Barndorff-Nielsen, O. E.; Jupp, P. E. Approximating exponential models. (English) Zbl 0721.62003 Ann. Inst. Stat. Math. 41, No. 2, 247-267 (1989). The authors describe Amari’s definition of locally approximating exponential models and provide some elementary properties and several examples of this type of approximation, which they call expected exponential approximation. The examples concern the von Mises distribution, nonlinear normal regression, and a simple model from quantum physics. Similar to Amari’s, but in the framework of observed rather than expected, likelihood geometry they construct observed exponential approximations of a statistical model. Elementary properties of the resulting observed exponential approximations are discussed. A more detailed discussion is given on the closeness of these approximations. The differences between the maximum likelihood estimators of the original model and the approximating models have the same convergence rate. Reviewer: S.Zwanzig (Berlin) Cited in 4 Documents MSC: 62A01 Foundations and philosophical topics in statistics 62F12 Asymptotic properties of parametric estimators Keywords:vector bundles; conditional inference; Amari’s definition of locally approximating exponential models; expected exponential approximation; von Mises distribution; nonlinear normal regression; quantum physics; likelihood geometry; observed exponential approximations; maximum likelihood estimators PDFBibTeX XMLCite \textit{O. E. Barndorff-Nielsen} and \textit{P. E. Jupp}, Ann. Inst. Stat. Math. 41, No. 2, 247--267 (1989; Zbl 0721.62003) Full Text: DOI References: [1] Amari, S.-I. (1987). Differential-geometrical theory of statistics?towards new developments, Differential Geometry in Statistical Inference, Lecture Notes?Monograph Series, 19-94, Institute of Mathematical Statistics, Hayward. [2] Barnard, G. A. (1971). Scientific inferences and day-to-day decisions, Foundations of Statistical Inference, (eds. V. P.Godambe and D. A.Sprott), 289-300, Holt, Rinehart and Winston of Canada, Toronto. [3] Barnard, G. A. (1982). Contribution to the discussion paper by D. V. Lindley, Internat. Statist. Rev., 50, 11-14. · doi:10.2307/1402449 [4] Barndorff-Nielsen, O. E. (1986). Likelihood and observed geometries, Ann. Statist., 14, 856-873. · Zbl 0632.62028 · doi:10.1214/aos/1176350038 [5] Barndorff-Nielsen, O. E. (1987). Differential and integral geometry in statistical inference, Differential Geometry in Statistical Inference, Lecture Notes-Monograph Series, 95-161, Institute of Mathematical Statistics, Hayward. [6] Barndorff-Nielsen, O. E. (1988). Parametric Statistical Models and Likelihood, Lecture Notes in Statistics, Springer, Heidelberg. · Zbl 0691.62002 [7] Dawid, A. P. (1975). Contribution to discussion of ?Defining the curvature of a statistical problem (with application to second order efficiency)? by B. Efron, Ann. Statist., 3, 1131-1134. [8] DiCiccio, T. J. (1984). On parameter transformations and interval estimation, Biometrika, 71, 477-485. · Zbl 0566.62022 · doi:10.1093/biomet/71.3.477 [9] Efron, B. (1975). Defining the curvature of a statistical problem (with application to second order efficiency), Ann. Statist., 3, 1189-1242 (with discussion). · Zbl 0321.62013 · doi:10.1214/aos/1176343282 [10] Husemoller, D. (1966). Fibre Bundles, McGraw-Hill, New York. · Zbl 0144.44804 [11] Ibragimov, I. A. and Has’minskii, R. Z. (1981). Statistical Estimation, Springer, Heidelberg. [12] LeCam, L. (1986). Asymptotic Methods in Statistical Decision Theory, Springer, New York. [13] Lichnerowicz, A. (1976). Global Theory of Connections and Holonomy Groups, Noordhoff International Publishing, Leyden. · Zbl 0337.53031 [14] Mardia, K. V. (1972). Statistics of Directional Data, Academic Press, New York. · Zbl 0244.62005 [15] Solmitz, E. T. (1964). Analysis of experiments in particle physics, Annu. Rev. Nucl. Sci., 14, 375-402. · doi:10.1146/annurev.ns.14.120164.002111 [16] Sprott, D. A. (1977). On the theory of statistical estimation, Math. Sci., 2, 127-137. · Zbl 0366.62032 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.