< List of probability distributions

The **half logistic distribution** is a one-parameter continuous probability distribution extensively used in reliability analysis. It is created by folding the standard logistic distribution, giving it its alternative name, the *folded logistic distribution*. In this article, we will delve deeper into what a half-logistic distribution is, its properties, and its applications.

## What is the half logistic distribution?

The half logistic distribution is a zero-mean logistic distribution restricted to positive values. The term originated with Balakrishnan [1], who termed the folding version of the standard logistic distribution, to the right of zero, the half logistic.

The standard half logistic distribution has a probability density function (PDF) given by [2]

The cumulative distribution function (CDF) is

## Applications

One of the most significant applications of the half-logistic distribution is in reliability analysis. The distribution is an excellent fit for modeling the failure times of components, and its decreasing density and increasing failure rate make it a popular choice in this field.

The reliability and hazard functions are

Reliability analysis is concerned with estimating the probability of a product or system functioning correctly over a specified period.

Another essential property of the half-logistic distribution is that it has a zero mean. This means that the distribution is symmetric about the origin, making it an ideal fit for modeling situations where the probability of an event occurring is equally likely to the probability of it not occurring. This symmetry is a unique property of the distribution and sets it apart from other distributions that are commonly used in reliability analysis.

Although widely used, the half logistic distribution does have a major disadvantage— it cannot model data with non-monotone failure rates and unimodal density [3]. To address this, many extensions have been proposed including the power half logistic [4], Olapade-half logistic, and the generalized half logistic. The generalized version has PDF

## Relationship to other distributions

The half-logistic distribution is related to several other distributions, including the Weibull distribution and the log-logistic distribution. The distribution’s relationship with these distributions makes it a versatile distribution that is applied in several areas, including finance, economics, and engineering. The distribution’s versatility arises from its ability to model both positively skewed and negatively skewed data.

In conclusion, the half-logistic distribution is a widely used probability distribution that is a popular choice in reliability analysis. Its unique properties, including its symmetric nature and zero mean, make it an appropriate fit for modeling failure times of components. It is a versatile distribution that has several applications across various fields. We hope this blog post has provided you with valuable insights into the half-logistic distribution and its properties.

## References

[1] Balakrishnan, N. (1985). Order statistics from the Half logistic distribution. J. Statist. Comp. Simul., 20, 287-309.

[2] Elsayed, A. Reliability Engineering. 2021. Wiley.

[3] 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

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