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 substantially in the presence of right-skewed distributions. In such settings, mean-based monitoring statistics are highly sensitive to tail behavior, which may result in delayed detection of small shifts or increased false alarm rates. This paper introduces a novel 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 enhancing sensitivity to persistent distributional shifts. Theoretical properties of the proposed monitoring statistic are investigated, and its run-length behavior is analyzed under non-normal distributions. The performance of the GR chart is evaluated through extensive Monte Carlo simulations and is compared with classical EWMA and CUSUM procedures under Gamma models. The results indicate that the proposed method provides a robust and stable alternative for monitoring skewed processes while maintaining competitive sensitivity to small location shifts. Overall, the GR control chart offers a distribution-aware and theoretically grounded framework for sequential monitoring in asymmetric stochastic environments.