Distributional inference

Abstract

The making of statistical inferences in distributional form is conceptionally complicated because the epistemic 'probabilities' assigned are mixtures of fact and fiction. In this respect they are essentially different from 'physical' or 'frequency-theoretic' probabilities. The distributional form is so attractive and useful, however, that it should be pursued. Our approach is In line with Walds theory of statistical decision functions and with Lehmann's books about hypothesis testing and point estimation: loss functions are defined, risk functions are studied, unbiasedness and equivariance restrictions are made, etc. A central theme is that the loss function should be 'proper'. This fundamental concept has been explored by meteorologists, psychometrists, Bayesian statisticians, and others. The paper should be regarded as an attempt to reconcile various schools of statisticians. By accepting what we regard 88 good and useful in the various approaches we are trying to develop a nondogmatic approach.Key Words & Phrases: decision theory, fiducial inference, foundations, invariance, unbiasedness. 1hmdUctm ' 0The making of statistical inferences is an uncertain affair. That is why different schools have emerged and the statistical controversy has appeared. We worry about this controversy, because the lack of consensus revealed by it impedes the respectabdity of our science. It is true that the debate between the various schools is less heated than it used to be. The statistical community has started to realize that every school has something useful to say and that the making of applications requires an eclectic attitude. Some situations are dealt with most appropriately by using a data-analytic 0 WS. 1995. PUblLbcd by BLd;nll P u b l i i IQ cow*y Rod. Oxford OX4 I f f , UK a d 238 M . * S U a r Canhidm MA 02142 USA.

LesP-values comme votes d'experts

“…Ce type d'inférence aété défendu dans de nombreux travaux (Kroese et al 1995). Il est particulièrement adapté aux applications qui ne réclament pas un choix définitif entre les deux hypothèses.…”

Estimating a frequency unseen: an application to ornithology

“…The ‘fiducial approach’ (see Kroese et al. , 1995 and Salomé , 1998) is based on the simple ‘classical‐statistical’ idea that the distribution function G m of the distributional inference Q ( m ) that we try to construct should be determined by identifying G m ( θ ) with the ‘most reasonable’ degree of belief α ( m ) in the truth of the hypothesis H: r 0 ≤ θ .…”

“…It refers to the quantification of (un)certainty, belief, etc., by using probabilistic terminology. This term was introduced by Kroese et al. (1995), because similar terms such as Bayesian inference, fiducial inference, etc., are too closely associated to a particular methodology of generating such distributional inferences (and probability statements).…”

An Optimum Property for Fiducial Inference

“…The same inference is obtained by the Bayesian argument if Lebesgue measure άθ is taken as prior. In [5] it is shown that the procedure defined by (2.3) is optimum in a decision-theoretic sense if the attention is restricted to translation-equivariant procedures.…”

“…It is interesting to restate Theorem 1 in terms of the decision-theoretic approach to distributional inference, which is discussed extensively in [1] and [5]. Central in this theory is a loss function L such that L (Θ, G z ) is the loss incurred if the distributional inference G x is made whereas θ is the true value.…”