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In relation to the following URL question, I would like to consider a proof for the one-sample case. https://stats.stackexchange.com/q/501493/401056

Definition

Consider the median of the average $$ \theta_{HL} = \mathrm{med}_{i \leq j} \left( \frac{X_i+X_j}{2}\right). $$ If the $X_j$ are I.I.D. according to a distribution $F(x-\theta)$ where $F$ has a density $f$ and is symmetric about $0$.

Theorem

$$ \sqrt n (\theta_{HL}-\theta) \overset{d}{\to} N\left(0, \frac{1}{12\left[\int f^2(x)dx \right]^2} \right) $$

Lehmann (1999) omits the proof. Can someone provide a proof or a hint?

ytnb
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    Lehmann give directions for a proof in Example 4.3.7, he even provides 2 references. Have you tried that ? – Gabriel Romon Nov 21 '23 at 10:13
  • Yes, I referred to “Theory of Point Estimation”(p.382-384) and “Statistical Inference Based on Ranks”(p.102), but the H-L estimator is only mentioned as an example of the R-estimator, and I did not understand the general theory of the R-estimator at all. I would like a direct explanation like in Ex.2.4.9. – ytnb Nov 21 '23 at 10:37

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