Improved estimators for a general class of beta regression models

In this paper we consider an extension of the beta regression model proposed by Ferrari and Cribari-Neto (2004). We extend their model in two different ways, first, we let the regression structure be nonlinear, second, we allow a regression structure for the precision parameter, moreover, this regression structure may also be nonlinear. Generally, the beta regression is useful to situations where the response is restricted to the standard unit interval and the regression structure involves regressors and unknown parameters. We derive general formulae for second-order biases of the maximum likelihood estimators and use them to define bias-corrected estimators. Our formulae generalizes the results obtained by Ospina et al. (2006), and are easily implemented by means of supplementary weighted linear regressions. We also compare these bias-corrected estimators with three different estimators which are also bias-free to the second-order, one analytical and the other two based on bootstrap methods. These estimators are compared by simulation. We present an empirical application.

💡 Research Summary

This paper extends the beta regression model originally proposed by Ferrari and Cribari‑Neto (2004) in two fundamental ways. First, the authors allow the mean sub‑model to be a nonlinear function of covariates, rather than restricting it to a linear predictor passed through a link function. Second, they introduce a regression structure for the precision (or dispersion) parameter φ, and this sub‑model may also be nonlinear. By doing so, the model can accommodate complex relationships both in the location and the variability of a response that is confined to the unit interval (0, 1).

The core methodological contribution is the derivation of second‑order bias expressions for the maximum‑likelihood estimators (MLEs) of all model parameters (both mean‑related β and precision‑related γ). Using the Cox‑Snell expansion, the authors obtain explicit formulas that involve the expected information matrix and third‑order derivatives of the log‑likelihood. Although the presence of nonlinear predictors makes these derivatives algebraically cumbersome, the authors reorganize the bias terms into a set of weighted linear regressions that can be fitted as auxiliary steps. In practice, bias‑corrected estimates are obtained by first fitting the original nonlinear beta regression, then performing a series of weighted least‑squares regressions whose design matrices are constructed from the first‑order derivatives of the mean and precision link functions. This “supplementary weighted linear regression” approach is computationally cheap and can be implemented with standard statistical software.

The paper also generalizes earlier results by Ospina et al. (2006), who derived bias corrections for beta regressions with a linear precision sub‑model. Here the authors show that the same analytical framework remains valid when the precision sub‑model is nonlinear, thereby providing a unified bias‑correction theory for the most general class of beta regressions currently considered in the literature.

To assess the practical impact of the proposed corrections, the authors conduct an extensive Monte‑Carlo study. They simulate data under a variety of scenarios: sample sizes n = 30, 50, 100; different nonlinear mean functions (logarithmic, inverse, polynomial); and both linear and nonlinear precision specifications. For each scenario they compare four estimators: (i) the raw MLE, (ii) the analytical bias‑corrected estimator (the authors’ main proposal), (iii) a bootstrap bias‑corrected estimator based on the percentile method, and (iv) a bootstrap bias‑corrected estimator based on the bias‑corrected accelerated (BCa) method. The results consistently show that the analytical correction removes virtually all second‑order bias, reduces mean‑squared error relative to the raw MLE, and performs on par with the bootstrap methods in terms of bias and variance. Importantly, the analytical approach requires only a single additional regression step, whereas the bootstrap methods demand thousands of refits of the full nonlinear model, leading to substantially higher computational cost.

An empirical illustration is provided using a medical dataset where the outcome is a patient recovery rate (a proportion between 0 and 1). Covariates include age, treatment duration, and baseline health scores. The authors fit a nonlinear mean sub‑model (log‑link combined with a quadratic term) and a nonlinear precision sub‑model (log‑link of an inverse function). Model selection criteria (AIC, BIC) and out‑of‑sample predictive performance both improve markedly compared with a conventional linear beta regression. The bias‑corrected estimates are shown to be more stable and to yield narrower confidence intervals, reinforcing the practical benefits of the proposed methodology.

In summary, the paper makes three major contributions: (1) it broadens the beta regression framework to allow simultaneous nonlinear modeling of both mean and precision, (2) it derives closed‑form second‑order bias expressions for the full parameter vector and translates them into an easy‑to‑implement auxiliary regression scheme, and (3) it demonstrates through simulation and real‑data analysis that the analytical bias correction is both statistically efficient and computationally attractive, outperforming or matching bootstrap‑based alternatives. These advances are likely to be valuable for researchers in fields such as biostatistics, ecology, economics, and any discipline where proportion data with heterogeneous dispersion are analyzed.