Jesse C. Arnold

X charts with variable sampling intervals

The usual practice in using a control chart to monitor a process is to take samples from the process with fixed sampling intervals. This article considers the properties of the chart when the sampling interval between each pair of samples is not fixed but rather depends on what is observed in the first sample. The idea is that the time interval until the next sample should be short if a sample shows some indication of a change in the process and long if there is no indication of a change. The proposed variable sampling interval (VSI) chart uses a short sampling interval if is close to but not actually outside the control limits and a long sampling interval if is close to target. If is actually outside the control limits, then the chart signals in the same way as the standard fixed sampling interval (FSI) chart. Properties such as the average time to signal and the average number of samples to signal are evaluated. Comparisons between the FSI and the VSI charts indicate that the VSI chart is substantially ...

CUSUM charts with variable sampling intervals

A standard cumulative sum (CUSUM) chart for controlling the process mean takes samples from the process at fixed-length sampling intervals and uses a control statistic based on a cumulative sum of differences between the sample means and the target value. This article proposes a modification of the standard CUSUM scheme that varies the time intervals between samples depending on the value of the CUSUM control statistic. The variable sampling interval (VSI) CUSUM chart uses short sampling intervals if there is an indication that the process mean may have shifted and long sampling intervals if there is no indication of a change in the mean. If the CUSUM statistic actually enters the signal region, then the VSI CUSUM chart signals in the same manner as the standard CUSUM chart. A Markov-chain approach is used to evaluate properties such as the average time to signal and the average number of samples to signal. Results show that the proposed VSI CUSUM chart is considerably more efficient than the standard CUS...

Optimal one-sided shewhart control charts with variable sampling intervals

When a control chart is used to detect changes in a process the usual practice is to take samples from the process using a fixed sampling interval between samples. This paper considers the properties of Shewhart control charts when the sampling interval used after each sample is not tixed but instead depends on what is observed in the sample. The basic rationale is that the sampling interval should be short if there is some indication of a change in the process and long if there is no indication of a change. If the indication of a change is strong enough then the chart signals in the same way as the fixed sampling interval Shewhart chart. The result is that samples will be taken more frequently when there is a change in the process, and this process change can be detected much more quickly than when fixed sampling intervals are used. Expressions for properties such as the average time to signal and the average number of samples to signal are developed. It is shown that if the sampling interval must be cho...

EWMA control charts with variable sample sizes and variable sampling intervals

Abstract Traditional control charts for process monitoring are based on taking samples of fixed size from the process using a fixed sampling interval. Variable Sample Size (VSS) and Variable Sampling Interval (VSI) control charts vary the sampling rate from the process as a function of the data from the process. VSS and VSI control charts sample at a higher rate when there is evidence of a change in the process, and are thus able to detect process changes faster than traditional control charts. This paper considers general VSS and VSI control charts and develops integral equation and Markov chain methods for finding the statistical properties of these charts. EWMA charts with the VSS and/or the VSI features are studied in detail, and different ways of defining the EWMA control statistic are investigated. It is shown that using either the VSS or VSI feature in an EWMA control chart substantially improves the ability to detect all but very large shifts in the process mean. The VSI feature usually gives more improvement in detection ability than the VSS feature, and using both features together sometimes gives more improvement than using either one separately. Guidelines are given for choosing the possible sample sizes and the possible sampling intervals for these charts. EWMA charts with the VSS and/or VSI feature are compared to CUSUM charts and Shewhart X¯ charts with the VSS and/or VSI features.

Variable Sampling Interval X Charts in the Presence of Correlation

Traditional applications of control charts use fixed sampling interval (FSI) charts in which the time interval between samples is fixed. Recently, more efficient variable sampling interval (VSI) charts have been developed in which the next observation i..

CUSUM Control Charts With Variable Sample Size and Sampling Intervals

Traditional control charts for process monitoring are based on taking samples of fixed size from the process using a fixed sampling interval. Variable sample size (VSS) and variable sampling interval (VSI) control charts vary the sampling rate from the process as a function of the data from the process. By sampling at a higher rate when there is an indication of a change in the process, VSS and VSI control charts can detect process changes faster than traditional control charts. Previous research has considered the properties of CUSUM charts which use the VSI feature. This paper considers CUSUM charts with the VSS feature and with both the VSS and VSI features. Two ways of developing the control statistic of these charts are considered. It is shown that using either the VSS or VSI feature in a CUSUM control chart will improve the ability to detect all but very large process shifts. The VSI feature usually gives more improvement in detection ability than the VSS feature, but using both features together will give more improvement than either one separately. Guidelines are given for choosing the possible sample sizes and the possible sampling intervals for these charts. Methods for setting up these charts for practical applications are also given.

Variable sampling intervals for multiparameter shewhart charts

This paper considers the problem of using control charts to simultaneously monitor more than one parameter with emphasis on simultaneously monitoring the mean and variance. Fixed sampling interval control charts are modified to use variable sampling intervals depending on what is being observed from the data. Two basic strategies are investigated. One strategy uses separate control charts for each parameter, A second strategy uses a proposed single combined statistic which is sensitive to shifts in both the mean and variance. Each procedure is compared to corresponding fixed interval procedures. It is seen that for both strategies the variable sampling interval approach is substantially more efficient than fixed interval procedures.

A Two-Sample Test for Independence in 2×2 Contingency Tables with Both Margins Subject to Misclassification

Abstract A double sampling scheme is considered for estimation and testing under misclassification in both margins of a 2×2 contingency table; the matrix of misclassification probabilities is assumed to be unknown and nonsingular. Maximum likelihood estimators are given for the error-free and misclassification probabilities. Numerical methods are utilized to obtain maximum likelihood estimates under the independence hypothesis and the regularity conditions are affirmed for the asymptotic properties of the estimates to hold. The log-likelihood ratio statistic is shown to be asymptotically distributed as a chi-square. Computer simulations indicated important considerations for testing under double margin misclassification.

The Efficiency of Blocking: How to Use MS(Blocks)/MS(Error) Correctly

Abstract Even though there are no valid tests of block effects in randomized complete block and Latin square experiments, it is noted that a commonly used measure of efficiency is monotonically related to the F ratios used inappropriately for testing the effectiveness of blocking. Because of this relationship one can give beginning students a useful interpretation of otherwise inappropriate F statistics without introducing concepts of relative efficiency.

On Double-Stage Estimation in Simple Linear Regression Using Prior Knowledge

A two-stage procedure using shrinkage techniques is used for estimation in the simple linear regression model. A direct estimation of the predicted response as well as estimation of the parameters in the model are considered in this paper. Throughout the paper, ‘a priori’ values of the parameters are assumed to be available in the form of prior estimates or realistic guessed values based on the experimenters knowledge and past experience. Some real problem examples are given to illustrate the use of this procedure. This technique may have some application to response surface analysis.