Geodesic
On estimation of a location parameter in presence of an ancillary component
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 1, pp. 172-176

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If $(X, Y)$ is an observation with distribution function $F(x-\theta,y)$, $\sigma^{2}=\textrm{var}(X)$, $\rho=\textrm{corr}(X,Y)$ and $I$ is the Fisher information on $\theta$ in $(X,Y)$, then $I\ge\{\sigma^2(1-\rho^2)\}^{-1}$. The equality sign holds under conditions closely related to the conditions for linearity of the Pitman estimator of $\theta$ from a sample from $F(x-\theta,y)$. The results are extensions of earlier results for the case when only the informative component $X$ is observed.
Keywords: Fisher information, Pitman estimator. 
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     title = {On estimation of a location parameter in presence of an ancillary component},
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A. M. Kagan; C. R. Rao. On estimation of a location parameter in presence of an ancillary component. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 1, pp. 172-176. http://geodesic.mathdoc.fr/item/TVP_2005_50_1_a11/