Monitoring mean of INAR(1) process with discrete mixture exponential innovations

This paper presents a discrete counterpart of the mixture exponential distribution, namely discrete mixture exponential distribution, by utilizing the survival discretization method. The moment generating function and associated moment measures are discussed. The distribution’s hazard rate function can assume increasing or decreasing forms, making it adaptable for diverse fields requiring count data modeling. The paper delves into two parameter estimation methods and evaluates their performance through a Monte Carlo simulation study. The applicability of this distribution extends to time series analysis, particularly within the framework of the first-order integer-valued autoregressive process. Consequently, an INAR(1) process with discrete mixture exponential innovations is proposed, outlining its fundamental properties, and the performance of conditional maximum likelihood and conditional least squares estimation methods is evaluated through a simulation study. Real data analysis showcases the proposed model’s superior performance compared to alternative models. Additionally, the paper explores quality control applications, addressing serial dependence challenges in count data encountered in production and market management. As a result, the INAR(1)DME process is employed to explore control charts for monitoring autocorrelated count data. The performance of two distinct control charts, the cumulative sum chart and the exponentially weighted moving average chart, are evaluated for their effectiveness in detecting shifts in the process mean across various designs. A bivariate Markov chain approach is used to estimate the average run length and their deviations for these charts, providing valuable insights for practical implementation. The nature of design parameters to improve the robustness of process monitoring under the considered charts is examined through a simulation study. The practical superiority of the proposed charts is demonstrated through effective modeling with real data, surpassing competing models.