# Is the Hodges-Lehmann estimator 'optimal' for estimating the location parameter of Logistic distribution?

Is the Hodges-Lehmann estimator $$\hat\theta_{HL}=\operatorname{median}\limits_{1\le i\le j\le n}\left\{\frac{X_i+X_j}{2}\right\}$$ in some sense 'optimal' for estimating the location parameter $$\theta$$ in a $$\text{Logistic}(\theta,1)$$ distribution?

The asymptotic relative efficiency (ARE) of $$\hat\theta_{HL}$$ with respect to the sample mean $$\overline X_n$$ based on a sample of size $$n$$ is known to be $$\operatorname{ARE}(\hat\theta_{HL},\overline X_n)=\frac{\pi^2}9(>1)$$. On the other hand, ARE of the sample median $$\widetilde{X_n}$$ with respect to $$\overline X_n$$ is $$\operatorname{ARE}(\widetilde{X_n},\overline X)=\frac{\pi^2}{12}(<1)$$. This perhaps shows that $$\hat\theta_{HL}$$ is 'better' than sample median when compared to $$\overline X_n$$; it is certainly more efficient than $$\overline X_n$$ unlike the sample median.

If we consider the $$\text{Laplace}(\theta,1)$$ distribution, then $$\operatorname{ARE}(\hat\theta_{HL},\overline X_n)=1.5$$ whereas $$\operatorname{ARE}(\widetilde{X_n},\overline X_n)=2$$. Here sample median performs better than $$\hat\theta_{HL}$$ (both are more efficient than $$\overline X_n$$) and is perhaps optimal. I can somewhat justify this based on the fact that sample median is MLE of $$\theta$$.

The full set of order statistics is minimal sufficient for both the Logistic and Laplace location families, but I am not sure what a 'good' estimator for location looks like in the Logistic family.