Stephen M. Scariano

THE ROBUSTNESS OF THE SYNTHETIC CONTROL CHART TO NON-NORMALITY

The Synthetic control chart has recently been introduced as an improvement over the standard Shewhart control chart for detecting changes in the mean of a normally distributed process. The salient features of the Shewhart chart and the Conforming Run Length chart are integrated to produce the Synthetic control chart. In many practical instances, the Synthetic chart is superior to the Shewhart chart in terms of quicker detection of out-of-control status when the process data are normally distributed. The robustness of the Synthetic chart to violations of the normality assumption is the central theme of this study. We find that in-control average run lengths for the Synthetic chart are reasonably close to the normal theory values when there is moderate nonnormality or when the sample size n is large. Additionally, out-of-control average run lengths are comparable to the corresponding normal theory values for a variety of non-normal distributions.

The Effects of Violations of Independence Assumptions in the One-Way ANOVA

Abstract This article focuses on the relationship between true Types I and II error probabilities and the effects of departures from independence assumptions on hypothesis testing in the oneway analysis of variance. A method for constructing a useful class of nonidentity error correlation matrices suitable for studying this relationship is offered and explored. Special emphasis is placed on the numerical features of this relationship that can be easily exploited in the classroom. The perspective is adaptable to more complicated designs including regression models.

The Generalized Synthetic Chart

Abstract The Generalized Synthetic chart is presented and mathematical expressions for its average run length and variance of the run length are developed. The methodology is applied to the EWMA and CUSUM charts and near-optimization procedures are discussed. The synthetic EWMA and CUSUM charts are compared with their standard counterparts, the original synthetic chart, and the Shewhart chart. Significant improvements in detecting power are reported.

A Note on the Lower-Sided Synthetic Chart for Exponentials

The synthetic control chart for exponential data is discussed and an expression is derived for its average run length, as well as its design parameters. The synthetic control chart for exponentials is shown analytically to be a two-in-a-row rule. This chart is compared with the Shewhart chart for individuals and with the worst-case, lower-sided exponential EWMA and CUSUM charts. While the synthetic control chart for exponentials outperforms the Shewhart chart for individuals, the EWMA and CUSUM charts are shown to be far superior in detecting decreases in the exponential mean.

The Synthetic t and Synthetic EWMA t Charts

Abstract The t and the exponentially weighted moving average (EWMA) t charts were introduced in quality control literature for cases when users are not able to accurately estimate the process standard deviation. In this paper we present the synthetic versions of the t and the EWMA-1 charts, and determine near-optimized control limits for these charts. The new charts are shown to have improved performance properties over the original t and EWMA-t charts, while maintaining the desirable feature that the process standard deviation does not need to be estimated.

Reconciling the Integral Equation and Markov Chain Approaches for Computing EWMA Average Run Lengths

Abstract The integral equation and Markov chain approaches for computing average run lengths for two-sided exponentially weighted moving average control charts are studied. For the integral equation approach, the choice of numerical method can greatly ease the burden of computation. Gaussian quadrature is recommended when the underlying process data arise from a distribution whose support is the entire real line; however, the Collocation method is to be preferred when the support is finite or semi-infinite. Results for EWMA average run length calculations are given for process data following normal, gamma, t, and uniform distributions. Ultimately, the Markov chain approach is shown to be equivalent to a special case of the integral equation method.

Joint Monitoring of the Mean and Variance of Combined Control Charts with Estimated Parameters

Joint , two-sided (CUSUM, S2), and (EWMA, S2) control charts are numerically compared when (i) process parameters are known and (ii) process parameters are estimated from retrospective data. In both cases, equations for the conditional and unconditional run length distributions are developed, and expressions for the average run lengths (ARL), the second moment of the run length (SMRL), and the standard deviation of the run lengths (SDRL) are derived for these charts. In-control and out-of-control ARLs and SDRLs are tabulated and compared for a variety of design parameters for each chart. Numerical results and practical recommendations are given.

Testing regression function adequacy with correlation and without replication

The well known pure error-lack of fit test which can be used to assess the adequacy of a proposed linear regression model requires replication and assumes that the error structure is . This procedure is generalized to provide a test for lack of fit for the 2 case of nonreplication and error structure for certain known positive definite correlation matrices V. Included in the class of applicable correlation matrices are the cases of intraclass correlation and equicorrelation. The critical points of the F distribution can be used to provide a test of the exact desired size.

A four-moment approach and other practical solutions to the behrens-f1sher problem

Fixed sample size approximately similar tests for the Behrens-Fisher problem are studied and compared with various other tests suggested in current sttistical methodelogy texts. Several fourmoment approxiamtely similar tests are developed and offered as alternatives. These tests are shown to be good practical solutions which are easily implemented in practice.

Computing Average Run Lengths for the MaxEWMA Chart

Abstract The MaxEWMA chart has recently been introduced as an alternative to control charting procedures that are designed to jointly detect changes in the mean and standard deviation of a normally distributed process. Here, a method for computing both in-control and out-of-control average run lengths for purposes of effectively designing this chart is offered. Design strategies are considered, numerical results to aid the design effort are given, and examples are discussed.