Hosoya, Yuzo Information amount and higher-order efficiency in estimation. (English) Zbl 0703.62031 Ann. Inst. Stat. Math. 42, No. 1, 37-49 (1990). Summary: By means of second-order asymptotic approximation, the paper clarifies the relationship between the Fisher information of first-order asymptotically efficient estimators and their decision-theoretic performance. It shows that if the estimators are modified so that they have the same asymptotic bias, the information amount can be connected with the risk based on convex loss functions in such a way that the greater information loss of an estimator implies its greater risk. The information loss of the maximum likelihood estimator is shown to be minimal in a general set-up. A multinomial model is used for illustration. Cited in 1 Document MSC: 62F12 Asymptotic properties of parametric estimators 62C99 Statistical decision theory Keywords:second order Kullback risk; efficiency; information loss; second-order asymptotic approximation; Fisher information of first-order asymptotically efficient estimators; asymptotic bias; convex loss functions; maximum likelihood estimator; multinomial model PDFBibTeX XMLCite \textit{Y. Hosoya}, Ann. Inst. Stat. Math. 42, No. 1, 37--49 (1990; Zbl 0703.62031) Full Text: DOI References: [1] Akahira, M. and Takeuchi, K. (1981). Asymptotic Efficiency of Statistical Estimators: Concept and Higher Order Asymptotic Efficiency, Lecture Notes in Statistics, Springer, New York. · Zbl 0463.62026 [2] Akahira, M., Hirakawa, F. and Takeuchi, K. (1988). Second and Third Order Asymptotic Completeness of the Class of Estimators, Probability Theory and Mathematical Statistics, Lecture Notes in Mathematics 1299, 11-27, Springer, New York. · Zbl 0641.62021 [3] Bickel, P. J., G?tze, F. and Van, Zwet, W. R. (1985). A simple analysis of third-order efficiency of estimates, Proc. Berkeley Conference in Honor of J. Neyman and J. Kiefer, Vol. II, (eds. L. M.Le, Cam and R. A., Olshen), 749-768, Wadsworth, Monterey. · Zbl 1373.62093 [4] Efron, B. (1982). Maximum likelihood and decision theory, Ann. Statist., 10, 340-356. · Zbl 0494.62004 · doi:10.1214/aos/1176345778 [5] Fisher, R. A. (1925). Theory of statistical estimation. Proc. Camb. Phil. Soc., 22, 700-725. · JFM 51.0385.01 · doi:10.1017/S0305004100009580 [6] Ghosh, J. K. and Subramanyam, K. (1974). Second order efficiency of maximum likelihood estimators, Sankhy? Ser. A., 36, 325-358. · Zbl 0321.62035 [7] Ghosh, J. K., Sinha, B. K. and Wieand, H. S. (1980). Second-order efficiency of the MLE with respect to any bowl-shaped loss function, Ann. Statist., 8, 506-521. · Zbl 0436.62031 · doi:10.1214/aos/1176345005 [8] Hosoya, Y. (1979). High-order efficiency in the estimation of linear processes, Ann. Statist., 7, 516-530. · Zbl 0406.62067 · doi:10.1214/aos/1176344673 [9] Hosoya, Y. (1988). The second-order Fisher information. Biometrika, 75, 265-274. · Zbl 0638.62020 · doi:10.1093/biomet/75.2.265 [10] Kullback, S. (1959). Information Theory and Statistics, Wiley, New York. · Zbl 0088.10406 [11] Peers, H. W. (1978). Second-order sufficiency and statistical invariants, Biometrika, 65, 489-496. · Zbl 0395.62005 · doi:10.1093/biomet/65.3.489 [12] Pfanzagl, J. and Wefelmeyer, W. (1978). A third-order optimum property of the maximum likelihood estimator. J.Multivariate Anal., 8, 1-29. · Zbl 0376.62019 · doi:10.1016/0047-259X(78)90016-7 [13] Rao, C. R. (1962). Efficient estimates and optimum inference procedures in large sample. J. Roy. Statist. Soc. Ser. B, 24, 26-72. · Zbl 0138.13103 [14] Takeuchi, K. (1982). Higher order asymptotic efficiency of estimators in decision procedures, Proc. III Purdue Symp. Decision Theory and Related Topics, Vol. II, 351-361, Academic Press. · Zbl 0583.62027 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.