> [!tldr] Hodges-Lehmann Estimators > > Hodges-Lehmann estimators derives a point estimate from a statistical test, without assuming the population distribution. > [!definition|*] Hodges-Lehmann Estimators > Given a hypothesis test about a parameter $\theta$ with hypotheses $H_0, H_1$, the **Hodges-Lehmann estimator** of $\theta$ is $\tilde{\theta}$ that makes the test least significant against $H_{0}:\theta=\tilde{\theta}$: $\tilde{\theta}=\underset{t}{\arg\min}~\mathbb{P}[H_{0}:\theta=t \text{ is rejected} ~|~ H_{0}]$ > > For tests using p-values $p_{H_{0}}$, the estimator is then $\tilde{\theta}=\underset{t}{\arg\max}\,p_{\{ H_{0}:\theta=t \}}.$ ^dfb102 For example, to estimate the location $\tilde{\Delta}$ from a p-value-based test, the HL estimator is $\tilde{\Delta}:= \underset{\Delta_{0}}{\arg\max} \,\,\mathbb{P}[\text{as extreme as observed }\,|\,\Delta=\Delta_{0}].$ - In a two-sample $(\mathbf{X}, \mathbf{Y})$ z-test, the HL estimator is $\tilde{\Delta}=\bar{Y} - \bar{X}$.