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On New Generalized Logistic Distributions and Applications Barreto FHS, Mota JMA and Rathie PN*

Abstract

In 2006, Rathie and Swamee had proposed a generalization of the logistic distribution which is more flexible and multimodal. This work presents an addition of a new parameter to increase the flexibilization of the distribution as well as an asymmetric distribution using the Azzalini method, adding another parameter of asymmetry. Five data sets (Human Body Fat Index, HIV, Precipitation, pH Concentration, Relative Humidity) are analysed by applying the new distributions. The estimation of the parameters of the new distributions and mixture of the normals was accomplished by the automaximum likelihood method. Due to complex mathematical resources required to calculate the estimates of the new distributions, we use interactive numerical methods such as L-BFGS-B, BFGS, SANN etc. using an adaptive barrier algorithm added to enforce the constraint and an adapted function that searches for global maximum of a very complex non-linear objective function to initial values of the algorithm of estimation. All computational work was implemented in software R. In most cases, we use the Hartigan’s test to reject unimodality. Using the KolmogorovSmirnov test at significance level of 5% and applying various criteria, such as Mean Square Error, Mean Absolute Deviation and Maximum Deviation, to indicate the best fit. The classical and general method for multimodal adjustment is a mixture of distributions, in particular, the mixture of the normal distributions because the normal distribution presents good mathematical properties. In the case of mixture of the normals, we use EM algorithm to calculate the estimates. We also use Akaike Information Criterion and Bayesian Information Criterion as selection criteria to highlight the best distribution, in both cases, comparing them with the mixture of normal distributions to illustrate the applicability of the results derived in this paper.

Barreto FHS, Mota JMA and Rathie PN

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