A Bimodal Weibull Distribution: Properties and Inference

TL;DR

This paper introduces a new bimodal Weibull distribution with added bimodality parameter, analyzes its properties, and demonstrates its effectiveness in modeling data using maximum likelihood and heuristic optimization methods.

Contribution

It proposes a novel bimodal Weibull distribution derived via quadratic transformation, expanding modeling flexibility and providing new tools for data analysis.

Findings

The distribution exhibits desirable bimodal properties.

Parameter estimation is effectively performed using maximum log-q likelihood.

Real data applications confirm its modeling capability.

Abstract

Modeling is a challenging topic and using parametric models is an important stage to reach flexible function for modeling. Weibull distribution has two parameters which are shape $\alpha$ and scale $\beta$. In this study, bimodality parameter is added and so bimodal Weibull distribution is proposed by using a quadratic transformation technique used to generate bimodal functions produced due to using the quadratic expression. The analytical simplicity of Weibull and quadratic form give an advantage to derive a bimodal Weibull via constructing normalizing constant. The characteristics and properties of the proposed distribution are examined to show its usability in modeling. After examination as first stage in modeling issue, it is appropriate to use bimodal Weibull for modeling data sets. Two estimation methods which are maximum $\log_q$ likelihood and its special form including objective functions $\log_q(f)$ and $\log(f)$ are used to estimate the parameters of shape, scale and bimodality parameters of the function. The second stage in modeling is overcome by using heuristic algorithm for optimization of function according to parameters due to fact that converging to global point of objective function is performed by heuristic algorithm based on the stochastic optimization. Real data sets are provided to show the modeling competence of the proposed distribution.