On the design of nonparametric runs‐rules schemes using the Markov chain approach

The aim of this paper is to highlight some concerns about the partly inaccurate manner in which the currently available 2‐of‐2 and 2‐of‐3 simple and improved runs‐rules charts based on the sign and the signed‐rank statistics were designed using Markov chain matrix. Because of the memory‐less property of the Markov chains, the empirical average run‐length (ARL) values were not affected; but the design structure of the matrix makes it difficult to formulate the general expressions of the ARL explicitly. Also, the dimension (and consequently, the simplicity) of the transition probability matrices (TPMs) and the false alarm rates expressions were affected negatively. Thus, we present some zero‐ and steady‐state formulae that make it easier to construct some key elements of the run‐length distribution (including the TPMs) of the simple and improved 2‐of‐(H + 1) runs‐rules charts based on the sign and the signed‐rank statistics, for any positive integer value H > 0, not just H = 1 and 2, as currently available for these monitoring schemes, so that any interested reader may use these formulae to obtain the corresponding empirical study.