The assumption that the output can be modelled by a sequence of independent random variables is standard in statistical process control. However, if the random variables are not independent but correlated the performance of the control schemes may be affected significantly. A typical effect of correlation is for instance a decrease in the in-control average run length (ARL) of a standard scheme — yielding to a higher false alarm rate, as reported by several numerical studies in the SPC literature. Moreover, W. Schmid and collaborators proved in a series of papers that, under mild conditions, the presence of correlation leads to a decrease of the survival function of the in-control R.L (and therefore of the in-control ARL) of several modified schemes for correlated output, if we falsely assume that the underlying process is a sequence of independent random variables.

Bearing in mind that the use of the ARL to measure the ability to detect a process shift gives an incomplete picture of how a control scheme performs, establishing stochastic ordering results in the line of work pioneered by W. Schmid — as opposed to numerical results, organized in tables and graphs — provides a qualitative basis for a more objective assessment of the impact of correlation in the performance of quality control schemes.

In this paper we establish stochastic ordering results concerning the (in-control and out-of-control) RL of residual schemes for the mean of stationary autoregressive processes of order 1 or 2.