# Variable dispersion beta regressions with parametric link functions

## TL;DR

This paper introduces a new class of beta regression models with parametric link functions for continuous data in (0,1), allowing flexible modeling of mean and dispersion with covariates and detailed inference procedures.

## Contribution

It proposes a novel regression framework with parametric link functions for beta-distributed data, including estimation, inference, diagnostics, and practical application.

## Key findings

- Link functions include symmetric and asymmetric Aranda-Ordaz types.
- Maximum likelihood estimation jointly estimates regression and link parameters.
- Simulation study shows good finite sample performance.

## Abstract

This paper presents a new class of regression models for continuous data restricted to the interval $(0,1)$, such as rates and proportions. The proposed class of models assumes a beta distribution for the variable of interest with regression structures for the mean and dispersion parameters. These structures consider covariates, unknown regression parameters, and parametric link functions. Link functions depend on parameters that model the relationship between the random component and the linear predictors. The symmetric and assymetric Aranda-Ordaz link functions are considered in details. Depending on the parameter values, these link functions refer to particular cases of fixed links such as logit and complementary log-log functions. Joint estimation of the regression and link function parameters is performed by maximum likelihood. Closed-form expressions for the score function and Fisher's information matrix are presented. Aspects of large sample inferences are discussed, and some diagnostic measures are proposed. A Monte Carlo simulation study is used to evaluate the finite sample performance of point estimators. Finally, a practical application that employs real data is presented and discussed.