Improving estimation for beta regression models via EM-algorithm and related diagnostic tools

ABSTRACT

In this paper we propose an alternative procedure for estimating the parameters of the beta regression model. This alternative estimation procedure is based on the EM-algorithm. For this, we took advantage of the stochastic representation of the beta random variable through ratio of independent gamma random variables. We present a complete approach based on the EM-algorithm. More specifically, this approach includes point and interval estimations and diagnostic tools for detecting outlying observations. As it will be illustrated in this paper, the EM-algorithm approach provides a better estimation of the precision parameter when compared to the direct maximum likelihood (ML) approach. We present the results of Monte Carlo simulations to compare EM-algorithm and direct ML. Finally, two empirical examples illustrate the full EM-algorithm approach for the beta regression model. This paper contains a Supplementary Material.

KEYWORDS:

• Bounded data
• diagnostic tools
• latent variables
• inference
• maximum likelihood estimators

Acknowledgments

We thank the referee for his/her comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. If $\mathbf{A}$ is an $n\times p\times q$ array and $\mathbf{B}$ is an $m\times n$ matrix, then $\mathbf{C}\mathbf{=}\mathbf{\left[}\mathbf{A}\mathbf{\right]}\mathbf{\left[}\mathbf{B}\mathbf{\right]}$ is called the bracket product of $\mathbf{A}$ and $\mathbf{B}$, that is an $m\times p\times q$ array with elements $Y_{tij}=\sum _{k=1}^n{B}_{tk}{A}_{kij}$.

Additional information

Funding

We also would like to thank the financial support from FAPEMIG (Brazil) and CNPq (Brazil).