Stochastic Ordering in the Performance Analysis of Control Charts for Binomial AR(1) Processes

The first-order integer-valued autoregressive (AR(1)) binomial process proposed by Al-Osh and Alzaid (Commun Statist Stochas Models 7:261–282, 1991) can be used to model, for example, autocorrelated counts of nonconforming items in random samples of fixed size n in a quality control setting. In this paper, we make use of stochastic ordering to prove that the binomial AR(1) process—with mean np and autocorrelation parameter \(M/n\) —is a discrete-time Markov chain governed by a totally positive of order 2 \((\mbox{TP}_2)\) transition probability matrix. We also resort to stochastic ordering to compare transition probability matrices referring to pairs of independent binomial AR(1) processes with different values of the parameter p (respectively, M). We assess the impact of these results, namely, on the stochastic properties of the run length of modified charts for monitoring binomial AR(1) counts.