A General Class of Trimodal Distributions: Properties and Inference

TL;DR

This paper introduces a new class of trimodal probability distributions achieved through transformations, focusing on the Gaussian case, and demonstrates their effectiveness in modeling real trimodal data sets using computational methods.

Contribution

It proposes a novel class of trimodal distributions based on transformations, with detailed analysis of the trimodal Gaussian model and application to real data.

Findings

The new distributions can effectively model trimodal data.

Analytical properties of the trimodal Gaussian are derived.

Real data sets demonstrate the model's practical utility.

Abstract

The modality is 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 arbitratry 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 by 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.