A combined SR-CUSUM procedure for detecting common changes in panel data

Abstract

A procedure based on the sum of N Shiryayev–Roberts processes is proposed to detect common changes in panel data and shown to perform better for small portions of changed panels. The change-point for each panel is estimated by using the CUSUM process calculated in parallel. The changed panels are isolated by using the scores formed by the post-change parameter estimations and the common change point is then estimated from the isolated changed panels. A real example is used for illustration. An adaptive detection procedures is also proposed when the unknown post-change parameters are estimated adaptively in each panel.     

Cumulative sum schemes for surgical performance monitoring

Summary.  The standard cumulative sum (CUSUM), risk-adjusted CUSUM and Shiryayev–Roberts schemes for monitoring surgical performance are compared. We find that both CUSUM schemes are comparable in run length performance except when there is a high heterogeneity of surgical risks, in which case the risk-adjusted CUSUM scheme is more sensitive in detecting a shift in surgical performance. The Shiryayev–Roberts scheme is found to be less sensitive compared with the CUSUM schemes in detecting a deterioration in surgical performance. Using the Markov chain method, the exact average run length of a standard CUSUM scheme can be computed whereas the average run length of a risk-adjusted CUSUM scheme is approximated. For a risk-adjusted CUSUM scheme, the accuracy of the average run length depends on the fineness of the discretization of CUSUM values, which relies on the chart limit, shift to be detected optimally and in-control surgical risk distribution. A sensitivity analysis shows that the risk-adjusted CUSUM and Shiryayev–Roberts schemes still perform moderately well in detecting a deterioration and an improvement in surgical performances respectively even though there is a misspecification of the in-control surgical risk distribution. In general, the run length performance of the Shiryayev–Roberts scheme is comparatively less sensitive to a misspecification of the in-control surgical risk distribution.     

An extension of a change-point problem

We consider a specific classification problem in the context of change-point detection. We present generalized classical maximum likelihood tests for homogeneity of the observed sample in a simple form which avoids the complex direct estimation of unknown parameters. This paper proposes a martingale approach to transformation of test statistics. For sequential and retrospective testing problems, we propose the adapted Shiryayev–Roberts statistics in order to obtain simple tests with asymptotic power one. An important application of the developed methods is in the analysis of exposure's measurements subject to limits of detection in occupational medicine.     

Asymptotics of the run lengths of two control charts

The CUSUM control chart proposed by Page is a widely used in monitoring the quality of manufacturing processes. The Shiryayev-Roberts (S-R) control chart due to Shiryayev (1963) and Roberts (1988) is one of its competitors, This paper is concerned with the distribution properties of the run lengths of these two control charts. In context of continuous time, we first give the expansions of the higher moments of these run lengths. Then, we show that the asymptotic distributions of these run lengths are either some exponential distributions, or the distribution of the suprema of a standard Brownian motion, or some normal distributions, according to whether the μ<δ/2,μ =δ/2 and μ>δ/2. Here δ is the reference value of the above charts. Some similar results are also obtained in the context of discrete time.     

Guaranteed testing for epidemic changes of a linear regression model

The objective of this paper is to propose and examine a class of generalized maximum likelihood asymptotic power one tests for detection of various types of changes in a linear regression model. The proposed retrospective tests are based on martingales structures Shiryayev–Roberts statistics. This approach is widely known in a sequential analysis of change point problems as an optimal method of detecting a change in distribution. Guaranteed non-asymptotic upper bounds for the significance levels of the considered tests are presented.Simulated data sets are used to demonstrate that the proposed tests can give good results in practice.     

A mixed GWMA–CUSUM control chart for monitoring the process mean

The memory-type control charts are widely used in the process and service industries for monitoring the production processes. The reason is their sensitivity to quickly react against the small process disturbances. Recently, a new cumulative sum (CUSUM) chart has been proposed that uses the exponentially weighted moving average (EWMA) statistic, called the EWMA–CUSUM chart. Similarly, in order to further enhance the sensitivity of the EWMA–CUSUM chart, we propose a new CUSUM chart using the generally weighted moving average (GWMA) statistic, called the GWMA–CUSUM chart, for efficiently monitoring the process mean. The GWMA–CUSUM chart encompasses the existing CUSUM and EWMA–CUSUM charts. Extensive Monte Carlo simulations are used to explore the run length profiles of the GWMA–CUSUM chart. Based on comprehensive run length comparisons, it turns out that the GWMA–CUSUM chart performs substantially better than the CUSUM, EWMA, GWMA, and EWMA–CUSUM charts when detecting small shifts in the process mean. An illustrative example is also presented to explain the implementation and working of the EWMA–CUSUM and GWMA–CUSUM charts.     

Monitoring parameter shift with Poisson integer-valued GARCH models

This study examines the statistical process control chart used to detect a parameter shift with Poisson integer-valued GARCH (INGARCH) models and zero-inflated Poisson INGARCH models. INGARCH models have a conditional mean structure similar to GARCH models and are well known to be appropriate to analyzing count data that feature overdispersion. Special attention is paid in this study to conditional and general likelihood ratio-based (CLR and GLR) CUSUM charts and the score function-based CUSUM (SFCUSUM) chart. The performance of each of the proposed methods is evaluated through a simulation study, by calculating their average run length. Our findings show that the proposed methods perform adequately, and that the CLR chart outperforms the GLR chart when there is an increased shift of parameters. Moreover, the use of the SFCUSUM chart in particular is found to lead to a lower false alarm rate than the use of the CLR chart.     

A a comparison of the markov chain and the integral equation approaches for evaluating the run length distribution of quality control charts

Two methods that are often used to evaluate the run length distribution of quality control charts are the Markov chain and integral equation approaches. Both methods have been used to evaluate the cumulative sum (CUSUM) charts and the exponentially weighted moving average (EWMA) control charts. The Markov chain approach involves "discretiz-ing" the possible values which can be plotted. Using properties of finite Markov chains, expressions for the distribution of the run length, and for the average run length (ARL), can be obtained. For the CUSUM and EWMA charts there exist integral equations whose solution gives the ARL. Approximate methods can then be used to solve the integral equation. In this article we show that if the product midpoint rule is used to approximate the integral in the integral equation, then both approaches yield the same approximations for the ARL. In addition we show that the recursive expressions for the probability functions are the same for the two approaches. These results establish the integral equation approach as preferable whenever an integral equation can be found     

Weighted adaptive multivariate CUSUM charts with variable sampling intervals

The adaptive multivariate CUSUM (AMCUSUM) chart has received considerable attention because of its superior sensitivity against a range of mean shift sizes than that of the conventional non-adaptive multivariate CUSUM (MCUSUM) chart. Recently, weighted AMCUSUM (WAMCUSUM) charts with a fixed sampling interval (FSI) have been proposed, called the WAMCUSUM-FSI charts, which provide more sensitivity than the AMCUSUM-FSI charts. In this paper, we extend this work and propose WAMCUSUM charts with variable sampling interval (VSI), named the WAMCUSUM-VSI charts, for efficiently monitoring the mean of a multivariate normally distributed process. The Monte Carlo simulation method is used to compute the average time to signal (ATS) and the adjusted ATS (AATS) profiles of the existing and proposed charts. It is found that the WAMCUSUM-VSI charts perform substantially and nearly uniformly better than the WAMCUSUM-FSI charts in terms of the ATS and AATS performance criterion. An example is given to explain the implementation of the WAMCUSUM charts with fixed and VSIs.     

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.     

Average Most Powerful Tests for a Segmented Regression

To improve the goodness of fit between a regression model and observations, the model can be complicated; however, that can reduce the statistical power when the complication does not lead significantly to an improved model. In the context of two-phase (segmented) logistic regressions, the model evaluation needs to include testing for simple (one-phase) versus two-phase logistic regression models. In this article, we propose and examine a class of likelihood ratio type tests for detecting a change in logistic regression parameters that splits the model into two-phases. We show that the proposed tests, based on Shiryayev–Roberts type statistics, are on average the most powerful. The article argues in favor of a new approach for fixing Type I errors of tests when the parameters of null hypotheses are unknown. Although the suggested approach is partly based on Bayes–Factor-type testing procedures, the classical significance levels of the proposed tests are under control. We demonstrate applications of the average most powerful tests to an epidemiologic study entitled “Time to pregnancy and multiple births.”     

A Combination of CUSUM Charts for Monitoring a Zero-Inflated Poisson Process

The Zero-inflated Poisson distribution (ZIP) is used to model the defects in processes with a large number of zeros. We propose a control charting procedure using a combination of two cumulative sum (CUSUM) charts to detect increases in the parameters of ZIP process, one is a conforming run length (CRL) CUSUM chart and another is a zero truncated Poisson (ZTP) CUSUM chart. The control limits of the control charts are obtained using both Markov chain-based methods and simulations. Simulation experiments show that the proposed method outperforms an existing method. Finally, a real example is presented.     

Control charts based on dependent count data with deflation or inflation of zeros

The process of serially dependent counts with deflation or inflation of zeros is commonly observed in many applications. This paper investigates the monitoring of such a process, the first-order zero-modified geometric integer-valued autoregressive process (ZMGINAR(1)). In particular, two control charts, the upper-sided and lower-sided CUSUM charts, are developed to detect the shifts in the mean process of the ZMGINAR(1). Both the average run length performance and the standard deviation of the run length performance of these two charts are investigated by using Markov chain approaches. Also, an extensive simulation is conducted to assess the effectiveness or performance of the charts, and the presented methods are applied to two sets of real data arising from a study on the drug use.     

Computation of the run-length percentiles of CUSUM control charts under changes in variances

In this paper, the run-length distributions of cumulative sum (CUSUM) charts for monitoring mean changes under normal distributions have been investigated thoroughly. However, there are few studies devoted to the analysis of the run-length distributions of CUSUM charts under changes in process variances. Motivated by this, this paper develops a fast and accurate algorithm based on piecewise collocation method for computing the run-length distributions of CUSUM scale charts. It is shown that the proposed method can provide more accurate approximation to the run-length distribution than the conventional Gauss-type quadrature-based methods applied to the CUSUM location charts. Some computational aspects for facilitating computation load are discussed, including the alternative formulation based on matrix decomposition and the geometric approximation to the distribution of large run lengths.     

A binomial CUSUM chart for detecting large shifts in fraction nonconforming

This article studies a unique feature of the binomial CUSUM chart in which the difference (d _t ?d ₀) is replaced by (d _t ?d ₀)² in the formulation of the cumulative sum C _t (where d _t and d ₀ are the actual and in-control numbers of nonconforming units, respectively, in a sample). Performance studies are reported and the results reveal that this new feature is able to increase the detection effectiveness when fraction nonconforming p becomes three to four times as large as the in-control value p ₀. The design of the new binomial CUSUM chart is presented along with the calculation of the in-control and out-of-control Average Run Lengths (ARL₀ and ARL₁).     

CUSUM control schemes for Gaussian processes

A CUSUM control scheme for detecting a change point in a Gaussian process is derived. An upper and a lower bound for the distribution of the run length and for its moments is given. In a Monte Carlo study the average run length (ARL) of this chart is compared with the ARL of two other CUSUM procedures which are based on approximations to the sequential probability ratio, and, moreover, with EWMA schemes for autocorrelated data. Results on the optimal choice of the reference value are presented. Furthermore it is investigated how these charts behave if the model parameters are estimated.     

A method for analyzing supersaturated designs inspired by control charts

The identification of active effects in supersaturated designs (SSDs) constitutes a problem of considerable interest to both scientists and engineers. The complicated structure of the design matrix renders the analysis of such designs a complicated issue. Although several methods have been proposed so far, a solution to the problem beyond one or two active factors seems to be inadequate. This article presents a heuristic approach for analyzing SSDs using the cumulative sum control chart (CUSUM) under a sure independence screening approach. Simulations are used to investigate the performance of the method comparing the proposed method with other well-known methods from the literature. The results establish the powerfulness of the proposed methodology.     

A Markov-chain method for computing the run-length distribution of the self-starting cumulative sum scheme

We propose an analytic method for computing the run-length distribution of the cumulative sum (CUSUM) of Q statistics. The method is based on a model in which the operation of this CUSUM is embedded in a nonstationary, discrete-time Markov chain. The calculations of the method agree closely with those of Monte Carlo simulation, supporting the method's accuracy. Our results facilitate understanding the effectiveness of the CUSUM of Q statistics in detecting process mean shifts.     

Evaluation of CUSUM Charts for Finite-Horizon Processes

This article analyses and evaluates the properties of a CUSUM chart designed for monitoring the process mean in short production runs. Several statistical measures of performance that are appropriate when the process operates for a finite-time horizon are proposed. The methodology developed in this article can be used to evaluate the performance of the CUSUM scheme for any given set of chart parameters from both an economic and a statistical point of view, and thus, allows comparisons with various other charts.     

Effect of Inspection Error on Out-of-Control Average Run Length of CUSUM Charts

The cumulative sum (CUSUM) chart is commonly used for detecting small or moderate shifts in the fraction of defective manufactured items. However, its construction relies on the error-free inspection assumption, which can seldom be met in practice. In this article, we discuss the construction of an upward CUSUM chart in the presence of inspection error, study the effects of inspection error on the out-of-control ARL of the CUSUM chart, and present a formula for determining the sampling size that compensates for the effect of inspection error on the out-of-control ARL.