Bimodal Regression Model

Guillermo Domingo Martinez,Hugo Salinas,Heleno Bolfarine

doi:10.15446/rce.v40n1.51738

Abstract

Regression analysis is a technique widely used in different areas ofhuman knowledge, with distinct distributions for the error term. Itis the case, however, that regression models with bimodal responsesor, equivalently, with the error term following a bimodal distribution are notcommon in the literature, perhaps due to the lack of simple to dealwith bimodal error distributions. In this paper we propose a simpleto deal with bimodal regression model with a symmetric-asymmetricdistribution for the error term for which for some values of theshape parameter it can be bimodal. This new distribution containsthe normal and skew-normal as special cases. A realdata application reveals that the new model can be extremely usefulin such situations.

Highlights

To study the relationship between variables in different areas of human knowledge, linear and nonlinear regression models have been substantially used
It is typically considered that the error term follows a normal distribution more general symmetric error distributions have been considered
2μr (0), if r is even, 2μr(0) + 2μr(β, α), if r is odd, where μr(λ, α) = 2αcα zrφ(z) {Φ(z)}α−1 Φ(λz)dz. In addition to these results, these authors have shown that the information matrix is nonsingular at the vicinity of symmetry, that is, α = 1 and λ = 0. This leads to large sample normal distribution for the maximum likelihood estimators for which the asymptotic covariance matrix is the inverse of the Fisher information matrix

Summary

Introduction

To study the relationship between variables in different areas of human knowledge, linear and nonlinear regression models have been substantially used. One situation in which we encounter an anomaly in the error term of the model occurs when it is of interest to explain the fat percentage in the human body as a function of the individual weight It is the case, that given inherent gender peculiarities, the exclusion of the gender variable can lead to a bimodal error distribution model. A viable alternative to this situation is to use a mixture of normal distributions for the error term. We suggest using the symmetric-asymmetric bimodal alpha-power model, considered in Bolfarine, Martínez and Salinas (2012), to adjust data with a linear relation. A real application considered in Section 5 illustrates the fact that the model considered can outperform traditional symmetric models that have been previously considered in the literature, in the mixture of normals

The Bimodal Symmetric-Asymmetric Alpha-Power Distribution

The Multiple Regression Model With ABPN Errors

Likelihood and Score Functions

The Observed and Expected Information Matrices

Bimodal Symmetric Case

Large Sample Distribution of the MLE for the ABPN Model

Simulation Study

Numerical Illustration

Final Discussion