On Estimation of a Location Parameter in the Presence of an Ancillary Component

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If $(X, Y)$ is an observation with distribution function $F(x-\theta,y)$, $\sigma^{2}={\rm var}(X)$, $\rho={\rm corr}(X,Y)$ 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.