Revisiting the Hodges-Lehmann estimator in a location mixture model: is asymptotic normality good enough? (Q1684149)

The author considers the two-component location mixture model \[ F(x)=\pi G(x-\mu_1)+(1-\pi)G(x-\mu_2), \] where \(\pi\in [0,1], -\infty<\mu_1<\mu_2<+\infty,\) and \(G\) is a symmetric distribution around zero. The plug-in estimator \[ D^2_n(\theta)=\int_{R}[\pi(1-F_n(\mu_1-t)-F_n(\mu_1+t))+(1-\pi)(1-F_n(\mu_2-t)-F_n(\mu_2+t))]^2dt \] for this model is constructed, where \(F_n\) is the empirical distribution. Asymptotical properties of its estimator \(\hat\theta_n\) are studied. The author gives sufficient conditions on the symmetric distribution for asymptotic normality of \(\hat\theta_n\) to hold. In case the symmetric distribution admits a log-concave density, the paper's assumptions are automatically satisfied. As examples, three distributions are considered: standard Gaussian, the uniform on \([0,1]\) and the double exponential on \(R\) ones.