Functional Data Regression on Distribution‐Valued Data via Logarithm Derivative Quantile Transformation

In this paper, we introduce a new regression method tailored for data presented as distributions. Building on the latest advancements in Distributional Data Analysis (DDA), we propose a new regression model based on a transformation of quantile functions using Logarithmic Derivative Quantile (LDQ) functions. For each distributional variable (where ), we model the LDQ functions as functional data by applying smoothing B‐splines at the points corresponding to the distributions' quantiles. The main contribution is the development of a regression model that considers functional regression coefficients. This allows for the consideration of distribution characteristics such as position, variability, and shape. Another contribution is the development of a robust procedure based on trimming distributions to reduce the instability of the tails and make more effective predictions. The proposed approach is corroborated by real environmental data. Cross‐validation and bootstrap techniques have been employed to assess the effectiveness of both the new regression model and its robust variant.