Marcos Orellana-Panchame

This article establishes a mechanistic foundation for the Rayleigh distribution in event-time analysis by deriving it from a decay process governed by a linearly increasing hazard under assumptions of independence and homogeneity. More generally, when the hazard is sufficiently smooth, vanishes at the origin, and has a positive initial slope, the Rayleigh distribution emerges as the leading-order approximation at early times. A Monte Carlo sensitivity analysis comprising 114,000 realizations across sample sizes from $n=50$ to $\mathrm{10,000}$ quantifies the effects of departures from hazard linearity and sampling variability. Increasing observed $R_{\lambda }^2$ is associated with decreasing Kolmogorov–Smirnov distances and greater consistency between independent estimates of the Rayleigh parameter, although its interpretation depends on sample size and does not permit establishing a universal threshold. A data-driven methodology is then proposed to operationally identify approximately linear hazard regimes and model the corresponding event times using a truncated Rayleigh distribution. Application to tuberculosis survival in guinea pigs, COVID-19 survival in healthy young men, and primate interspike intervals identified regimes up to $T^{*}=255$ days, 26 days, and $18.5$ ms, with $R_{\lambda }^2=0.95$, $0.94$, and $0.73$, respectively. Within these regimes, the model showed strong quantile agreement ($R_{\mathrm{QQ}}^2=0.97$, $0.99$, and $0.98$) and relative discrepancies between parameter estimates of $0.08$, $0.08$, and $0.19$. These findings support a mechanistically grounded, data-driven framework for identifying and modeling early-time Rayleigh behavior in event-time data.