Estimation and Prediction Using Belief Functions: Application to Stochastic Frontier Analysis

Abstract

We outline an approach to statistical inference based on belief functions. For estimation, a consonant belief functions is constructed from the likelihood function. For prediction, the method is based on an equation linking the unobserved random quantity to be predicted, to the parameter and some underlying auxiliary variable with known distribution. The approach allows us to compute a predictive belief function that reflects both estimation and random uncertainties. The method is invariant to one-to-one transformations of the parameter and compatible with Bayesian inference, in the sense that it yields the same results when provided with the same information. It does not, however, require the user to provide prior probability distributions. The method is applied to stochastic frontier analysis with cross-sectional data. We demonstrate how predictive belief functions on inefficiencies can be constructed for this problem and used to assess the plausibility of various assertions.

Frequency-calibrated belief functions: Review and new insights

“…Use of a structural equation. In this section, we propose a new general method to compute a predictive belief function at a given confidence level, by adapting the method described in [25]. For simplicity, we assume here that X and Y are independent, but the approach can easily be extended to relax this assumption.…”

“…where U is a pivotal random variable with known distribution [37,18,25]. Equation (9) can be obtained by inverting the cdf of Y .…”

“…, u N of U . The degree of belief Bel Y |x (A) in A can then be estimated by the proportion of u i such that ϕ(C α (x), u i ) ⊆ A (see [25] for more details on this method).…”

“…We will assume that the reader already has some familiarity with the theory of belief functions. A concise exposition of the main relevant notions can be found in [25], for instance. The three approaches mentioned will then be described sequentially.…”

Evidential Classification of Incomplete Data via Imprecise Relabelling: Application to Plastic Sorting

“…Thus, imprecision is better captured in this framework compared to the probability framework where equiprobability and imprecision are confused. The recent growing interest to this theory has allowed to develop techniques to resolve a diverse panel of problems such as estimation [11,16], standard classification [9,32], or even hierarchical classification [1,23].…”