Abstract
It seems to be generally known that the proof of the continuity part of Theorem 1 in Hodges' and Lehmann's paper (1963) is incorrect. The fact that the theorem is incorrect is--perhaps--not so well known. We show this by constructing independent real random variables $X_1,\cdots, X_n$, each having the same non-atomic symmetric distribution, and an odd translation invariant estimator $h(X_1,\cdots, X_n)$ such that $P(h(X_1,\cdots, X_n) = 0) > 0. h$ may be chosen symmetric provided $n \geqq 3$.
Citation
Erik N. Torgersen. "A Counterexample on Translation Invariant Estimators." Ann. Math. Statist. 42 (4) 1450 - 1451, August, 1971. https://doi.org/10.1214/aoms/1177693260
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