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.