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Abstract:
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Interval-valued data are different from conventional data in that they have inherit internal structures. This work aims to develop the mathematical theory for the likelihood-based modelling of interval-valued random variables. We first study the topological property and the measurability of the space of intervals, which lead to the distribution functions and density functions. Then, we propose two types of models, the descriptive model and the generative model. In particular, we focus on the generative model, of which the sequence of latent variables are exchangeable. It results in the mixture model, which can account for both intra- and inter- interval variations. The asymptotic properties of mixture models have been studied and their connections to descriptive models are emphasised. We illustrate the proposed models through simulated studies and one application of analysing the aggregated interval-valued data from the credit card customers.
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