Inverse Gaussian cumulative sum control charts for location and shape

Abstract

Cumulative sum (CUSUM) control charts are very effective at detecting persisting special causes. The most common CUSUM chart assumes that the process measurement being monitored follows the normal distribution. Many industrial problems yield measures with skewed, positive distributionsÐexamples are component reliabilities, times to completion of tasks and insurance claims. Non-normal measures such as these should not be monitored using procedures based on the normal distribution. The inverse Gaussian distribution provides a¯exible distribution that can be used to model positive skew quantities, and therefore provides an effective framework for statistical process control on processes producing such measures. This paper de®nes the optimal CUSUM control chart schemes for location and shape of the inverse Gaussian distribution and evaluates its performance in detecting step changes in each of these parameters. The inverse Gaussian distribution has been shown to be a good ®t to a long record of task completion times on a General Motors assembly line. We extend this application, showing how our CUSUMS may be used to detect changes in the distribution of these task completion times.

Instability and change detection in exponential families and generalized linear models, with a study of Atlantic tropical storms

“…Some researchers have treated special cases in the EF CUSUM family, including Hawkins and Olwell (1997) on detecting known location and shape change in inverse gamma distribution, Hawkins and Zamba (2005) on change point detection in unknown mean and variance for normal distribution, Watkins et al (2008) used negative binomial CUSUM to study outbreaks of Ross River virus disease and compared it to Early Aberration Reporting System CUSUM algorithms, Wu et al (2008) studied large shifts in fraction non-conforming, and Lucas (1985) improved the Poisson CUSUM with fast initial response (FIR) and introduced the two-in-a-row rule to robust CUSUM. Healy (1987) discussed shift in mean and covariance for multivariate normal distribution using CUSUM, Alwan (2000) proposed transformation to normality to deal with EF CUSUM chart, Severo and Gama (2010) discussed using Kalman filter and CUSUM to detect residual mean and variance in the regression model, and Qiu and Hawkins (2001) used a rank-based CUSUM procedure to deal with multivariate measurements without normality assumption.…”

Inverse Gaussian Control Charts for Monitoring Process Variability

“…An approximate distribution of the weighted sum of sample variances is used in Aminzadeh (1993) to obtain the control limits of the EWMA chart for detecting shifts in the variability of the inverse Gaussian process. The CUSUM chart for monitoring changes in the variability of the IGð, Þ process when both the parameters and are known is constructed in Hawkins and Olwell, (1997) and Edgeman (1996). The CUSUM chart for monitoring process variability with unknown and does not seem to be available for the IGð, Þ process.…”

“…The lead spread dimension is critical for successful soldering of an IC onto an electronic circuits board. Another strong argument for using the IGð, Þ distribution (see Hawkins and Olwell, 1997) is that the sampling distributions of the maximum likelihood (ML) estimator or uniformly minimum variance unbiased (UMVU) estimators of its parameters , and À1 are known (see Folks and Chhikara, 1978) and easy to work with.…”

GLM‐based control charts for the inverse Gaussian distributed response variable

“…The sampling distributions of IG location and scale parameter were easy to derive. Therefore, control charts for the parameters of IG distribution have been proposed in several articles (cf Edgeman, Edgeman, Edgeman, Hawkins and Olwell, and Sim).…”