Alternative to Detecting Changes in the Mean of an Autoregressive Fractionally Integrated Process with Exponential White Noise Running on the Modified EWMA Control Chart

Round 1

Reviewer 1 Report

The authors propose a modified EWMA control chart for detecting small-to-moderate shifts in the process mean of an autoregressive fractionally integrated (ARFI(p, d)) process with exponential white noise running.  The statistical performances of the traditional EWMA is evaluated with the proposed modified EWMA control chart in terms of the average run length (ARL).  The accuracies of the traditional and proposed method were identical for various out-of-control situations and long-term memory processes; the standard deviations of the run length and median run lengths also yielded the same results.  The authors have published similar works on modified EWMA control charts, as they cite.

Comment 1. The manuscript adds an incremental improvement to the study of modified EWMA control chart in the process mean of an autoregressive process.  The authors indicate that the strengths of the method are the formula is simple to calculate and the computational time is significantly reduced.  They illustrate the importance of computational speed.  However, the improvement from 3-4 seconds in previous approaches for 0.01 seconds needs further discussions for applications, e.g., self-driving cars, robotics, AI, extremely fast production processes, etc.  The monthly incidence of pneumonia per 100,000 people is probably not the best example when the main improvement is computational speed, i.e., the data are collected quite slowly.  The daily Netflix Inc. stock price is probably a better example in the present world of automated training using machine learning algorithms, etc.  Some additional text in the discussion is required and additional text on the limitations of the method needs to be added to the conclusions.

Comment 2: Is the LCL for Figure 2a 2b actually zero?  Or is there no LCL?  Also in Figures 2a and 2b, the horizontal axis is not defined or labeled.  Some additional discussion on the harmonic nature of the out of control points in Figure 2a needs further discussion.

Comment 3: The paper by ‘Supharakonsakun Y, Areepong Y, Sukparungsee S. The performance of a modified EWMA control chart for monitoring autocorrelated PM2.5 and carbon monoxide air pollution data. PeerJ. 2020 Dec 15;8:e10467. doi: 10.7717/peerj.10467. PMID: 33362964; PMCID: PMC7747693.’ needs to be cited and included in the discussion.

Author Response

1.Long-memory processes are found in many fields, such as hydrology or even in economics. An important of this study is that the proposed ARL using the exact formula in this situation which is provided to efficacy of the proposed explicit formulas with processes involving variety of real data.

2. For application, we focused on the one-sided upper modified EWMA control chart for detecting shifts in the mean of a process in order to the observation are from exponential distributed .

3.The authors have cited the paper by ‘Supharakonsakun Y, Areepong Y, Sukparungsee S. In addition, for long-memory processes, we use the advanced concept of fractional integration when modeling, which has derived from the development of Supharakonsakun et al. which we already included in the discussion.

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors in this paper first modify the EWMA scheme for an ARFI(p,d) process with an exponential white noise and then drive the exact formula for computing its performance measure using integral equations.

This paper is very well presented and its novelty is acceptable. However, the authors are advised to improve the paper's English writing, preferably by a native English speaker. Also, there are too many formulas in the main text. Please move as much as you can to the appendix section.

Author Response

1.The details of the calculation are shown in Appendixes A-C. The authors have only shown the important formulas in the main text.

2.The paper’s English writing has been proofread by a native English speaker.

Author Response File: Author Response.pdf