Generalized logistic distribution and its regression model
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(2020) 7:7
RESEARCH
Open Access
Generalized logistic distribution and its regression model Mohammad A. Aljarrah1*, Felix Famoye2 and Carl Lee2 * Correspondence: aljarrah@ymail. com 1 Department of Mathematics, Tafila Technical University, Tafila 66110, Jordan Full list of author information is available at the end of the article
Abstract A new generalized asymmetric logistic distribution is defined. In some cases, existing three parameter distributions provide poor fit to heavy tailed data sets. The proposed new distribution consists of only three parameters and is shown to fit a much wider range of heavy left and right tailed data when compared with various existing distributions. The new generalized distribution has logistic, maximum and minimum Gumbel distributions as sub-models. Some properties of the new distribution including mode, skewness, kurtosis, hazard function, and moments are studied. We propose the method of maximum likelihood to estimate the parameters and assess the finite sample size performance of the method. A generalized logistic regression model, based on the new distribution, is presented. Logistic-log-logistic regression, Weibull-extreme value regression and log-Fréchet regression are special cases of the generalized logistic regression model. The model is applied to fit failure time of a new insulation technique and the survival of a heart transplant study. Keywords: Beta-family, Symmetric distribution, Hazard function, Moments, Censored data 2010 Mathematics subject classification: 62E15, 62F10, 62 J12, 62P10
Introduction The use of logistic distribution in various disciplines can be found in (Johnson et al. 1995) and the references therein. The logistic distribution has the cumulative distribution function (CDF) defined as x − μ − 1 ; − ∞ < x; μ < ∞; σ > 0: F ðxÞ ¼ 1 þ exp − σ
ð1Þ
Note that the logistic distribution is the limiting distribution of the average of largest and smallest values of random samples of size n from a symmetric distribution of exponential type (Gumbel 1958). The CDF of the standard logistic distribution is F(y) = (1 + e−y)−1, − ∞ < y < ∞. The standard logistic density function with kurtosis 4.2 is symmetric about zero, and is more peaked and has heavier tails than the normal density function. These properties make logistic distribution a popular choice for fitting symmetric non-normal data. © The Author(s). 2020 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statuto
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