Phase II change-point estimation in Log–Bilal regression profiles

In many process control applications, the quality of a product or service is described by a relationship between a proportion-type response on the interval (0, 1) and one or more explanatory variables, commonly referred to as a regression profile. When the response is a proportion or rate, normality is violated by bounded support and skewness, making Gaussian-based monitoring inadequate. Phase II change-point estimation for such regression profiles has received limited attention. This paper develops a Phase II monitoring procedure for Log–Bilal regression profiles subject to a single step change in the regression coefficients at an unknown profile index. The change time τ is estimated by maximizing a before/after Log– Bilal log-likelihood, and the resulting estimator is embedded in a Hotelling’s T􀀑 chart with an empirically calibrated upper control limit to attain the target in-control ARL. A Monte Carlo study with 10,000 replications per scenario evaluates the performance of the procedure under individual and simultaneous coefficient shifts. The results show timely detection and accurate localization of the change point, with decreasing dispersion of 􀀒̂ as shift magnitude increases. A likelihood-ratio confidence set for τ is also constructed, and a real-data example based on residual-chlorine profiles demonstrates the practical applicability of the method.