Steven E. Rigdon

A multivariate exponentially weighted moving average control chart

A multivariate extension of the exponentially weighted moving average (EWMA) control chart is presented, and guidelines given for designing this easy-to-implement multivariate procedure. A comparison shows that the average run length (ARL) performance of this chart is similar to that of multivariate cumulative sum (CUSUM) control charts in detecting a shift in the mean vector of a multivariate normal distribution. As with the Hotellings χ2 and multivariate CUSUM charts, the ARL performance of the multivariate EWMA chart depends on the underlying mean vector and covariance matrix only through the value of the noncentrality parameter. Worst-case scenarios show that Hotellings χ2 charts should always be used in conjunction with multivariate CUSUM and EWMA charts to avoid potential inertia problems. Examples are given to illustrate the use of the proposed procedure.

The Performance of Exponentially Weighted Moving Average Charts With Estimated Parameters

The exponentially weighted moving average (EWMA) control chart is typically designed assuming that standards are given for the process parameters. In practice, the parameters are rarely known, and control charts are constructed using estimates in place of the parameters. This practice can affect the control charts run-length performance in both in- and out-of-control situations. Specifically, estimation can lead to substantially more frequent false alarms and yet reduce the sensitivity of the chart to detecting process changes. In this article, the run-length distribution of the EWMA chart with estimated parameters is derived. The effect of estimation on the performance of the chart is discussed in a variety of practical scenarios.

Poisson EWMA Control Charts

An exponentially weighted moving average control chart for monitoring Poisson data is introduced. The charting procedure is evaluated using a Markov chain approximation, and its average run length is compared to other procedures for Poisson data. Figure..

The Run Length Distribution of the CUSUM with Estimated Parameters

The CUSUM control chart is a popular method used to monitor the performance of production processes. The performance of the CUSUM is generally evaluated with the assumption that the process parameters are known. In practice, the parameters are rarely known and are frequently replaced with estimates from an in-control reference sample. We discuss the run length distribution of the CUSUM with estimated parameters and provide a method for approximating this distribution and moments. We evaluate the performance of the CUSUM with estimated parameters in a variety of practical situations.

The Power Law Process: A Model for the Reliability of Repairable Systems

The power law process, often misleadingly called the Weibull process, is a useful and simple model for describing the failure times of repairable systems. We present elementary properties of the power law process, such as point estimation of unknown par..

Parameter Estimation in Latent Trait Models.

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parametersθ=(θ1, …,θn) and item parametersβ=(β1, …,βk), whereβj may be vector valued. Withθ considered a random sample from a prior distribution with parameterφ, the estimation of (θ, β) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.

An integral equation for the in-control average run length of a multivariate exponentially weighted moving average control chart

An integral equation is given for the in-control average run length of a multivariate exponentially weighted moving average control chart. This integral equation is used to determine the appropriate upper control limits for various values of the smoothing constant r, the dimension p of the measured quality characteristic, and the desired in-control average run length L o Tables are given which allow a user to select the appropriate value of the upper control limit given these three conditions.

Properties of the T2 Control Chart When Parameters Are Estimated

Moments of the run length distribution are often used to design and study the performance of quality control charts. In this article the run length distribution of the T2 chart for monitoring a multivariate process mean is analyzed. It is assumed that the in-control process observations are iid random samples from a multivariate normal distribution with unknown mean vector and covariance matrix. It is shown that the in-control run length distribution of the chart does not depend on the unknown process parameters. Furthermore, it is shown that the out-of-control run length distribution of the chart depends only on the statistical distance between the in-control and out-of-control mean vectors. It follows that a performance analysis can be given without knowledge of the in-control values of the parameters or their estimates. The performance of charts constructed using traditional F-distribution–based control limits is studied. Recommendations are given for sample size requirements necessary to achieve desired performance. Corrected control limits are given for designing charts with estimated parameters when large sample sizes are not available.

A double-integral equation for the average run length of a multivariate exponentially weighted moving average control chart

The multivariate exponentially weighted moving average control chart is a control charting scheme that uses weighted averages of previously observed random vectors. This scheme, which is defined using Z0 = [mu]0, Zi = rXi + (1 - r)Zi - 1 (i [greater-or-equal, slanted] 1), where X1, X2, ... denote the vector-valued output of a process, can be used to detect shifts in the process mean vector more quickly, on the average, than the usual Hotelling T2 chart. We prove that for the special case [mu] = 0, [Sigma] = I, the average run length (ARL) depends on the initial value z0 for the MEWMA statistic only through its magnitude and the angle it makes with the mean vector. This theorem is then used to derive an integral equation of the ARL. This integral equation involves a double integral, and the unknown function is a function of two variables. ARLs can be obtained by approximating the solution to the integral equation. Previously, simulation was needed to approximate the ARLs.

Statistical Inference for a Modulated Power Law Process

The modulated power law process (MPLP) is a three-parameter stochastic point process model that can be used to describe the failure times of a repairable system. While the power law process implies that a system is in exactly the same condition just after a repair as it was just before the failure, the MPLP allows for the system to be affected by the failure and repair. We describe an algorithm for obtaining the maximum likelihood estimates (MLEs) of the three model parameters. Asymptotic results are used to give approximate confidence intervals and hypothesis tests for the parameters. A simulation study indicates that the confidence intervals and hypothesis tests have approximately the nominal level.