On the Shrinkage Estimation of Variance and Pitman Closeness Criterion for Large Samples G´ erard Biau 1 Universit´ e Pierre et Marie Curie 2 & Ecole Normale Sup´ erieure 3 , France gerard.biau@upmc.fr Yannis G. Yatracos Cyprus University of Technology, Cyprus 4 yannis.yatracos@cut.ac.cy Abstract For a large class of distributions and large samples, it is shown that estimates of the variance σ 2 and of the standard deviation σ are more often Pitman closer to their target than the corresponding shrinkage estimates which improve the mean squared error. Our results indi- cate that Pitman closeness criterion, despite its controversial nature, should be regarded as a useful and complementary tool for the evalu- ation of estimates of σ 2 and of σ. Index Terms — Variance estimation, standard deviation, shrinkage, Pitman closeness, mean squared error. 2010 Mathematics Subject Classification : 62F10, 62C15. 1 Introduction Given two estimates ˆ θ 1 and ˆ θ 2 of an unknown parameter θ, Pitman [13] has suggested in 1937 that ˆ θ 1 should be regarded as a “closer” estimate of θ if P  | ˆ θ 2 - θ| > | ˆ θ 1 - θ|  > 1/2. 1 Corresponding author. 2 Research partially supported by the French National Research Agency under grant ANR-09-BLAN-0051-02 “CLARA”. 3 Research carried out within the INRIA project “CLASSIC” hosted by Ecole Normale Sup´ erieure and CNRS. 4 Research supported by a start-up grant from Cyprus University of Technology. 1