Bayesian Current Record Analysis for Mixed Rayleigh–Weibull Distributions with Application to Saudi Humidity Data

Abstract

This paper develops a comprehensive Bayesian framework for analyzing current records arising from a two-component mixture of Rayleigh and Weibull distributions. Current records capture the most recent extreme observations in sequential data and are particularly valuable in industrial reliability monitoring, where only record-breaking events are retained. We derive closed-form expressions for the probability density and cumulative distribution functions of both upper and lower current records under the mixed Rayleigh–Weibull model. A key theoretical contribution is the establishment of identifiability conditions for this mixture,ensuring unique parameter estimation when the Weibull shape parameter is not equal to 2. Parameter estimation is addressed via the Expectation-Maximization (EM) algorithm for maximum likelihood and via Markov Chain Monte Carlo (MCMC) methods for Bayesian inference. Furthermore, we develop both non-Bayesian and Bayesian prediction intervals for future current records and record ranges, with the Bayesian approach fully accounting for parameter uncertainty. Extensive simulation studies across three representative configurations demonstrate that the Bayesian estimators exhibit lower bias and root-mean-squared error, and provide coverage closer to the nominal level than maximum likelihood, especially for small sample sizes. The methodology is applied to daily humidity data from the western region of Saudi Arabia (92 days in 2025), where the mixture model provides an excellent fit (Kolmogorov–Smirnov p-value 0.8765). Results confirm that the unified framework offers superior flexibility in capturing complex data behaviors compared to single-distribution approaches, making it a valuable tool for industrial and environmental monitoring under the Kingdom’s Vision 2030.