On some properties of the bimodal normal distribution and its bivariate version

TL;DR

This paper explores the mathematical properties, estimation methods, and bivariate extension of the bimodal normal distribution, including proofs, asymptotic analysis, and simulation studies.

Contribution

It provides new theoretical insights into the bimodal normal distribution and introduces a bivariate version with analysis of its properties and estimation techniques.

Findings

Proof of bimodality and identifiability established

Maximum likelihood estimates analyzed and evaluated

Bivariate distribution properties and estimation performance studied

Abstract

In this work, we derive some novel properties of the bimodal normal distribution. Some of its mathematical properties are examined. We provide a formal proof for the bimodality and assess identifiability. We then discuss the maximum likelihood estimates as well as the existence of these estimates, and also some asymptotic properties of the estimator of the parameter that controls the bimodality. A bivariate version of the BN distribution is derived and some characteristics such as covariance and correlation are analyzed. We study stationarity and ergodicity and a triangular array central limit theorem. Finally, a Monte Carlo study is carried out for evaluating the performance of the maximum likelihood estimates.