The **half-logistic distribution** is a one parameter, continuous probability distribution created by folding the standard logistic distribution (this leads to its alternate name, the *folded logistic distribution*) [1]; It is equivalent to a logistic distribution with only positive values and a zero mean.

This widely used classical distribution is popular for modeling failure times of components; It has decreasing density and increasing failure rate [2] with a reliability function of R(t) = 2 / (1 + e^{bt}).

## PDF of the Half-Logistic Distribution

The probability density function for the half-logistic distribution is [3]

for x ≥ 0.

## Extensions of the Half-Logistic Distribution

Although widely used in a variety of fields, the half logistic distribution does have one major disadvantage— it cannot model data with non-monotone failure rates and unimodal density [2]. To address this, many extensions have been proposed including the *power half logistic* [4], *Olapade-half logistic*, and the *generalized half logistic*, each with their own PDFs. For example, the generalized version has PDF

## References

[1]Balakrishnan, N. (1991). Handbook of the Logistic Distribution. Taylor & Francis.

[2] Muhammad, M. & Liu, L. A New Extension of the Generalized Half Logistic Distribution with Applications to Real Data. Entropy (Basel). 2019 Apr; 21(4): 339.

Published online 2019 Mar 28. doi: 10.3390/e21040339

[3] LPPF. Retrieved January 1, 2022 from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.217.7879&rep=rep1&type=pdf

[4] Krishnarani S.D. On a Power Transformation of Half-Logistic Distribution. J. Probab. Stat. 2016 doi: 10.1155/2016/2084236.

[5] Olapade A.K.A. The Type I Generalized Half Logistic Distribution. J. Iran. Stat. Soc. 2014;13:69–82.

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