearman's rho in detecting first‐order autoregressive dependence, despite the fact that these two statistics are asymptotically equivalent under the null hypothesis, as well as under local alternatives.     

Confidence Intervals for Gini's Diversity Measure and Shannon's Entropy Using Adjusted Proportions

Abstract

Asymptotic confidence intervals are given for two functions of multinomial outcome probabilities: Gini's diversity measure and Shannon's entropy. “Adjusted” proportions are used in all asymptotic mean and variance formulas, along with a possible logarithmic transformation. Exact confidence coefficients are computed in some cases. Monte Carlo simulation is used in other cases to compare actual coverages to nominal ones. Some recommendations are made.     

Economic design based on a multivariate Bayesian VSI chart with a dual control limit

In this paper, a multivariate Bayesian variable sampling interval (VSI) control chart for the economic design and optimization of statistical parameters is designed. Based on the VSI sampling strategy of a multivariate Bayesian control chart with dual control limits, the optimal expected cost function is constructed. The proposed model allows the determination of the scheme parameters that minimize the expected cost per time of the process. The effectiveness of the Bayesian VSI chart is estimated through economic comparisons with the Bayesian fixed sampling interval and the Hotelling's T² chart. This study is an in-depth study on a Bayesian multivariate control chart with variable parameter. Furthermore, it is shown that significant cost improvement may be realized through the new model.     

On the Bivariate Kummer-Beta Type IV Distribution

Recent studies have shown the X-bar control chart with variable sampling interval detects shifts in the process mean faster than the traditional X-bar chart. These studies are usually based on the assumption that the process data are independently and normally distributed. However, many situations in practice violate these assumptions. In this study, a methodology is developed to economically design a variable sampling interval X-bar control chart that takes into consideration correlated non normal sample data. An example is provided to illustrate the solution procedure. A sensitivity analysis on the input parameters (i.e., the cost and the process parameters) is performed taking into account the non normality and the correlation on the optimal design of the chart.