The Integral Equation Approach for Solving the Average Run Length of EWMA Procedure for Autocorrelated Process


  • Suvimol Phanyaem Department of Applied Statistics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand


Autoregressive integrated moving average, exponentially weighted moving average, Fredholm integral equation, explicit formula


The main goal of statistical process control (SPC) is to improve the capacity of the process. One of quality tools is the control chart and it is usually designed under the assumption that the observations are independent and identically distributed. However, some characteristics of the production process especially processes that are continuously produced such as chemical processes, the process is autocorrelated in various time series models. In this paper, we will focus on an autoregressive integrated moving average, ARIMA(p,d,q) model. The performance of control chart is evaluated in terms of the average run length. This paper aims to solve explicit formulas and develop numerical integration for the average run length of the exponentially weighted moving average (EWMA) control chart. The accuracy of the proposed formulas is established by comparing them to the numerical integration method. A comparison of the results from explicit formula and numerical integration shows that the absolute percentage difference is less than 0.1% In terms of computational time, the explicit formula can reduce the computational time better than the numerical integration.


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