Maximum leave-one-out likelihood estimation for location parameter of unbounded densities

Maximum likelihood estimation of a location parameter fails when the density have unbounded mode. An alternative approach is considered by leaving out a data point to avoid the unbounded density in the full likelihood. This modification give rise to the leave-one-out likelihood. We propose an ECM algorithm which maximises the leave-one-out likelihood. It was shown that the estimator which maximises the leave-one-out likelihood is consistent and super-efficient. However, other asymptotic properties such as the optimal rate of convergence and asymptotic distribution is still under question. We use simulations to investigate these asymptotic properties of the location estimator using our proposed algorithm.