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Statistica Sinica 21 (2011), 475-494



Jianchun Zhang and Chuanhai Liu

Purdue University

Abstract: The work of A. P. Dempster in 1960s extending Fisher's fiducial inference for parametric inference using multivalued mapping, and that of G. Shafer in 1970s on the assessment and combination of evidence led to what is now known as the Dempster-Shafer (DS) theory of belief functions. However, application of DS for parametric inference has been limited due, perhaps, to its computational difficulty, non-uniqueness, and lack of frequency properties. In this paper, we return to Dempster's original approach to constructing belief functions for parametric inference, called basic DS models (BDSMs), which are usual probability models on the space of the so-called focal elements. We propose to modify BDSMs by enlarging focal elements to obtain belief functions that have desired frequency properties. We call our method Weak Belief (WB). When it enlarges the focal elements no more than necessary, the method of WB is called Maximal Belief (MB). The MB method is illustrated with two examples: $(i)$ inference about a binomial proportion, and $(ii)$ inference about the number of outliers $(\mu_i\neq 0)$ based on the observed data $X_1,\ldots,X_n$ with the model $X_i\stackrel{ind}{\sim}N(\mu_i,1)$.

Key words and phrases: Belief functions, fiducial inference, frequentist evaluation, hypothesis testing, maximal belief, predictive random sets.

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