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 (Formula presented.) (where (Formula presented.)), 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.