Preprint Article Version 1 NOT YET PEER-REVIEWED

# An Explicit Expression of Average Run Length of Exponentially Weighted Moving Average Control Chart with ARIMA (p,d,q)(P, D, Q)L Models

Version 1 : Received: 14 August 2016 / Approved: 15 August 2016 / Online: 15 August 2016 (10:43:14 CEST)

How to cite: Areepong, Y.; Sukparungsee, S. An Explicit Expression of Average Run Length of Exponentially Weighted Moving Average Control Chart with ARIMA (p,d,q)(P, D, Q)L Models. Preprints 2016, 2016080146 (doi: 10.20944/preprints201608.0146.v1). Areepong, Y.; Sukparungsee, S. An Explicit Expression of Average Run Length of Exponentially Weighted Moving Average Control Chart with ARIMA (p,d,q)(P, D, Q)L Models. Preprints 2016, 2016080146 (doi: 10.20944/preprints201608.0146.v1).

## Abstract

In this paper we propose the explicit formulas of Average Run Length (ARL) of Exponentially Weighted Moving Average (EWMA) control chart for Autoregressive Integrated Moving Average: ARIMA (p,d,q) (P, D, Q)L process with exponential white noise. To check the accuracy, the ARL results were compared with numerical integral equations based on the Gauss-Legendre rule. There was an excellent agreement between the explicit formulas and the numerical solutions. Additionally, we compared the computational time between our explicit formulas for the ARL with the one obtained via Gauss-Legendre numerical scheme. The computational time for the explicit formulas was approximately one second that is much less than the numerical approximations. The explicit analytical formulas for evaluating ARL0 and ARL1 can produce a set of optimal parameters which depend on the smoothing parameter (λ) and the width of control limit (H), for designing an EWMA chart with a minimum ARL1.

## Subject Areas

exponentially weighted moving average control chart (EWMA); autoregressive integrated moving average (ARIMA); average run length (ARL)