Estimates with asymptotically uniformly minimal $d$-risk

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].