Robust inference in inflated beta regression

Technical Summary: Robust Inference in Inflated Beta Regression

Problem Statement
The inflated beta regression model is a standard tool for analyzing continuous proportions that include boundary values (zeros or ones), such as infection rates or budget allocations. While widely adopted, the standard inference relies on Maximum Likelihood Estimation (MLE). As established in the literature for both binary regression (the discrete component) and beta regression (the continuous component), MLE is highly sensitive to outliers. Outliers can severely distort parameter estimates and lead to misleading conclusions. Existing robust estimators for standard beta regression (e.g., MDPDE and SMLE) or binary data often fail to be well-defined for all parameter values, particularly when dealing with unbounded beta densities, or they do not address the specific structural complexity of the inflated model, which combines a discrete mixture component with a continuous beta component.

Methodology
The authors propose a framework for robust inference in zero-or-one inflated beta regression models by extending robust estimation techniques to both the discrete and continuous parts of the model simultaneously.

1. Model Structure: The response variable $Y$ follows a zero-or-one inflated beta distribution, a mixture of a degenerate variable at $c \in \{0, 1\}$ and a beta-distributed variable on $(0, 1)$. The model parameters are linked to covariates via link functions for the probability of the boundary ($\vartheta$), the conditional mean ($\mu$), and the precision ($\phi$).
2. Robust Estimators: The paper introduces two specific estimators, M-LSE and M-LME, which solve a system of estimating equations $U(y; \upsilon) = 0$. These estimators treat the discrete and continuous components separately but within a unified framework:
   • Discrete Part: The authors employ the Minimum Density Power Divergence Estimator (MDPDE) for the binary component (modeling the probability of observing a zero or one). This approach downweights observations with large discrepancies between observed and predicted probabilities.
   • Continuous Part: To address the issue of unbounded beta densities where standard robust estimators (SMLE and MDPDE) may not be well-defined, the authors utilize the logit transformation of the response variable ($Y^* = \log(Y/(1-Y))$).
     • M-LSE: Combines MDPDE for the discrete part with the Logit Surrogate Maximum Likelihood Estimator (LSMLE) for the continuous part.
     • M-LME: Combines MDPDE for the discrete part with the Logit Minimum Density Power Divergence Estimator (LMDPDE) for the continuous part.
   • By working on the logit scale, the estimators ensure the density class is closed under power transformations, guaranteeing the estimators are well-defined for all parameter values.
3. Data-Driven Tuning: A key methodological contribution is the adaptation of a data-driven algorithm (originally proposed for beta regression) to select the tuning constant $\alpha$ for both components. The algorithm seeks the smallest $\alpha$ (closest to MLE) that maintains stability in the estimates, ensuring full efficiency in the absence of contamination while providing robustness when outliers are present.
4. Inference: The authors derive the asymptotic normality of the proposed estimators and develop robust Wald-type tests for hypothesis testing. They also provide a method for calculating standard errors that accounts for the joint nature of the inference, even though point estimates are computed separately.

Key Contributions

• Extension to Inflated Models: This work is the first to propose robust inference methods specifically for zero-or-one inflated beta regression models, addressing the unique challenges posed by the mixture of discrete and continuous components.
• Well-Defined Estimators: By utilizing the logit transformation and the specific constructions of LSMLE and LMDPDE, the proposed estimators overcome the limitations of previous robust methods (MDPDE/SMLE) which were not well-defined for unbounded beta densities.
• Unified Framework: The paper provides a complete inferential framework, including point estimation, asymptotic variance-covariance matrices, and robust Wald-type tests, implemented in an R package.
• Adaptive Tuning: The application of a data-driven algorithm to select tuning constants for both the discrete and continuous parts allows the method to automatically revert to MLE when data is clean, balancing efficiency and robustness.

Empirical Results
The authors evaluated the performance of M-LSE and M-LME through simulation studies and a real data application:

• Simulations: In scenarios with 5% contamination in the discrete part, the continuous part, or both, standard MLE exhibited severe bias and high Root Mean Squared Error (RMSE). In contrast, the robust estimators maintained low bias and RMSE, performing similarly to MLE in uncontaminated data (where the algorithm selected $\alpha \approx 0$) and significantly outperforming MLE in contaminated scenarios (with TMSE ratios up to 34 times lower than MLE).
• Hypothesis Testing: The robust Wald-type tests maintained empirical rejection rates close to the nominal 5% level under contamination, whereas standard Wald tests suffered from inflated Type I errors (rejection rates near 100%) or zero power.
• Real Data Application: The method was applied to COVID-19 case fatality rates in 200 Brazilian cities. The data contained a prominent outlier (a city with a single case and one death). The MLE results were heavily influenced by this observation, changing the significance and magnitude of covariate effects (e.g., HDI). The robust estimators (M-LSE) downweighted the outlier, producing results consistent with an analysis where the outlier was manually removed, demonstrating the method's ability to provide reliable inference without arbitrary data deletion.

Significance
The paper claims that while flexible models exist to handle outliers, they often introduce unnecessary complexity and lack interpretability. The proposed approach retains the interpretability and structure of the standard inflated beta model while replacing the estimation process with a robust alternative. The authors emphasize that their work fills a critical gap in the literature by providing the first robust inference methods for inflated models of bounded continuous data. The simulation results and real-world application demonstrate that the proposed estimators effectively mitigate the influence of outliers, preventing misleading conclusions that frequently arise from standard maximum likelihood inference in the presence of contaminated data.