An advantage of methods that base inference on a posterior distribution is credible regions are readily obtained. Except in well-specified situations, however, there no guarantee such will achieve the nominal frequentist coverage probability, even approximately. To overcome this difficulty, we propose general strategy introduces an additional scalar tuning parameter to control spread, and develop algorithm chooses so corresponding region achieves probability.

Bayesian posterior distributions are widely used for inference, but their dependence on a statistical model creates some challenges. In particular, there may be lots of nuisance parameters that require prior and computations, plus potentially serious risk misspecification bias. Gibbs distributions, the other hand, offer direct, principled, probabilistic inference quantities interest through loss function, not model-based likelihood. Here we provide simple sufficient conditions establishing...

The Dempster--Shafer (DS) theory is a powerful tool for probabilistic reasoning based on formal calculus combining evidence. DS has been widely used in computer science and engineering applications, but yet to reach the statistical mainstream, perhaps because belief functions do not satisfy long-run frequency properties. Recently, two of authors proposed an extension DS, called weak (WB) approach, that can incorporate desirable properties into framework by systematically enlarging focal...

The inferential models (IM) framework provides prior-free, frequency-calibrated, posterior probabilistic inference. key is the use of random sets to predict unobservable auxiliary variables connected observable data and unknown parameters. When nuisance parameters are present, a marginalization step can reduce dimension variable which, in turn, leads more efficient For regular problems, exact be achieved, we give conditions for marginal IM validity. We show that our approach inference...

Accurate prediction of future claims is a fundamentally important problem in insurance. The Bayesian approach natural this context, as it provides complete predictive distribution for claims. classical credibility theory simple approximation to the mean that point predictor, but ignores other features distribution, such spread, would be useful decision making. In article, we propose Dirichlet process mixture log-normals model and discuss theoretical properties computation corresponding...

Summary The inferential model (IM) framework provides valid prior-free probabilistic inference by focusing on predicting unobserved auxiliary variables. But, efficient IM-based can be challenging when the variable is of higher dimension than parameter. Here we show that features are often fully observed and, in such cases, a simultaneous reduction and information aggregation achieved conditioning. This proposed conditioning strategy leads to IM casts new light Fisher's notions sufficiency,...

Generalized Bayes posterior distributions are formed by putting a fractional power on the likelihood before combining with prior via Bayes's formula. This power, which is often viewed as remedy for potential model misspecification bias, called learning rate, and number of data-driven rate selection methods have been proposed in recent literature. Each these proposals has different focus, target they aim to achieve, makes them difficult compare. In this paper, we provide direct head-to-head...

Mixture models have received considerable attention recently and Newton [Sankhyā Ser. A 64 (2002) 306–322] proposed a fast recursive algorithm for estimating mixing distribution. We prove almost sure consistency of this estimate in the weak topology under mild conditions on family densities being mixed. This depends data ordering permutation-invariant modification is proposed, which an average original over permutations sequence. Rao–Blackwell argument used to probability alternative...

In the frequentist program, inferential methods with exact control on error rates are a primary focus. The standard approach, however, is to rely asymptotic approximations, which may not be suitable. This paper presents general framework for construction of procedures based plausibility functions. It shown that function-based tests and confidence regions have desired properties in finite samples---no large-sample justification needed. An extension proposed method also given problems...

Satellite conjunction analysis is the assessment of collision risk during a close encounter between satellite and another object in orbit. A counterintuitive phenomenon has emerged literature, namely, probability dilution, which lower quality data paradoxically appear to reduce collision. We show that dilution symptom fundamental deficiency probabilistic representations statistical inference, there are propositions will consistently be assigned high degree belief, regardless whether or not...

P-values are a mainstay in statistics but often misinterpreted.We propose new interpretation of p-value as meaningful plausibility, where this is to be interpreted formally within the inferential model framework.We show that, for most practical hypothesis testing problems, there exists an such that corresponding plausibility function, evaluated at null hypothesis, exactly p-value.The advantages representation notion consistent with way practitioners use and interpret p-values, calculation...

In the context of predicting future claims, a fully Bayesian analysis – one that specifies statistical model, prior distribution, and updates using Bayes's formula is often viewed as gold-standard, while Bühlmann's credibility estimator serves simple approximation. But those desirable properties give solution its elevated status depend critically on posited model being correctly specified. Here we investigate asymptotic behavior posterior distributions under misspecified our conclusion...

Here we explore general asymptotic properties of Predictive Recursion (PR) for nonparametric estimation mixing distributions. We prove that, when the mixture model is mis-specified, estimated converges almost surely in total variation to that minimizes Kullback-Leibler divergence, and a bound on (Hellinger contrast) rate convergence obtained. Simulations suggest this nearly sharp minimax sense. Moreover, identifiable, sure weak distribution estimate follows. PR assumes support known. To...

Prediction of future observations is a fundamental problem in statistics. Here we present general approach based on the recently developed inferential model (IM) framework. We employ an IM-based technique to marginalize out unknown parameters, yielding prior-free probabilistic prediction observables. Verifiable sufficient conditions are given for validity our IM prediction, and variety examples demonstrate proposed method's performance. Thanks its generality ease implementation, expect that...

Accurate estimation of value-at-risk (VaR) and assessment associated uncertainty is crucial for both insurers regulators, particularly in Europe. Existing approaches link data VaR indirectly by first linking to the parameter a probability model, then expressing as function that parameter. This indirect approach exposes insurer model misspecification bias or inefficiency, depending on whether finite- infinite-dimensional. In this paper, we directly via what call discrepancy function, leads...