Abstract—The objective of this paper will propose four types of copulas on the CUSUM control chart when observations are exponential distribution. We use the Monte Carlo simulation to investigate the value of Average Run Length (ARL). The dependence of random variables are used and measured by Kendall’s tau in each copula. The numerical results show that negative dependence Normal copula is better than the others. For positive dependence, in the case of two parameter shifts, Normal copula is better than others and Gumbel copula is better than others in the case of one parameters shift. Index Terms—Copula, Average run length, CUSUM chart, Monte Carlo simulation I. INTRODUCTION TATISTICAL Process Control (SPC) is a method for monitoring, controlling and improving quality of production in many areas of applications. These areas are in industry, finance and economics, health care, environment sciences and other fields. Control charts are statistical and visual tools designed to detect shifts in a process and they are designed and evaluated under the assumption that the observations are from processes are independent and identically distributed (i.i.d.). A Univariate control chart is devised to monitor the quality of a single process characteristic [1]. However, modern process often monitor more than one quality characteristic and they are referred to as multivariate statistical process control charts. Multivariate statistical process control (MSPC) is one of the most rapidly developing sections of statistical process control [2] and lead to an interest in the simultaneous inspection of several related quality characteristics [3, 4]. There are multivariate extensions for all kinds of univariate control charts, such as multivariate Shewhart control chart, multivariate exponentially weighted moving average control chart (MEWMA) and multivariate cumulative sum control Manuscript received January 10, 2015; revised February 3, 2015. This work was supported in part by the Graduate Colleges, King Mongkut’s University of Technology North Bangkok. S. Kuvattana is with the Department of Applied Statistics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand (e-mail: sasigarn2010@gmail.com). S. Sukparungsee is with the Department of Applied Statistics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand (e-mail: swns@kmutnb.ac.th). P. Busababodhin is with the Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham, 41150, Thailand (e-mail: piyapatr99@gmail.com). Y. Areepong is with the Department of Applied Statistics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand (e-mail:yupaporna@kmutnb.ac.th). chart (MCUSUM) [5]. Multivariate Shewhart control chart is used to detect large shifts in the mean vectors. The MEWMA and MCUSUM are commonly used to detect small or moderate shifts in the mean vectors [6]. Most of multivariate detection procedures are based on a multi-normality assumption and independence but many processes are often non-normality and correlated. Many multivariate control charts are the lack of the related joint distribution and copula can specify this property. Copulas introduced by Sklar [7], are useful devices which give a representation of a multivariate distribution function in terms of its univariate marginal distribution [8]. The copula approach has become a popular tool for modeling nonlinearity, asymmetricality and tail dependence in several fields [9] and it can be used in the study of dependence or association between random variables. Copulas modeling can estimate joint distribution of nonlinear outcomes and explain the dependence structure among variables through the joint distribution by eliminating the effect of univariate marginals. A bivariate copula is the simplest case for the description of dependent random variables and it can apply to control chart. Recently, several papers use copula in control chart such as, copula based on bivariate ZIP control chart [10, 11], copula Markov CUSUM chart [12], Shewhart control charts for autocorrelated and normal data [13], new control chart based on a nonparametric Kendall’s tau statistics [14], non- normal multivariate cases for the Hotelling 2 T control chart [15] and bivariate copula on the Shewhart control chart [16]. This paper presents the work on the CUSUM control chart when observations are exponential distribution with the means shifts and use a bivariate copula function for specifying dependence between random variables. II. THE MULTIVARIATE CUMULATIVE SUM CONTROL CHART The multivariate cumulative sum (MCUSUM) control chart is the multivariate extension of the univariate cumulative sum (CUSUM) chart. The MCUSUM chart was initially proposed by Crosier [17]. The MCUSUM chart may be expressed as follows: 1/2 1 , 1, 2, [( , ) 3 ] t t t C t         -1 t-1 t S X a (S +X -a) (1) where covariance    and t S are the cumulative sums expressed as: 1 , if ( )1 , if t t t t t t C k k C k C                   0 S S X a (2) Efficiency of Bivariate Copula on the CUSUM Chart Sasigarn Kuvattana, Saowanit Sukparungsee, Piyapatr Busababodhin, and Yupaporn Areepong S Proceedings of the International MultiConference of Engineers and Computer Scientists 2015 Vol II, IMECS 2015, March 18 - 20, 2015, Hong Kong ISBN: 978-988-19253-9-8 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) IMECS 2015