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Fiducial and Structural Statistical Inference

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Fiducial and Structural Statistical Inference Feudalism Rader T 1971 The Economics of Feudalism. Gordon & Breach, 1. A Simple Example New York Roth P 1863 FeudalitaW t und Unterthan!erband. Bo$ hlau, Weimar, Consider a variable y that is directly available or has Germany arisen by some preliminary reduction process and Sander P 1906 Feudalstaat und BuW rgerliche Verfassung. Ein suppose that y measures θ in an unbiased manner and Versuch uW ber das Grundproblem der deutschen Verfassungs- has error that is normal with known variance σ#. Then geschichte. Bath, Berlin, Germany ! we can say for example that P(y ! θj1.64σ!; θ) l 95 Schluchter W 1979 Die Entwicklung des okzidentalen Ration- percent. alismus. Eine Analyse !on Max Webers Gesellschaftsge- schichte. Mohr (Paul Siebeck), Tu$ bingen, Germany Parenthetically, we note that the preliminary re- Schreiner K 1983 ‘Grundherrschaft’. Entstehung und Bede- duction could have occurred as part of the underlying utungswandel eines geschichtswissenschaftlichen Ordnungs- investigation or as part of some subsequent simplifi- und Erkla$ rungsbegriffs. In: Patze H (ed.) Die Grundherrschaft cation of the statistical model by one of the common im Mittelalter. Thorbecke, Sigmaringen, Germany pp. 11–74 statistical reduction methods, sufficiency or con- Sidney P 1961 Feudalism and Liberty: Articles and Addresses. ditionality. If the reduction is to a sufficient statistic, Johans Hopkins Press, Baltimore then the conditional distribution describing possible Stephenson C 1948 Medie!al Feudalism. Cornell University antecedent data has no dependence on the parameter Press, Ithaca, NY and the model for the sufficient statistic is used in place Strayer J R 1965 Feudalism. Van Nostrand, Princeton Sweezy P u. a. 1976 The Transition from Feudalism to Capitalism. of the original model. If the reduction is by con- Verso, London ditionality then there is typically an ancillary variable Sydney H 1921\1969 The Fall of Feudalism in France. Barnes & with a distribution free of the parameter and the given Noble, New York model for the possible original data is replaced by the van Horst Bartel 1969 SachwoW rterbuch der Geschichte Deut- conditional model given the ancillary (supportive) schlands und der deutschen Arbeiterbewegung Dietz, Berlin, variable; see also Statistical Sufficiency. Germany Vol. 1, pp. 582–6 With a data value y! the fiducial methodology Von Gierke O 1868\1954 Das Deutsche Genossenschaftsrecht. would take the above probability expression and Graz Akademische Druck-und Verlangs-Anstadt, Graz, substitute the value y! and then treat θ as the variable Austria Vol. 1 for the probability statement, thus giving a 95 percent Ward J O 1985 Feudalism: Comparati!e Studies. Sydney Asso- ! ! ciation for Social Studies in Society & Culture, Sydney fiducial probability statement P(θ " y k1.64σ!; y ) Weber M 1921\1972 Wirtschaft und Gesellschaft. Grundriß der l 95 percent. The structural approach would consider # !erstehenden Soziologie, 5th edn. Mohr (Paul Siebeck), Tu$ b- the normal (0; σ!) distribution for the error e l ykθ ingen and might record for example the probability ! Wehler H-U 1987 Deutsche Gesellschaftsgeschichte, Vol.1: Vom P(e ! 1.64σ!) l 95 percent; then with data value y Feudalismus des Alten Reichs bis zur Defensi!en Modern- the probability statement would be applied to the isierung der ReformaW ra 1700–1815. Beck, Mu$ nchen, Germany error e l y!kθ giving the structural probability = Wunder H 1974 Der Feudalismus-Begriff. Uberlegungen zu P(y!kθ ! 1.64σ ; y!) l 95 percent or equivalently Mo$ glichkeiten der historischen Begriffsbildung. In: Wunder H ! ! ! (ed.) Feudalismus. Nymphenburger Verlag, Mu$ nchen, Ger- P(θ " y k1.64σ!; y ) l 95 percent. In a somewhat many related manner the Bayesian methodology might use a Wunder H 1989 Artikel ‘Feudalismus’. In: Sautier R H (ed.) uniform prior cdθ and obtain a posterior distribution ! ! Lexikon des Mittelalters. Artemis, Mu$ nchen, Germany, Vol. for θ that would have P(θ " y k1.64σ!; y ) l 95 4, pp. 411–15 percent. For this simple example the three methods give the same result at the 95 percent level and also at K. Schreiner other levels, thus saying essentially that with data value y! the probability distribution describing the ! # unknown θ is normal (y ; σ!). With more complicated models the results from the Fiducial and Structural Statistical three methodologies can differ and philosophical Inference arguments concerning substance and relative merits arise. However, for one straightforward generaliza- Fiducial inference is a statistical approach to interval tion the methods remain in agreement: the normal estimation first advocated by R. A. Fisher as an distribution can be replaced by some alternative alternative to the then dominant method of inverse distribution form; this is discussed in some detail in probability, i.e., using Bayes’ Theorem. Considerable Sect. 2. effort has gone into formalizing Fisher’s notions using such concepts as statistical invariance and pivotal quantities. This entry describes elements of the fiducial approach and relates them to other currently more 2. Fiducial Probability widelyusedstatisticalapproachestoinference.Section1 introduces some basic inferential ideas via a simple Fisher (1922, 1925; see also Fisher, Ronald A example. (1890–1962)) had already introduced most of the 5616 Fiducial and Structural Statistical Inference fundamental concepts of statistical theory, such as with a sample y ,…, y from the normal (µ; σ#) " n #! sufficiency, likelihood, efficiency, exhaustiveness distribution, the reduction would be to t(y) l (y` , sy). (minimal sufficiency), when he chose (Fisher 1930) to The methodology then suggests the use of a pivotal address directly the aspiration mentioned above. He quantity p l p(t; θ) with a fixed distribution and a took Laplace and Gauss to task for ‘fall(ing) into error one-one relationship between any two of p, t, θ; recall on a question of prime theoretical importance’ by more generally that a pivotal quantity is a function of adopting the Bayesian approach that ‘Bayes (had) the variable and parameter that provides a measure of tentatively wished to postulate in a special case’ and departure of variable value from parameter value, and which was published posthumously (Bayes, ibid). Be has a fixed distribution which allows an assessment of then proposed in a restricted context the fiducial an observed departure. Thus for the example a natural method, as discussed. pivotal is Neyman and Pearson (1933) then gave a math- ematical formulation of fiducial probability that be- 1 y` µ (n 1) s# 5 came known as confidence intervals. Fisher (1956) 2 k # k y 6 3 z l , χ l 7 however treated Neyman and Pearson’s formulation σ\Nn n−" σ# as a ‘misconception having some troubling conse- 4 8 quences …’; logical and philosophical arguments be- tween the two sides were intense for many years. In which has independent components, normal (0, 1) and particular the slight to Laplace and Gauss may well chi-square with nk1 degrees of freedom. Fisher (1956, have affected the views of the more mathematical p. 172), however, rather deviously rejected this as a participants. legitimate part of the fiducial methodology, but he was Fisher (1930) entitled his paper ‘Inverse Probability’ somewhat less explicit about what would be legitimate. and examined a statistic t(y) whose distribution This pivotal reduction procedure is now however a depended on a single parameter θ. Let P l F (t; θ) be rather familiar component of standard inference the distribution function of t, and let P itself be what theory and in particular of confidence theory. we might now call a p-value for assessing θ; of course The final step is to invert the pivotal quantity, that in the usual continuous case P has the uniform is, to insert the observed values for the variables and distribution on (0, 1). ‘If now we give P any particular then transform the distribution of the pivotal quantity value such as 0.95, we have … the perfectly objective to the parameter. For the example this gives fact that in 5 percent of samples’ t ‘will exceed the 95 percent value corresponding to the actual value of µ y z (n 1)−"/#χ s Nn θ ….’ Then to ‘any value of’ t ‘there will moreover be l ` ko \ k n−"q y\ usually a particular value of θ to which it bears this σ# l (nk1) s#\χ# relationship; we may call this the ‘fiducial 5 percent y n−" value of θ’ corresponding to’ the given t. This led to # Neyman and Pearson’s (ibid) confidence methodology where the y` , sy have their observed values. We can then but Fisher treated this as a misconception and he write followed different directions and interpretations for the fiducial methodology; for a view on related µ l y` kts \Nn approaches see Estimation: Point and Inter!al. For this y present simple case with scalar t and scalar θ, there seems little difference between the fiducial and the where t is Student with nk1 degrees of freedom. This confidence approaches and interpretations. fiducial calculation closely parallels that for the confi- This discussion effectively ascribes a distribution to dence approach, except that the limits here are θ based on an observed t; this is called the fiducial calculated from the Student distribution for µ rather distribution for θ. Just as the density of t for given θ is than from the Student distribution of the pivotal t. In obtained as (c\ct)F (t; θ) so also the fiducial density is quite wide generality fiducial regions can correspond obtained as (kc\cθ)F (t;θ); the negative sign is in- to confidence regions; it is just a matter of whether the consequential and is merely the result of F (t; θ) being limits are calculated before or after the data are examined typically for the case that is increasing with observed, a non-issue from the Bayesian viewpoint.
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