  Histogram-valued variables are a particular kind of variables studied in
Symbolic Data Analysis where to each entity under analysis corresponds a
distribution that may be represented by a histogram or by a quantile function.
Linear regression models for this type of data are necessarily more complex
than a simple generalization of the classical model: the parameters cannot be
negative still the linear relationship between the variables must be allowed to
be either direct or inverse. In this work we propose a new linear regression
model for histogram-valued variables that solves this problem, named
Distribution and Symmetric Distribution Regression Model. To determine the
parameters of this model it is necessary to solve a quadratic optimization
problem, subject to non-negativity constraints on the unknowns; the error
measure between the predicted and observed distributions uses the Mallows
distance. As in classical analysis, the model is associated with a
goodness-of-fit measure whose values range between 0 and 1. Using the proposed
model, applications with real and simulated data are presented.
