Uniform distribution width estimation from data observed with Laplace additive error

A one-dimensional problem of a uniform distribution width estimation from data observed with a Laplace additive error is analyzed. The error variance is considered as a nuisance parameter and it is supposed to be known or consistently estimated before. It is proved that the maximum likelihood estimator in the described model is consistent and asymptotically efficient and sufficient conditions for its existence are given. The method of moment estimator is also analyzed in this model and compared with the maximum likelihood estimator theoretically and in simulations. Finally, one real-world example illustrates the possibility for applications in two-dimensional problems.