  There is a growing need for the ability to analyse interval-valued data.
However, existing descriptive frameworks to achieve this ignore the process by
which interval-valued data are typically constructed; namely by the aggregation
of real-valued data generated from some underlying process. In this article we
develop the foundations of likelihood based statistical inference for random
intervals that directly incorporates the underlying generative procedure into
the analysis. That is, it permits the direct fitting of models for the
underlying real-valued data given only the random interval-valued summaries.
This generative approach overcomes several problems associated with existing
methods, including the rarely satisfied assumption of within-interval
uniformity. The new methods are illustrated by simulated and real data
analyses.
