Monitoring of simple linear profiles and change point estimation in the presence of within-profile ARMA autocorrelation

In statistical process control applications, the quality of certain processes or products can be accurately described by either a univariate or multivariate distribution. Nonetheless, in certain instances, the quality of a process or product can be defined by a profile, which represents the relationship between independent and response variables. Numerous studies have examined the monitoring of simple linear profiles that incorporate uncorrelated observations. Nevertheless, in practice, this assumption is seldom met as a result of spatial autocorrelation or time collapse, which can result in unsatisfactory outcomes. In numerous studies, the autocorrelation structure between observations is modeled as a first-order autoregressive ( ) model. However, a wide range of autocorrelation between observations might not be modeled by  models. Therefore, this paper examines a simple linear profile and assumes an autoregressive moving average  autocorrelation structure between each observation, which is more flexible than  models. It is assumed that in each profile, random errors follow an  model. This article mainly focuses on the Phase II monitoring of simple linear profiles, with a particular emphasis on the estimation of change points, which can lead to substantial reductions in time and cost. This paper aims to estimate the change point for each simple linear profile that possesses an autocorrelation structure of . To achieve this, a maximum likelihood estimator is developed. Simulation experiments are conducted to compare Hotelling's  control chart with the proposed control chart. Additionally, the proposed change point estimator is compared to one of the built-in estimators for exponentially weighted moving average ( ) control charts. The results demonstrate that the proposed estimator has accurately estimated the change point regardless of the shift size and the  coefficients, and it outperforms the built-in control chart estimator in terms of accuracy.