Gompertz-Rayleigh distribution

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The Gompertz-Rayleigh distribution is an extension of the Rayleigh distribution that allows for better modeling of highly-skewed (off-center) datasets compared to compound distributions [1].

A similarly named but unrelated distribution is a generalized Gompertz-Rayleigh distribution, proposed by Bradley [2] 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 [3], Merovci & Eltabal’s Weibull-Rayleigh [4], and Ahmad et al.’s transmuted Rayleigh distribution [5], but Mohammed et. al [6] proposed adding the Gompertz-G family of distributions to have more flexibility and greater skewness.

The Gompertz-G family of distributions was introduced by Mohammed et al. in a manner similar to others recently developed. The mathematical properties of the new family are available from a linear mixture of exponentiated-G densities and the creators claim two main advantages in these recent methods:

  • They create distributions with several known particular cases.
  • The added parameters may provide better fits of the generated distributions to data sets and they can have physical interpretations.

Gompertz-Rayleigh Distribution PDF / CDF

The Gompertz-Rayleigh distribution (GomRD) has the probability density function (PDF):

PDF of the Gompertz-Rayleigh distribution

and cumulative distribution function (CDF):

CDF for GomRD

Where x > 0; θ,  α, β > 0.


[1] 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.

[2] Bradley, D. et al. (1984). A generalized Gompertz-Rayleigh model as a survival distribution. Mathematical Biosciences Volume 70, Issue 2, August 1984, Pages 195-202

[3] Kundu, D. & Raqab, M. Generalized Rayleigh Distribution: Different methods of estimations. Comp. Stat. & Data Anal. 2005; 49:187-200

[4] Merovci, F & Eltabal, I. Weibull-Rayleigh distribution: Theory and applications. Appl. Math & Inf. Sci. 2015; 9(5): 1-11.

[5] Ahmad et al. Transmuted inverse Rayleigh distribution: A generalization of the inverse Rayleigh distribution. Math Theo. & Mod. 2015; 4(7):90-98.

[6] Mohamed et. al. Gompertz-G family of distributions. November 2016Journal of Statistical Theory and Practice. 11(1)https://www.researchgate.net/publication/311105965_The_Gompertz-G_family_of_distributions