Geodesic
Estimates with asymptotically uniformly minimal $d$-risk
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 3, pp. 609-618 Cet article a éte moissonné depuis la source Math-Net.Ru

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The definition of a decision function with asymptotically ($n\to\infty$) uniformly minimal $d$-risk is presented in the framework of the general theory of statistical inference. Using this definition, we prove that the maximum likelihood estimate has asymptotically uniformly minimal $d$-risk. This extends one result by I. N. Volodin and A. A. Novikov [Theory Probab. Appl., 38 (1994), pp. 118–128] for shrinking priors to the general class of continuous distributions. The proof uses the asymptotic representation of the posterior risk function, as obtained in [A. A. Zaikin, J. Math. Sci. (N.Y.), 229 (2018), pp. 678–697].
Keywords: $d$-risk, posterior risk asymptotics, maximum likelihood estimate. 
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A. A. Zaikin. Estimates with asymptotically uniformly minimal $d$-risk. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 3, pp. 609-618. http://geodesic.mathdoc.fr/item/TVP_2018_63_3_a10/

[1] I. N. Volodin, A. A. Novikov, “Statistical estimators with asymptotically minimum $d$-risk”, Theory Probab. Appl., 38:1 (1993), 118–128 | DOI | MR | Zbl

[2] A. A. Zaikin, “Asymptotic expansion of posterior distribution of parameter centered by a $\sqrt n$-consistent estimate”, J. Math. Sci. (N. Y.), 229:6 (2018), 678–697 | DOI | MR | Zbl

[3] I. N. Volodin, S. V. Simushkin, “O $d$-aposteriornom podkhode k probleme statisticheskogo vyvoda”, 3-ya mezhdunarodnaya Vilnyusskaya konferentsiya po teorii veroyatnostei i matematicheskoi statistike, Tezisy dokladov, v. 1, In-t matem. i kibern. AN Lit. SSR, Vilnyus, 1981, 100–101

[4] I. N. Volodin, “Guaranteed statistical inference procedures (determination of the optimal sample size)”, J. Soviet Math., 44:5 (1989), 568–600 | DOI | Zbl

[5] S. V. Simushkin, I. N. Volodin, “Statistical inference with a minimal $d$-risk”, Probability theory and mathematical statistics (Tbilisi, 1982), Lecture Notes in Math., 1021, Springer, Berlin, 1983, 629–636 | DOI | MR

[6] I. N. Volodin, S. V. Simushkin, “Unbiasedness and Bayesness”, Soviet Math. (Iz. VUZ), 31:1 (1987), 1–8 | MR | Zbl

[7] I. A. Ibragimov, R. Z. Has'minskiĭ, Statistical estimation. Asymptotic theory, Appl. Math., 16, Springer-Verlag, New York–Berlin, 1981, vii+403 pp. | MR | MR | Zbl | Zbl

[8] I. A. Ibragimov, R. Z. Khas'minskii, “Asymptotic behavior of statistical estimators in the smooth case. I. Study of the likelihood ratio”, Theory Probab. Appl., 17:3 (1973), 445–462 | DOI | MR | Zbl

[9] I. A. Ibragimov, R. Z. Khas'minskii, “Asymptotical behaviour of some statistical estimators. II. Limiting theorems for the a posteriory density and Bayes' estimators”, Theory Probab. Appl., 18:1 (1973), 76–91 | DOI | MR | Zbl

[10] L. M. Le Cam, “On the asymptotic theory of estimation and testing hypotheses”, Proceedings of the 3rd Berkeley symposium on mathematical statistics and probability, v. 1, Contributions to the theory of statistics, Univ. of California Press, Berkeley, 1956, 129–156 | MR | Zbl