Monitoring count time series: Robustness to nonlinearity when linear models are utilized

Linear models are typically utilized for time series analysis as these are often simple to implement and interpret, as well as being useful in modeling many practical phenomena. Hence, most of the literature on control charts for monitoring time series also consider linearity of the data generating processes (DGP). In practice, however, a nonlinear DGP can be modeled as if it is linear, either due to overlook or illusion, when it is approximately linear. This study quantifies the effects of nonlinear DGPs, misspecified and modeled as linear, on the performance of Shewhart-type and cumulative sum (CUSUM) control charts for count time series data. Time series models for bounded and unbounded counts with several parametrizations are considered for studying the sensitivity of linear approximations to nonlinear DGPs. The Markov chain (MC) approach and simulations are used to compute the average run length (ARL) performance of the control charts. Robustness of the performance is evaluated with respect to the extent that the DGP violates linearity, with and without errors in parameter estimation. It is shown that the chart designs are, in general, robust to model misspecification when the parameters are specified. However, the CUSUM performance can be affected significantly if both the model is misspecified and the parameters are estimated. Real-world data implementations are provided to illustrate the sensitivity of the control charts' performances to the type of model and to the estimation approach.