Likelihood-Based Inference for the Asymmetric Exponentiated Bimodal Normal Model

Abstract

Asymmetric probability distributions have been widely studied by various authors in recent decades, who have introduced new families of flexible distributions in terms of skewness and kurtosis than the classical distributions known in statistical theory. Most of the new distributions fit unimodal data, others fit bimodal data, however, in the bimodal, singularity problems have been found in their information matrices in most of the proposals presented. In contrast, in this paper an extension of the family of alpha-power distributions was developed, which has a non-singular information matrix, based on the bimodal-normal and bimodal elliptic-skew-normal probability distributions. These new extensions model asymmetric bimodal data commonly found in various areas of scientific interest. The properties of these new probabilistic distributions were also studied in detail and the respective statistical inference process was carried out to estimate the parameters of these new models. The stochastic convergence for the vector of maximum likelihood estimators could be found due to the non-singularity of the expected information matrix in the corresponding support.