A Unimodal/Bimodal Skew/Symmetric Distribution Generated from Lambert’s Transformation

Abstract

The generalized bimodal distribution is especially efficient in modeling univariate data exhibiting symmetry and bimodality. However, its performance is poor when the data show important levels of skewness. This article introduces a new unimodal/bimodal distribution capable of modeling different skewness levels. The proposal arises from the recently introduced Lambert transformation when considering a generalized bimodal baseline distribution. The bimodal-normal and generalized bimodal distributions can be derived as special cases of the new distribution. The main structural properties are derived and the parameter estimation is carried out under the maximum likelihood method. The behavior of the estimators is assessed through simulation experiments. Finally, two applications are presented in order to illustrate the utility of the proposed distribution in data modeling in different real settings.

Rotation Periods, Inclinations, and Obliquities of Cool Stars Hosting Directly Imaged Substellar Companions: Spin–Orbit Misalignments Are Common

“…The LGB distribution has four parameters: the location parameter μ (defined for all real numbers), the scale parameter σ (defined for positive numbers), and the shape parameters γ (defined over [0, 2)) and α (defined over (0, e), where e is Euler's constant). Depending on these parameters, the LGB distribution can be unimodal, bimodal, symmetric, or asymmetric; examples of the diversity of shapes can be seen in Figure 1 of Iriarte et al (2021). γ controls the bimodality of the distribution, and α determines the relative heights of the bimodal peaks or unimodal asymmetry.…”

An Asymmetric Bimodal Double Regression Model

“…Other practical examples of bimodality in data can be seen in [4][5][6]. In the literature, there are many proposals discussing bimodal distributions; e.g., the works of [7][8][9][10][11][12]. Bimodal data can be fitted by a mixture of two unimodal distributions.…”