A Median-Centered Sequential Monitoring Scheme Based on Golden Ratio Weighting for Skewed Distributions

Detecting small location shifts in stochastic processes is a fundamental problem in sequential statistical monitoring. Classical procedures such as Shewhart-type schemes, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) methods are known to perform well under normality or near-symmetry assumptions; however, their effectiveness may deteriorate in the presence of right-skewed distributions. In such settings, mean-based monitoring statistics can be highly sensitive to tail behavior, which may lead to delayed detection of small shifts or unstable false alarm performance. This paper introduces a monitoring scheme referred to as the Golden Ratio (GR) control chart, designed for detecting small location shifts in right-skewed distributions. The proposed method is constructed using a median-centered statistic combined with a geometrically decaying weighting mechanism derived from the golden ratio. Unlike classical time-based weighting schemes, the GR chart assigns weights according to the rank-based distance from the sample median, thereby attenuating the influence of isolated extreme observations while preserving sensitivity to persistent distributional changes. The run-length performance of the proposed chart is investigated using Monte Carlo simulation experiments. All competing procedures are calibrated to achieve comparable in-control average run lengths. The GR chart is compared with classical EWMA and CUSUM charts under several skewed distributions, including Gamma, Lognormal, and Weibull models. Simulation results indicate that the proposed approach provides a robust and stable monitoring alternative for skewed processes. In particular, the GR chart demonstrates competitive performance for detecting small location shifts while reducing the influence of extreme observations commonly encountered in right-skewed environments.