On Hitting Times for Markov Time Series of Counts with Applications to Quality Control

Abstract

Examples of time series of counts arise in several areas, for instance in epidemiology, industry, insurance and network analysis. Several time series models for these counts have been proposed and some are based on the binomial thinning operation, namely the integer-valued autoregressive (INAR) model, which mimics the structure and the autocorrelation function of the autoregressive (AR) model. The detection of shifts in the mean of an INAR process is a recent research subject and it can be done by using quality control charts. Underlying the performance analysis of these charts, there is an indisputable popular measure: the run length (RL), the number of samples until a signal is triggered by the chart. Since a signal is given as soon as the control statistic falls outside the control limits, the RL is nothing but a hitting time. In this paper, we use stochastic ordering to assess: – the ageing properties of the RL of charts for the process mean of Poisson INAR(1) output; – the impact of shifts in model parameters on this RL. We also explore the implications of all these properties, thus casting interesting light on this hitting time for a Markov time series of counts.