Simultaneous monitoring of origin and scale in left-bounded processes via depth

A novel notion of depth for the origin and scale of a distribution with left-bounded support, like those modelling lifetime data, is presented. This origin-scale depth, evaluated on parameter estimates, serves as plotting statistic of a Shewhart-type chart for rational samples of a fixed size for which three variations are considered: a nonparametric version, built without distributional assumptions, and parametric and semiparametric versions, built for shifted exponential processes (paradigmatic model governed by origin and scale parameters). An EWMA scheme based on the rank induced by the origin-scale depth is also presented and, together with the Shewhart-type chart, compared with other existing monitoring proposals. The new proposals are competitive when the reference distribution is a shifted exponential one, while outperform them for other reference distributions.