Revisiting the Hodges-Lehmann estimator in a location mixture model: Is asymptotic normality good enough?

We derive the exact limit distribution of the Hogdes–Lehmann estimator of [1] which they considered in the semi-parametric model of a location mixture of symmetric distributions. We give sufficient conditions on the true symmetric component for the weak convergence to hold. As already expected by [1], the limit distribution is that of a three-dimensional centered Gaussian distribution. The variance–covariance matrix can be calculated using the known covariance structure of a standard Brownian Bridge. The examples we used to illustrate the theory indicate that the estimator is not to be advocated when the mixture components are not well separated.