Half-Logistic Distribution

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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 + ebt).

PDF of the Half-Logistic Distribution

PDF of the Half-Logistic Distribution
Half-logistic PDF.

The probability density function for the half-logistic distribution is [3]
pdf half-logistic distribution 1

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
generalized pdf


[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=
[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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