On the shrinkage estimation of variance

Biau, Gérard; Yatracos, Yannis G.

Journal de la société française de statistique, Tome 153 (2012) no. 1, pp. 5-21.

Nous montrons dans cet article que certains estimateurs de la variance ${\sigma }^2$ et de l’écart type $\sigma$ sont plus souvent proches de leur cible au sens de Pitman que les estimateurs correspondants obtenus par “shrinkage”, pourtant connus pour améliorer l’erreur quadratique moyenne. Nos résultats sont valables asymptotiquement et pour une grande famille de lois de probabilité. Ils indiquent en particulier que le critère de proximité de Pitman, malgré sa nature controversée, devrait être envisagé comme un outil utile à l’évaluation de la qualité des estimateurs de ${\sigma }^2$ et de $\sigma$.

For a large class of distributions and large samples, it is shown that estimates of the variance ${\sigma }^2$ and of the standard deviation $\sigma$ are more often Pitman closer to their target than the corresponding shrinkage estimates which improve the mean squared error. Our results indicate that Pitman closeness criterion, despite its controversial nature, should be regarded as a useful and complementary tool for the evaluation of estimates of ${\sigma }^2$ and of $\sigma$.

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