Flexible modeling of nonnegative continuous data: Box-Cox symmetric regression and its zero-adjusted extension

TL;DR

The paper tackles modeling of skewed positive data with a substantial mass at zero by introducing Box-Cox symmetric (BCS) distributions and their zero-adjusted extension (ZABCS). It develops regression formulations that link the scale and dispersion parameters to covariates via monotone links, and implements a two-stage maximum likelihood estimation that separates the discrete zero component from the continuous positive part. Key contributions include a formalization of BCS regression and ZABCS regression, diagnostic tools such as quantile residuals and local influence measures, and an R package named BCSreg that provides flexible fitting and diagnostics. The empirical application to basic education expenditure demonstrates the method’s ability to capture both the zero-inflation and the heavy-tailed, right-skewed positive expenditures, with ZABCLOII offering superior fit among competitors. Overall, the work broadens the toolkit for analyzing zero-inflated positive data and supplies practical software for applied researchers and policy analysis, supported by simulations and diagnostics.

Abstract

The Box-Cox symmetric distributions constitute a broad class of probability models for positive continuous data, offering flexibility in modeling skewness and tail behavior. Their parameterization allows a straightforward quantile-based interpretation, which is particularly useful in regression modeling. Despite their potential, only a few specific distributions within this class have been explored in regression contexts, and zero-adjusted extensions have not yet been formally addressed in the literature. This paper formalizes the class of Box-Cox symmetric regression models and introduces a new zero-adjusted extension suitable for modeling data with a non-negligible proportion of observations equal to zero. We discuss maximum likelihood estimation, assess finite-sample performance through simulations, and develop diagnostic tools including residual analysis, local influence measures, and goodness-of-fit statistics. An empirical application on basic education expenditure illustrates the models' ability to capture complex patterns in zero-inflated and highly skewed nonnegative data. To support practical use, we developed the new BCSreg R package, which implements all proposed methods.

Figures (4)

• Figure 1: Distribution of positive education expenditures. The main panel shows the density histogram for individuals with positive expenditures only, while the inset bar chart reports the proportion of individuals with zero and positive education expenditure.
• Figure 2: Plots of the quantile residuals with simulated envelopes for the fits under the ZABCLOII (a), ZABCHP (b), ZABCT (c), and ZABCPE (d) models.
• Figure 3: Index plots of $|\bm{d}_{max}|$ (a) and $C_i$ (b).
• Figure 4: Index plot of the (randomized) quantile residual (a), quantile residual with simulated envelope (b), and $|\bm{d}_\text{max}|$ (local influence) under case-weight perturbation (c).

Theorems & Definitions (2)

• Definition 2.1
• Definition 2.2