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Theory & Methods: On Efron’s Parameterization
Author(s) -
Kumagai Etsuo,
Inagaki Nobuo,
Inoue Kiyoshi
Publication year - 2002
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/1467-842x.00235
Subject(s) - mathematics , trinomial , exponential family , counterexample , exponential function , mathematical analysis , distribution (mathematics) , exponential dichotomy , differential equation , natural exponential family , exponential distribution , plane (geometry) , geometry , statistics , combinatorics
This paper discusses a curved exponential family of distributions which is defined by a differential equation with respect to the expectation parameters in the two–dimensional exponential family. The differential equation considered here is the same as the one given by Efron (1975) for the trinomial distribution. This equation is extended here to a general exponential family, and called Efron’s parameterization in the two–dimensional exponential family. The solution of Efron’s parameterization is obtained explicitly in an exponential family, although Kumagai & Inagaki (1996) showed that there exists no proper solution of Efron’s equation for the trinomial distribution, in line with the counterexample given by Efron (1975 p. 1206). The paper gives some characterizations of Efron’s parameterization with special reference to Fisher’s circle model. The implications of these characterizations are the two–dimensional normal distribution and a spiral curve in the plane.