EWMA Schemes with Nonhomogeneous Transition Kernels

Abstract

In the last decades, various methods are applied for evaluating different performance measures of change-point detection schemes. In only some cases—recent examples are Chandrasekaran et al. (Chandrasekaran, S.; English, J.R.; Disney, R.L. Modeling and analysis of EWMA control schemes with variance-adjusted control limits. IIE Transactions 1995, 277, 282–290) and Steiner (Steiner, S.H. EWMA control charts with time-varying control limits and fast initial response. Journal of Quality Technology 1999, 31 (1), 75–86)—EWMA schemes with varying control limits are considered. However, EWMA charts with just these limits are sometimes more appropriate than those with fixed limits. Here, a computational approach is presented which allows to compute the usual performance measures with high precision. The main idea is connected to earlier results of Madsen and Conn (Madsen, R.W.; Conn, P.S. Ergodic behavior for nonnegative kernels. Ann. Probab. 1973, 1, 995–1013), Woodall (Woodall, W.H. The distribution of the run length of one-sided CUSUM procedures for continuous random variables. Technometrics 1983, 25, 295–301), and Waldmann (Waldmann, K.-H. Bounds for the distribution of the run length of geometric moving average charts. J. R. Stat. Soc., Ser. C, Appl. Stat. 1986a, 35, 151–158). Additionally, quantities as the steady-state ARL and the steady-state distribution of the chart statistic can be computed very precisely.

Keywords:

• Average run length
• Steady-state
• Conditional delay
• Run length distribution
• Positive integral kernels
• Varying control limits