Statistical issues in general (Laplace regression paired samples)
February 25, 2026 The authors explored new modeling for discrete paired data without normality assumptions. Skellam (1946) introduced the Skellam distribution as the difference of two Poisson random variables; Kozubowski and Inusah (2006) proposed the skew discrete Laplace (SDL) distribution based on the difference between two geometric variables. More recently, Conceição et al. (2021) proposed a zero-modified Skellam distribution and its associated regression model, also estimated via Bayesian methods. While this approach introduces flexibility in modeling the zero probability, the regression coefficients associated with this component cannot be directly interpreted in terms of the mean. Even though it has nice properties, the SDL distribution has received little attention in a regression context. One of its major limitations is that its mode is always zero, regardless of the parameter values, restricting its flexibility. In this paper, the authors proposed a simple generalization of the SDL distribution—referred to as the modified SDL distribution—whose mode is not constrained to zero. They also introduced a new regression model for integer-valued and paired discrete data, along with a parameterization based on the mean and dispersion parameters, enhancing the interpretability of the model. The authors listed the main advantages of the proposed regression framework which include in their own words:
1.Flexibility: The modified SDL distribution accommodates a wide range of distributional shapes, including left- and right-skewed distributions, as well as symmetric ones.