Bayesian framework for interval-valued data using Jeffreys’ prior and posterior predictive checking methods

Abstract

Interval-valued data have been widely used in both academic and industry research areas. With the development of computational statistics, the Bayesian methods are growing rapidly. Based on the existing theoretical researches, this paper tries to establish a Bayesian framework for interval-valued data, which contains the determination of Jeffreys’ prior, posterior inferences, and posterior predictive checking methods. First, Jeffreys’ prior for interval-valued data is proposed for the first time. Then, the posterior conditional distribution is derived, and posterior samples are obtained by using Gibbs sampling methods. Based on posterior samples, we obtain the Bayesian point and credible interval estimations for interval-valued data. Finally, we propose a posterior predictive checking method for interval-valued data based on novel test quantities. If the posterior predictive checking method shows that the data do not meet the distribution assumptions, we suggest the use of a data transformation. The effectiveness of our method is demonstrated by simulations and two cardiological data sets.

Keywords:

• Interval-valued data
• Bayesian method
• Jeffreys’ prior
• Posterior predictive checking method

Additional information

Funding

The research for this paper was supported by National Natural Science Foundation of China (Nos. 72071008 and 71771011). This work was supported by the Open Research Fund of Key Laboratory of Advanced Theory and Application in Statistics and Data Science (East China Normal University), Ministry of Education.