{"ID":3083735,"CreatedAt":"2026-06-05T06:46:15.197025399Z","UpdatedAt":"2026-06-07T06:54:00.442624098Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.06149","arxiv_id":"2606.06149","title":"Effect of the measurement errors on one-sided Synthetic-RZ control charts for monitoring the ratio of two normal variables","abstract":"In numerous industrial production settings, keeping track of the ratio formed by two normally distributed random variables is a task of considerable practical interest. The present work examines how measurement errors influence the behaviour of a pair of one-sided Synthetic control charts designed to monitor such a ratio (referred to here as Synthetic-RZ charts), with the analysis covering both the zero-state and the steady-state average run length ($ARL$). To incorporate measurement error into the operation of these charts, we adopt a linear covariate error model. We describe, step by step, how the parameters of the underlying model evolve as the process moves from an in-control to an out-of-control state, and we deliberately avoid the restrictive premise that the observed shift magnitude is unrelated to the measurement errors. The run length characteristics of the charts are obtained by means of a Markov chain formulation. A series of numerical experiments makes clear that measurement error erodes the detection capability of the charts. A particularly useful outcome of the investigation is that collecting several measurements on each inspected unit does not constitute an efficient remedy for the adverse influence of measurement error on the performance of the Synthetic-RZ charts.","short_abstract":"In numerous industrial production settings, keeping track of the ratio formed by two normally distributed random variables is a task of considerable practical interest. The present work examines how measurement errors influence the behaviour of a pair of one-sided Synthetic control charts designed to monitor such a rat...","url_abs":"https://arxiv.org/abs/2606.06149","url_pdf":"https://arxiv.org/pdf/2606.06149v1","authors":"[\"Kim Duc Tran\",\"Thi Hien Nguyen\",\"Kim Phuc Tran\"]","published":"2026-06-04T13:25:38Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}