New Classes of Priors Based on Stochastic Orders and Distortion Functions

Deriving Robust Bayesian Premiums Under Bands of Prior Distributions With Applications

“…Note that F π h (x) is again a distribution function for a particular distorted random variable, denoted by X π h , with density function π h . The following lemma, given in Arias-Nicolás et al (2016), formalizes the idea, in terms of the likelihood ratio order, that X π h gives more weight to higher (lower) risk events when h is convex (respectively, concave). The result is also a reformulation of Theorem 1 of Blazej (2008), which is a more general result stated in terms of weighted distributions for absolutely continuous distributions.…”

“…From Lemma 2, we have π h 1 ≤ lr π ≤ lr π h 2 . This led Arias-Nicolás et al (2016) to define the following class of priors.…”

“…random sample of size n from the baseline prior distribution π, which seem to be reasonable bounds for the confidence band. Other examples are given in Arias-Nicolás et al (2016) where distortions plays different roles. The distorted band satisfies some nice properties (see Arias-Nicolás et al, 2016).…”

“…Other examples are given in Arias-Nicolás et al (2016) where distortions plays different roles. The distorted band satisfies some nice properties (see Arias-Nicolás et al, 2016). For example, (1 − )π + π ∈ h 1 ,h 2 ,π , for all π ∈ h 1 ,h 2 ,π and for all 0 ≤ ≤ 1 (which is related to the -contamination classes).…”

“…The aim of this paper is to study the propagation of uncertainty from a class of priors recently introduced by Arias- Nicolás et al (2016), called the distorted band of priors, to the premiums (both the collective and the Bayesian). The distorted band of priors fulfills some desirable requirements: elicitation is easy, the prior uncertainty can be measured by different metrics, and the range of quantities of interest is easily obtained from the extremal members of the class.…”