Let $X = (X_1,\cdots, X_p)^t, p \geqq 3$, have density $f(x - \theta)$ with respect to Lebesgue measure. It is desired to estimate $\theta = (\theta_1,\cdots, \theta_p)^t$ under the loss $L(\delta - \theta)$. Assuming the problem has a minimax risk $R_0$, an estimator is defined to be tail minimax if its risk is no larger than $R_0$ outside some compact set. Under quite general conditions on $f$ and $L$, sufficient conditions for an estimator to be tail minimax are given. A class of good tail minimax estimators is then developed and compared with the best invariant estimator.
"Tail Minimaxity in Location Vector Problems and Its Applications." Ann. Statist. 4 (1) 33 - 50, January, 1976. https://doi.org/10.1214/aos/1176343346