The Gompertz-Rayleigh distribution is an extension of the Rayleigh distribution that allows for better modeling of highly-skewed datasets compared to compound distributions .
A similarly named but unrelated distribution is a generalized Gompertz-Rayleigh distribution, proposed by Bradley  as a potential survival distribution for modeling risk.
Many other generalized forms of the Rayleigh distribution have been proposed including Kundu & Raqab’s ‘s generalized Rayleigh , Merovci & Eltabal’s Weibull-Rayleigh , and Ahmad et al.’s transmuted Rayleigh distribution , but Mohammed et. al proposed adding the Gompertz-G family to have more flexibility and greater skewness.
Gompertz-Rayleigh Distribution PDF / CDF
The Gompertz-Rayleigh distribution (GomRD) has the probability density function (PDF):
and cumulative distribution function (CDF):
Where x > 0; θ, α, β > 0.
 Mohammed, F. et al. (2020). A study of some properties and goodness-of-fit of a Gompertz-Rayleigh model. Asian journal of probability and statistics, Page 18-31.
 Kundu, D. & Raqab, M. Generalized Rayleigh Distribution: Different methods of estimations. Comp. Stat. & Data Anal. 2005; 49:187-200
 Merovci, F & Eltabal, I. Weibull-Rayleigh distribution: Theory and applications. Appl. Math & Inf. Sci. 2015; 9(5): 1-11.
 Ahmad et al. Transmuted inverse Rayleigh distribution: A generalization of the inverse Rayleigh distribution. Math Theo. & Mod. 2015; 4(7):90-98.