Stochastic ordering in the performance analysis of control charts for binomial AR(1) processes
Morais, M. C.
Advanced Statistical Methods in Process Monitoring, Finance, and Environmental Science - Essays in Honour of Wolfgang Schmid, (2024), 155-172
https://link.springer.com/chapter/10.1007/978-3-031-69111-9_7
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 (TP2) 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.
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