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

The paper extends the beta regression model originally proposed by Ferrari and Cribari‑Neto (2004) in two important directions. First, the authors allow the mean sub‑model to be nonlinear, i.e., the predictor η₁ = g₁(μ) can be any smooth function f₁(x;β) rather than a linear combination of covariates. Second, they introduce a regression structure for the precision (or dispersion) parameter φ, again possibly nonlinear, η₂ = g₂(φ)=f₂(z;θ). This dual extension enables the modelling of heteroscedasticity and complex mean‑precision relationships that are common in proportion data confined to the unit interval.

Using the re‑parameterisation μ = p/(p+q) and φ = p+q, the likelihood for a sample y₁,…,yₙ∼Beta(μᵢ,φᵢ) is written in closed form. The score vector and Fisher information matrix are derived, showing that, unlike ordinary generalized linear models, the blocks corresponding to β and θ are not orthogonal (the cross‑information matrix W_{βθ}≠0). Consequently, the maximum‑likelihood estimators (MLEs) for β and θ are biased to order O(n⁻¹).

To correct this bias, the authors apply the general Cox‑Snell (1968) formula for the second‑order bias of multiparameter MLEs. After extensive algebraic manipulation, they obtain compact matrix expressions for the bias of β̂ and θ̂. These expressions involve the first three derivatives of the log‑likelihood, the digamma and trigamma functions, and the second‑order derivatives of the nonlinear predictors. Crucially, the bias can be computed by fitting auxiliary weighted linear regressions, so the correction is computationally inexpensive once the MLEs are available.

Three bias‑reduction strategies are compared:

1. Analytical correction – subtract the derived O(n⁻¹) bias from the MLEs (β̂_bc = β̂ − B(β̂), θ̂_bc = θ̂ − B(θ̂)).
2. Firth’s penalised likelihood – maximise ℓ* = ℓ + ½ log|K(ζ)|, which implicitly removes the first‑order bias.
3. Bootstrap correction – estimate bias non‑parametrically by resampling (standard bootstrap and percentile bootstrap).

Monte‑Carlo simulations are conducted for sample sizes n = 30, 50, 100, various link functions (logit, probit, complementary log‑log) and several nonlinear mean/precision specifications. Results show that the analytical correction consistently yields the smallest bias and mean‑squared error, especially when the precision parameter exhibits large variability. Firth’s method also reduces bias but sometimes inflates standard errors; bootstrap correction is computationally intensive but robust across model specifications.

An empirical illustration uses medical proportion data (e.g., patient recovery rates). The extended nonlinear‑precision beta regression attains lower AIC/BIC than the standard linear beta regression, and the bias‑corrected estimates provide tighter, more accurate confidence intervals, improving substantive interpretation.

In summary, the paper makes three substantive contributions: (i) a flexible beta regression framework that simultaneously accommodates nonlinear mean and precision sub‑models; (ii) explicit second‑order bias formulas for the MLEs, expressed in a form that can be implemented via auxiliary weighted regressions; and (iii) a thorough comparative study of analytical, penalised‑likelihood, and bootstrap bias‑reduction techniques. These results are especially valuable for researchers working with bounded outcomes in small to moderate samples, where conventional MLEs can be appreciably biased. The methodology broadens the applicability of beta regression in fields such as biostatistics, economics, and environmental science.