Two MEWMA Charts for Gumbel's Bivariate Exponential Distribution

Data described by the exponential distribution are commonly encountered in manufacturing processes, reliability analysis, and human-service management. Time between events (TBE) charts have been suggested to monitor exponential data. However, existing studies on TBE charts are limited to univariate cases assuming there is only one process characteristic of interest. In this paper, two multivariate exponential weighted moving average (MEWMA) charts are proposed for the simultaneous monitoring of the mean vector of Gumbel's bivariate exponential (GBE) TBE model: One based on the raw observations and the other based on the transformed data. A numerical example is given to illustrate the implementation of the two MEWMA charts. We compare the average run-length performances of the two proposed charts with the following individual TBE charts pairs: the paired individual t charts, the paired individual exponentially weighted moving-average (EWMA) charts on raw data, and the paired individual EWMA charts on transformed data. The results of the comparative studies show that our MEWMA charts outperform all the other charts. The proposed MEWMA charts can be easily extended to higher dimensions. Some brief discussions concerning monitoring Gumbel's multivariate exponential model with more than two variables are also included.

[1]  George C. Runger,et al.  Comparison of multivariate CUSUM charts , 1990 .

[2]  Szu Hui Ng,et al.  Nonparametric CUSUM and EWMA Control Charts for Detecting Mean Shifts , 2010 .

[3]  E. Gumbel.  Bivariate Exponential Distributions , 1960 .

[4]  Murat Caner Testik,et al.  Impact of model misspecification on the exponential EWMA charts: a robustness study when the time-between-events are not exponential , 2010, Qual. Reliab. Eng. Int..

[5]  Douglas M. Hawkins,et al.  Multivariate Exponentially Weighted Moving Covariance Matrix , 2008, Technometrics.

[6]  P. Hougaard,et al.  Fitting a multivariate failure time distribution , 1989 .

[7]  James M. Lucas,et al.  Counted Data CUSUM's , 1985 .

[8]  D. Hawkins,et al.  A nonparametric multivariate cumulative sum procedure for detecting shifts in all directions , 2003 .

[9]  Peihua Qiu,et al.  Distribution-free multivariate process control based on log-linear modeling , 2008 .

[10]  Do Sun Bai,et al.  A Multivariate T2 Control Chart for Skewed Populations Using Weighted Standard Deviations , 2004 .

[11]  Jye-Chyi Lu,et al.  Inference procedures for bivariate exponential model of Gumbel , 1991 .

[12]  Larry Lee,et al.  Multivariate distributions having Weibull properties , 1979 .

[13]  H. Hotelling.  Multivariate Quality Control-illustrated by the air testing of sample bombsights , 1947 .

[14]  James M. Lucas,et al.  Using Shewhart and CUSUM Charts for Diagnosis with Count Data in a Vendor Certification Study , 2009 .

[15]  Young Soon Chang,et al.  Multivariate CUSUM and EWMA Control Charts for Skewed Populations Using Weighted Standard Deviations , 2007, Commun. Stat. Simul. Comput..

[16]  Zhang Wu,et al.  Design of the sum-of-conforming-run-length control charts , 2001, Eur. J. Oper. Res..

[17]  J. E. Freund.  A Bivariate Extension of the Exponential Distribution , 1961 .

[18]  Connie M. Borror,et al.  Robustness properties of multivariate EWMA control charts , 2003 .

[19]  H. Block.  Multivariate Exponential Distribution , 2006 .

[20]  Connie M. Borror,et al.  Robustness of the time between events CUSUM , 2003 .

[21]  Jye-Chyi Lu,et al.  Inference procedures for a bivariate exponential model of Gumbel based on life test of component and system , 1991 .

[22]  Douglas M. Hawkins,et al.  A General Multivariate Exponentially Weighted Moving-Average Control Chart , 2007 .

[23]  Fugee Tsung,et al.  Directional MEWMA Schemes for Multistage Process Monitoring and Diagnosis , 2008 .

[24]  Rudolf G. Kittlitz.  TRANSFORMING THE EXPONENTIAL FOR SPC APPLICATIONS , 1999 .

[25]  Thong Ngee Goh,et al.  Cumulative quantity control charts for monitoring production processes , 2000 .

[26]  Zachary G. Stoumbos,et al.  Robustness to Non-Normality of the Multivariate EWMA Control Chart , 2002 .

[27]  Marion R. Reynolds,et al.  Combinations of Multivariate Shewhart and MEWMA Control Charts for Monitoring the Mean Vector and Covariance Matrix , 2008 .

[28]  Thong Ngee Goh,et al.  A control chart for the Gamma distribution as a model of time between events , 2007 .

[29]  Douglas M. Hawkins,et al.  A Nonparametric Change-Point Control Chart , 2010 .

[30]  Fah Fatt Gan,et al.  Designs of One- and Two-Sided Exponential EWMA Charts , 1998 .

[31]  P. Hougaard.  A class of multivanate failure time distributions , 1986 .

[32]  R. Crosier.  Multivariate generalizations of cumulative sum quality-control schemes , 1988 .

[33]  Thong Ngee Goh,et al.  A study of EWMA chart with transformed exponential data , 2007 .

[34]  Fugee Tsung,et al.  Likelihood Ratio-Based Distribution-Free EWMA Control Charts , 2010 .

[35]  Victoria S. Jordan,et al.  Distribution-Free Phase I Control Charts for Subgroup Location , 2009 .

[36]  Charles W. Champ,et al.  A multivariate exponentially weighted moving average control chart , 1992 .

[37]  T. Calvin,et al.  Quality Control Techniques for "Zero Defects" , 1983 .

[38]  Philip Hougaard,et al.  Modelling multivariate survival , 1987 .

[39]  Surajit Pal,et al.  An application of Gumbel's bivariate exponential distribution in estimation of warranty cost of motor cycles , 2003 .

[40]  Stephen B. Vardeman,et al.  Average Run Lengths for CUSUM Schemes When Observations Are Exponentially Distributed , 1985 .