The Log-Bimodal Asymmetric Generalized Gaussian Model with Application to Positive Data

Abstract

One of the most widely known probability distributions used to explain the probabilistic behavior of positive data is the log-normal (LN). Although the LN distribution is capable of adjusting data types, it is not always fully true that the model manages to adequately model the behavior of the response of interest since in some cases, the degree of skewness and/or kurtosis of the data are greater or less than those that the LN distribution can capture. Another peculiarity of the LN distribution is that it only fits unimodal positive data, which constitutes a limitation when dealing with data that present more than one mode (bimodality). On the other hand, the log-normal model only fits unimodal positive data and in reality there are multiple applications where the behavior of materials is bimodal. To fill this gap, this paper introduces a new probability distribution that is capable of fitting unimodal or bimodal positive data with a high or low degree of skewness and/or kurtosis. The new distribution is a generalization of the LN distribution. For the new proposal, its main properties are studied and the process of estimation of the parameters involved in the model is carried out from a classical perspective using the maximum likelihood method. An important feature of this distribution is the non-singularity of the Fisher information matrix, which guarantees the use of asymptotic theory to study the properties of the parameter estimators. A Monte Carlo type simulation study is carried out to evaluate the properties of the estimators and finally, an illustration is presented with a set of data related to the concentration of nickel in soil samples, allowing to show that the proposed distribution fits extremely well in certain situations.

Keywords

• bimodality
• generalized Gaussian model
• maximum likelihood estimation
• positive data