Extending normality: A case of unit distribution generated from the moments of the standard normal distribution

TL;DR

This paper introduces the Alpha-Unit distribution derived from the moments of the standard normal distribution, providing new estimation methods, properties, and real-world applications for modeling bounded data.

Contribution

It proposes the Alpha-Unit distribution based on normal moments, along with estimation techniques, properties, and validation through simulations and real data applications.

Findings

The Alpha-Unit distribution effectively models data with heavy tails and ranges greater than 0.4.

MLE and UMVUE estimators are statistically consistent and robust.

The model outperforms several existing unit distributions in real-world datasets.

Abstract

This article presents an important theorem, which shows that from the moments of the standard normal distribution one can generate density functions originating a family of models. Additionally, we discussed that different random variable domains are achieved with transformations. For instance, we adopted the moment of order two, from the proposed theorem, and transformed it, which allowed us to exemplify this class as unit distribution. We named it as Alpha-Unit (AU) distribution, which contains a single positive parameter $\alpha$ ($\text{AU}(\alpha) \in [0,1]$). We presented its properties and showed two estimation methods for the $\alpha$ parameter, the maximum likelihood estimator (MLE) and uniformly minimum-variance unbiased estimator (UMVUE) methods. In order to analyze the statistical consistency of the estimators, a Monte Carlo simulation study was carried out, where the robustness was demonstrated. As real-world application, we adopted two sets of unit data, the first regarding the dynamics of Chilean inflation in the post-military period, and the other regarding the daily maximum relative humidity of the air in the Atacama Desert. In both cases shown, the AU model is competitive, whenever the data present a range greater than 0.4 and extremely heavy asymmetric tail. We compared our model against other commonly used unit models, such as the beta, Kumaraswamy, logit-normal, simplex, unit-half-normal, and unit-Lindley distributions.