Distribution free testing for the family of Laplace distributions

The Khmaladze Transformation provides a general method for transforming a parametric empirical process into an asymptotically distribution free process. Distribution free goodness of fit statistics can then be obtained from this transformed process, i.e., their limit distribution does not depend on the parametric family under consideration. However, this transformation involves inversion of incomplete Fisher information matrices and if degeneracy is present it seems at first that the transformation is no longer well-defined. The Laplace families of distributions (i.e., symmetric and asymmetric Laplace) are practically important families where this apparent complication arises. We demonstrate that even though this component of the transformation appears problematic, the transformation is still well-defined. We derive the explicit form of this transformation for these families and demonstrate with simulations that goodness of fit statistics obtained, exhibit desired distribution freeness.