This article is cited in 3 papers

On the correctness of statistical point estimation problem

N. N. Čencov

Moscow

Abstract: 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 $I[P_N^*:P]$.

Received: 15.12.1979

 English version: Theory of Probability and its Applications, 1981, 26:1, 13–29

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