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Constructing likelihood functions for interval-valued random variables. (English) Zbl 1444.62139
The present work investigates statistical models for interval-valued data that are directly constructed from an assumed underlying data generating model $$f(x_1,\dots,x_m|\alpha)$$ and a data aggregation function $$\phi(.)$$ that maps the space of real-valued data to the space of intervals. Basics as random interval, descriptive model, containment functional, containment distribution function of a random interval, density function of a random interval and properties are presented in the second section of the paper. In the third section, one develops a generative model of the random interval. The likelihood functions for generative models are directly constructed from likelihood functions for the underlying real-valued data. Then, one focuses on a special case of the generative model for which the latent data points are exchangeable and this is the hierarchical generative model. Results for the convergence of the hierarchical generative models to valid descriptive models are proved. Further, in the fourth section of the paper, the obtained results are extended from interval to $$p$$-hyper-rectangles. In the Applications section, a direct comparison between descriptive and generative models is performed. There is an analysis of simulation results and estimates and one presents a practical example on “credit card data”. All proofs and additional theoretical introductory elements are resumed in the APPENDIX part.

##### MSC:
 62R07 Statistical aspects of big data and data science 62H12 Estimation in multivariate analysis
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