< List of probability distributions

The **Lomax distribution** (also called the *Pareto II distribution*) was first introduced by Lomax in 1987 [1] for the analysis of business failure lifetime data. This heavy-tailed distribution is often used in survival analysis, actuarial science, economics, and business. In this article, we will provide a brief overview of the Lomax distribution, including its definition, variants, and applications.

## What is the Lomax distribution?

The Lomax distribution is a shifted variant of the Pareto distribution. It is commonly used to model the lifetime of an object or the duration of an event. The distribution has a positive lower bound of zero, with probability density function (PDF)

where x, λ and κ are greater than zero and

- λ > 0 is the scale parameter
- κ > 0 is the shape parameter.

This distribution exhibits a heavy right tail, which implies that it has a higher probability of generating rare events. This heavy right tail is an essential feature that makes the Lomax useful for survival analysis and extreme value analysis.

The distribution’s cumulative distribution function (CDF) is defined as:

**F (x | λ, κ) = 1 – (λ / (x + λ)) ^{κ}**

The CDF explicitly shows the dependence of the distribution on the scale and shape parameters, which allows for the calculation of quantiles and probabilities. The parameter λ shifts the distribution horizontally while κ shapes the tail’s heaviness.

## Applications of the Lomax distribution

One of the most common applications of the Lomax distribution is in survival analysis, where it is used to model the lifetime of an object or event. This distribution is also commonly applied in actuarial science, where it is often used in risk modeling and insurance. Additionally, the Lomax distribution has found applications in economics and business, where it is used in forecasting and analyzing extreme events such as financial crashes.

## Variants

Many variants of the Lomax distribution exist, including Oguntunde et al’s generalization with increasing, decreasing, and constant failure rate [3] and the beta exponentiated Lomax — a five-parameter continuous probability distribution proposed by Mead in 2016 [4]. Other variants include

- Exponential Lomax [5]
- Exponentiated Lomax [6]
- Gamma Lomax [7]
- McDonald Lomax [8]
- Poisson Lomax [9]
- Power Lomax [10]
- Transmuted Lomax [11]
- Weibull Lomax [12]
- Weighted Lomax [13].

In conclusion, the Lomax distribution is a crucial probability distribution with many applications in survival analysis, actuarial science, economics, and business. It is a heavy-tailed distribution that exhibits a higher probability of generating rare events, making it useful in extreme value analysis. This distribution is defined by two parameters: λ, the scale parameter, and κ, the shape parameter. The Lomax distribution’s PDF and CDF formulas explicitly show the dependence of the distribution on these parameters, making it easy to calculate quantiles and probabilities. Understanding the Lomax distribution’s properties and applications is essential in many fields, and hopefully, this blog has provided a useful introduction.

## References

[1] K. Lomax. Business failures: another example of the analysis of failure data. J Am Stat Assoc. 1987;49:847–852

[2] Fuzzyrandom, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

[3] E. Oguntunde et al. “A New Generalization of the Lomax Distribution with Increasing, Decreasing, and Constant Failure Rate.” Modelling and Simulation in Engineering Volume 2017.

[4] M.E. Mead. On Five-Parameter Lomax Distribution:Properties and Applications. Pakistan Journal of Statistics and Operation Research. Vol 12. No.1. 2016.

[5] El-Bassiouny A. H., Abdo N. F., Shahen H. S. Exponential lomax distribution. *International Journal of Computer Applications*, 2015; 121(13).

[6] Abdul-Moniem, Ibrahim & Abdel-Hameed, H. (2012). On exponentiated Lomax distribution. International Journal of Mathematical Education. 33. 1-7.

[7] Gauss M. Cordeiro, Edwin M.M. Ortega & Božidar V. Popović (2015) The gamma-Lomax distribution, Journal of Statistical Computation and Simulation, 85:2, 305-319, DOI: 10.1080/00949655.2013.822869

[8] Artur J. Lemonte & Gauss M. Cordeiro (2013) An extended Lomax distribution, Statistics, 47:4, 800-816, DOI: 10.1080/02331888.2011.568119

[9] Bander Al-Zahrania, Hanaa Sagorb. The Poisson-Lomax Distribution. Revista Colombiana de Estadística Junio 2014, volumen 37, no. 1, pp. 225 a 245

[10] Rady, EH.A., Hassanein, W.A. & Elhaddad, T.A. The power Lomax distribution with an application to bladder cancer data. *SpringerPlus* **5**, 1838 (2016). https://doi.org/10.1186/s40064-016-3464-y

[11] Ashour, Samir & Eltehiwy, Mahmoud. (2013). Transmuted Lomax Distribution. American Journal of Applied Mathematics and Statistics. 1. 121-127. 10.12691/ajams-1-6-3.

[12] Tahir, M. & Cordeiro, Gauss & Mansoor, M. & Zubair,. (2014). The Weibull-Lomax distribution: Properties and Applications. Hacettepe University Bulletin of Natural Sciences and Engineering Series B: Mathematics and Statistics. 10.15672/HJMS.2014147465.

[13] Kilany, Neveen. (2016). Weighted Lomax distribution. SpringerPlus. 5. 10.1186/s40064-016-3489-2.