A note on p-values interpreted as plausibilities

P-values are a mainstay in statistics but are often misinterpreted. We propose a new interpretation of p-value as a 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 inferential model such that the corresponding plausibility function, evaluated at the null hypothesis, is exactly the p-value. The advantages of this representation are that the notion of plausibility is consistent with the way practitioners use and interpret p-values, and the plausibility calculation avoids the troublesome conditioning on the truthfulness of the null. This connection with plausibilities also reveals a shortcoming of standard p-values in problems with non-trivial parameter constraints.Comment: 13 pages, 1 figur

Random sets and exact confidence regions

An important problem in statistics is the construction of confidence regions for unknown parameters. In most cases, asymptotic distribution theory is used to construct confidence regions, so any coverage probability claims only hold approximately, for large samples. This paper describes a new approach, using random sets, which allows users to construct exact confidence regions without appeal to asymptotic theory. In particular, if the user-specified random set satisfies a certain validity property, confidence regions obtained by thresholding the induced data-dependent plausibility function are shown to have the desired coverage probability.Comment: 14 pages, 2 figure

Inferential models: A framework for prior-free posterior probabilistic inference

Posterior probabilistic statistical inference without priors is an important but so far elusive goal. Fisher's fiducial inference, Dempster-Shafer theory of belief functions, and Bayesian inference with default priors are attempts to achieve this goal but, to date, none has given a completely satisfactory picture. This paper presents a new framework for probabilistic inference, based on inferential models (IMs), which not only provides data-dependent probabilistic measures of uncertainty about the unknown parameter, but does so with an automatic long-run frequency calibration property. The key to this new approach is the identification of an unobservable auxiliary variable associated with observable data and unknown parameter, and the prediction of this auxiliary variable with a random set before conditioning on data. Here we present a three-step IM construction, and prove a frequency-calibration property of the IM's belief function under mild conditions. A corresponding optimality theory is developed, which helps to resolve the non-uniqueness issue. Several examples are presented to illustrate this new approach.Comment: 29 pages with 3 figures. Main text is the same as the published version. Appendix B is an addition, not in the published version, that contains some corrections and extensions of two of the main theorem

Inference about constrained parameters using the elastic belief method,” Internat

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