Mean regression for (0,1) responses via beta scale mixtures

TL;DR

This work addresses robust modeling of 0–1 bounded responses by extending classical beta regression through beta scale mixtures (BSM), which scale the beta dispersion with a mixing variable W drawn from a parametric family to capture heavy tails and skewness. A regression framework with a logit link μ(x;β) is developed, and four concrete mixtures—TPB, GB, LNB, and IGB—are presented to broaden the achievable moments beyond those of the standard beta. Parameter estimation combines direct maximum likelihood for GB, LNB, and IGB with an EM algorithm for TPB, and a sensitivity analysis demonstrates improved robustness to outliers. The methods are validated on simulated data and real datasets (Mock Jurors and Autologous PBSC recovery), where BSM models achieve superior fit (via AIC/BIC) and better robustness relative to classical beta regression and other bounded-domain models. The framework also invites extensions to fully covariate-dependent parameters and additional mixture choices, enhancing applicability across diverse unit-interval problems.

Abstract

To achieve a greater general flexibility for modeling heavy-tailed bounded responses, a beta scale mixture model is proposed. Each member of the family is obtained by multiplying the scale parameter of the conditional beta distribution by a mixing random variable taking values on all or part of the positive real line and whose distribution depends on a single parameter governing the tail behavior of the resulting compound distribution. These family members allow for a wider range of values for skewness and kurtosis. To validate the effectiveness of the proposed model, we conduct experiments on both simulated data and real datasets. The results indicate that the beta scale mixture model demonstrates superior performance relative to the classical beta regression model and alternative competing methods for modeling responses on the bounded unit domain.

Figures (11)

• Figure 1: Plots of the beta distribution \ref{['eq: pdf UB']} for varying parameter values.
• Figure 2: Plots of the TPB distribution \ref{['UB-BER:PDF']} for varying parameter values.
• Figure 3: Examples of behaviour of Skew($X$) (on the left) and Kurt($X$) (on the right), as function of $\mu$, for fixed $\phi=0.01$ and different values of $\theta$ of the TPB distribution \ref{['UB-BER:PDF']}, with $\theta_2=10$.
• Figure 4: Examples of behaviour of Skew($X$) (on the left) and Kurt($X$) (on the right), as function of $\mu$, for fixed $\phi=0.01$ and different values of $\theta$ of the TPB distribution \ref{['UB-BER:PDF']}, with $\theta_1=0.75$.
• Figure 5: Plot of the GB distribution \ref{['UB-UG: PDF']} for varying values of $\theta$, when $\mu=0.5$, and $\phi=0.3$

Theorems & Definitions (1)

• Definition 1