Simultaneous monitoring of origin and scale of a shifted exponential process with unknown and estimated parameters

The distribution of consumer lifetimes, high-voltage of current in semiconductor transistors, and the risk associated with monitoring health care often come with a threshold. A two-parameter (or shifted) exponential distribution is, in general, regarded as a better statistical model in such situations compared with a traditional (one-parameter) exponential model. Research on inferential problems associated with two-parameter exponential distributions, including monitoring schemes for the parameters of this model, is active. Currently, all existing monitoring schemes for origin and scale parameters of a shifted exponential distribution assume that the process parameters are known (Case-K). The actual values of the process parameters are, however, rarely known in practice. The traditional method of estimating parameters from a set of a (Phase-I) reference sample and plug them in for Phase-II monitoring affects the performance of a monitoring scheme. Skewed processes, like the two-parameter exponential process, exacerbate this problem. The present article shows that even a reference sample of size 50,000 cannot guarantee nominal in-control performances of monitoring schemes when the actual process parameters are unknown (Case-U). To address this problem, we develop monitoring schemes based on max and distance statistics for simultaneously monitoring the two parameters of a shifted exponential process in Case-U. We show that the proposed schemes perform well. We illustrate the practical application of the proposed procedures by analyzing data about the production of an electronic component.