Summary
The problem of estimation is considered to be that of making the most informative statement about an unknown parameterϑ. If previous empirical information aboutϑ is known in the form of a prior probability distribution, Bayes’s Theorem can be used to make probability statements aboutϑ. In the absence of such knowledge the fiducial method can sometimes be used to provide statements of probability. If, in a given case, neither of these methods is applicable use must be made of the likelihood function alone to provide a weaker measure of uncertainty. The controversies over the appropriate areas of application of these methods are discussed. Confidence intervals and the general concepts of point and interval estimation are considered and it is stated that in themselves they do not provide a reasonable solution to the problem of estimation considered above.
Zusammenfassung
Das Problem des Schätzens wird in der Aufgabe gesehen, die informations-reichste Aussage über einen unbekannten Parameter ϑ zu machen. Wenn bereits empirische Information über ϑ in Form einer Apriori-Wahrscheinlichkeitsverteilung vorliegt, so kann man das Bayessche Theorem anwenden, um Wahrscheinlichkeitsaussagen über ϑ zu bekommen. Wenn ein solches Wissen fehlt, kann die Fiduzialmethode manchmal zu Wahrscheinlichkeitsaussagen verhelfen. In allen übrigen Fällen, wo beide Methoden versagen, kann nur die Likelihoodfunktion zu einem allerdings schwächeren Ungewißheitsmaß führen. Die Kontroversen über die richtige Angrenzung der Anwendungsgebiete dieser Verfahren werden besprochen. Die Theorie der Konfidenzintervalle und die Theorie der Punkt- und Intervallschätzung werden betrachtet, und es wird festgestellt, daß sie als solche keine vernünftige Lösung des oben definierten Schätzproblems liefern.
Résumé
Le problème d’estimation est conçu comme celui-ci de faire l’énoncé le plus informatif sur un paramètre inconnuϑ. Si, préalablement, on possède une information empirique surϑ en forme d’une loi de probabilité à priori, on peut utiliser le théorème de Bayes pour faire un énoncé de probabilité surϑ. A défaut d’une telle connaissance, la méthode dite “fiduciale” (fiducial en anglais) peut, en certains cas, produire des énoncés de probabilité. Si aucune de ces deux méthodes ne peut être appliquée, seule la fonction de vraisemblance fournira un instrument, moins puissant, il est vrai, pour mesurer l’incertitude. Les controverses sur les domaines appropriés d’application de ces méthodes sont discutées. La théorie des intervalles de confiance et la théorie de l’estimation ponctuelle et de l’estimation par régions sont considérées, et il est constaté que, par elles-mêmes, elles ne fournissent pas une solution raisonnable du problème d’estimation défini ci-dessus.
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Sprott, D.A. Statistical estimation — Some approaches and controversies. Statistische Hefte 6, 97–111 (1965). https://doi.org/10.1007/BF02922288
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DOI: https://doi.org/10.1007/BF02922288