  Many statistical data are imprecise due to factors such as measurement
errors, computation errors, and lack of information. In such cases, data are
better represented by intervals rather than by single numbers. Existing methods
for analyzing interval-valued data include regressions in the metric space of
intervals and symbolic data analysis, the latter being proposed in a more
general setting. However, there has been a lack of literature on the
distribution-based inferences for interval-valued data. In an attempt to fill
this gap, we extend the concept of normality for random sets by Lyashenko
(1983) and propose a normal hierarchical model for random intervals. In
addition, we develop a minimum contrast estimator (MCE) for the model
parameters, which we show is both consistent and asymptotically normal.
Simulation studies support our theoretical findings, and show very promising
results. Finally, we successfully apply our model and MCE to a real dataset.
