It is shown that for Euclidean parameter spaces every sequence of Pitman estimates has local asymptotic minimax properties. The result generalizes a previous result of Hajek, 1972, which has been proved under the condition of local asymptotic normality. In the present paper it is only assumed that a sequence of experiments En, neℕ, converges weakly to a translation invariant limit experiment. According to LeCam, 1973 b, this is nearly the most general case which may occur. There are two main results. The first result states that Pitman estimates are minimax for translation invariant experiments. This improves a theorem of Girshick and Savage, 1951, which is restricted to location parameter experiments. In the second result we prove that the distributions of Pitman estimates for En, neℕ, converge weakly to the distribution of the Pitman estimate for the limit experiment. This improves previous assertions of this kind due to Ibragimov and Has'minskii, 1973, Inagaki and Ogata, 1975, or Grosmann, 1979, since the condition of weak convergence of experiments used here is considerably weaker than the invariance principles for likelihood processes used by these authors.