On Copula Based Serial Dependence in Statistical Process Control

Abstract

Copula is a distribution function on the unit hypercube with uniform margins. The margin is directly related to the stochastic behaviour of one variable, while joint distribution function covers the holistic character of more. In multivariate (and particularly bivariate) analysis, using copulas is an elegant way to solve the missing information problem between joint distribution function and the total of the margins. Hereby, the intention of this paper is twofold. Firstly, the paper intends to emphasize the advantages of copulas in practice. In order to encourage potential researchers to diversify their subject of work with these functions, authors give the essential introductory details for a clear understanding of copulas associated with their basic mathematical and statistical preliminaries. Secondly, the study exemplifies the practical usage of copulas in statistical process control area. In this context, process parameters are estimated in order to calculate the control limits of a typical Shewhart type control chart. Parameter estimation is performed by Maximum Likelihood Estimation (MLE) for the bivariate Clayton copula in univariate AR (1) time series with several different levels of high dependence. Since monitoring autocorrelated data in control charts is known as being one of the main causes of producing tighter control limits than required, false alarm rate may be increased and accordingly, the performance of control charts may be dramatically decreased. This study shows that copulas may alternatively be used for getting the same or little wider acceptable region between upper and lower limits. This recognition of the properness of copulas may help to decrease some of the negative effects of dependent data being monitored on charts for further studies.