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

In: Advanced Statistical Methods in Process Monitoring, Finance, and Environmental Science

Author

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  • Manuel Cabral Morais

    (Instituto Superior Técnico, Universidade de Lisboa, Department of Mathematics & CEMAT (Center for Computational and Stochastic Mathematics))

Abstract

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 $$M/n$$ —is a discrete-time Markov chain governed by a totally positive of order 2 ( TP 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.

Suggested Citation

  • Manuel Cabral Morais, 2024. "Stochastic Ordering in the Performance Analysis of Control Charts for Binomial AR(1) Processes," Springer Books, in: Sven Knoth & Yarema Okhrin & Philipp Otto (ed.), Advanced Statistical Methods in Process Monitoring, Finance, and Environmental Science, pages 155-172, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-69111-9_7
    DOI: 10.1007/978-3-031-69111-9_7
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