Inflated Beta Distributions

This paper considers the issue of modeling fractional data observed in the interval [0,1), (0,1] or [0,1]. Mixed continuous-discrete distributions are proposed. The beta distribution is used to describe the continuous component of the model since its density can have quite diferent shapes depending on the values of the two parameters that index the distribution. Properties of the proposed distributions are examined. Also, maximum likelihood and method of moments estimation is discussed. Finally, practical applications that employ real data are presented.

💡 Research Summary

The paper addresses a common problem in the analysis of fractional data that lie in the unit interval: the frequent occurrence of exact zeros and/or ones (inflation). Standard beta regression models, while flexible for interior values (0, 1), cannot accommodate point masses at the boundaries, leading to poor fit and biased inference when such inflation is present. To remedy this, the authors propose a family of “inflated beta” distributions that combine a continuous beta component with discrete point masses at 0 and/or 1. The general form of the probability function is

 f(y) = π₀ · I(y = 0) + π₁ · I(y = 1) + (1 − π₀ − π₁) · Beta(y; α, β),

where π₀ and π₁ are the inflation probabilities (π₀ ≥ 0, π₁ ≥ 0, π₀ + π₁ ≤ 1) and α, β > 0 are the usual beta shape parameters. Special cases include a zero‑inflated beta (π₁ = 0), a one‑inflated beta (π₀ = 0), and a full 0‑1‑inflated beta (both π₀ and π₁ positive).

The authors derive the first two moments of the mixture, showing that the mean is
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