The Asymmetric Bimodal Normal Distribution: A Tractable Mixture Model for Skewed and Bimodal Data

We study a parsimonious constrained two-component Gaussian mixture with symmetric locations ±λ and unequal weights controlled by α∈[−1,1]; we refer to this family as the asymmetric bimodal normal. The constraint eliminates label switching and yields an identifiable parametrization for λ>0, while noting the boundary degeneracy at λ=0 where α is not identifiable. We derive closed-form analytical expressions for the density and distribution functions, an equivalent constructive representation (useful for simulation and interpretation), explicit moment formulas, and conditions distinguishing unimodality from bimodality. For inference, we develop maximum likelihood estimation with observed information standard errors and provide numerically stable fits via a block-coordinate quasi-Newton routine using method of moments initial values. A Monte Carlo simulation study across representative parameter settings evaluates bias and root mean squared error, and examines the behavior of Hessian-based standard error estimates, highlighting regimes where the observed information becomes ill-conditioned under weak separation. Empirical analyses, chemical calibration deviations from the National Institute of Standards and Technology and a regression example with asymmetric errors, show competitive or superior fit and interpretability relative to skewed normal alternatives, asymmetric Laplace models, and unconstrained Gaussian mixtures, with consistent advantages under model comparison using the Akaike information criterion and the Bayesian information criterion.