A general class of trimodal distributions: properties and inference

Abstract

The modality is an important topic for modelling. Using parametric models is an efficient way when real data set shows trimodality. In this paper, we propose a new class of trimodal probability distributions, that is, probability distributions that have up to three modes. Trimodality itself is achieved by applying a proper transformation to density function of certain continuous probability distributions. At first, we obtain preliminary results for an arbitrary density function $g(x)$ and, next, we focus on the Gaussian case, studying trimodal Gaussian model more deeply. The Gaussian distribution is applied to produce the trimodal form of Gaussian known as normal distribution. The tractability of analytical expression of normal distribution and properties of the trimodal normal distribution are important reasons why we choose normal distribution. Furthermore, the existing distributions should be improved to be capable of modelling efficiently when there exists a trimodal form in a data set. After new density function is proposed, estimating its parameters is important. Since Mathematica 12.0 software has optimization tools and important modelling techniques, computational steps are performed using this software. The bootstrapped form of real data sets are applied to show the modelling ability of the proposed distribution when real data sets show trimodality.

Keywords:

• Class of distributions
• unimodality
• bimodality
• trimodality
• inference

Acknowledgments

We acknowledge the anonymous referees for their helpful comments, suggestions and references provided in their reports. R. V. thanks A. V. Medino, J. Roldan and E. M. M. Ortega for partial discussions of Theorem 3.4 and for general paper questions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES)—Finance Code 001.