An ARL Derivation Based on Explicit Formulas to Detect Shifts in the Mean of Seasonal Time-Series Models Running on a CUSUM Control Chart

Wilasinee  Peerajit

Authors

Wilasinee Peerajit Department of Applied Statistics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand

Keywords:

Long-memory SARFIMA process, exponential white noise, numerical integral equation (NIE) method

Abstract

Herein, we propose a derivation of the average run length (ARL) to detect shifts in the mean of seasonal time-series models running on a CUSUM control chart based on explicit formulas by solving the Fredholm integral equation of the second kind. The existence and uniqueness of the solution were guaranteed by applying Banach’s fixed-point theorem. We applied the explicit formulas for the ARL to detect changes in the mean of a long-memory SARFIMA (p, d, q) x (P, D, Q)_s process with exponential white noise running on a one-sided CUSUM control chart. When comparing the performance of the proposed method with the ARL obtained via the standard numerical integral equation (NIE) method, the former performed better for small-to-moderate shifts in the process mean under the same conditions and circumstances. Moreover, its percentage accuracy (a benchmark for the ARL) was greater than 95%, thereby indicating excellent agreement between the two methods. In addition, the computer processing time for the proposed method was considerably faster than that required for the NIE method. Hence, this explicit formulas approach provides a novel means of determining the ARL for detecting changes in the mean of a long-memory SARFIMA (p, d, q) x (P, D, Q)_s process with exponential white noise running on a one-sided CUSUM control chart. Finally, the proposed and NIE methods were applied to applications with real-life data to determine their efficacy.

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