On the correctness of statistical point estimation problem

N. N. Čencov

Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 1, pp. 15-31

Résumé

Strongly consistent (in the sense of convergence in variation) decision procedures $\Pi=\{\Pi_N\}$ for the statistical point estimation problem are considered. We prove that the statistical problem of estimation the probability distribution $P$ on $E=\{x\colon 0\le x\le 1\}$ by means of independent $P$-distributed bservations $x_i^*$ ($i=1,\dots,N$, $N\to\infty$) without additional a priori information about $P$ is incorrect in this sense. The unknown $P$ being a priori absolutely continuous, the problem turns out to be correct [15]. However this modified problem is found not to admit the uniformly consistent decision procedures. Also it does not admit the procedures with vanishing (at $N\to\infty$) supremum of the risk, when a loss function is given by a Kullback information deviation $I[P_N^*:P]$.