On the one parameter unit-Lindley distribution and its associated regression model for proportion data

TL;DR

This paper introduces the unit-Lindley distribution derived from the Lindley distribution, explores its properties, and develops a regression model for proportion data that outperforms Beta regression in practical applications.

Contribution

The paper proposes the new unit-Lindley distribution, derives its properties, and develops a regression model for proportion data with bias correction and covariate incorporation.

Findings

The distribution belongs to the exponential family.

The model provides analytical bias correction for estimators.

It outperforms Beta regression in real data fitting.

Abstract

In this paper considering the transformation $X=\frac{Y}{1+Y}$, where $Y \sim\text{Lindley}(\theta)$, we propose the unit-Lindley distribution and investigate some of its mathematical properties. A important fact associated with this new distribution is that is possible to obtain the analytical expression for bias correction of the maximum likelihood estimator. Moreover, it belongs to the exponential family. This distribution allows us to incorporate covariates directly in the mean and consequently to quantify the influence on the average of the response variable. Finally, a practical application is present and it is shown that our model fits much better than the Beta regression.