About the power of asymptotically optimal test in the case of Laplace distribution

In the present paper we heuristically obtain a formula (see (3.1)) for the limit of the difference between the power of the asymptotically optimal test and the power of the asymptotically most powerful (AMP) test for the case of Laplace distribution. This difference has the order $n^{-1/2}$ due to the nonregularity of the Laplace distribution, in contrast to regular laws for which this order equals $n^{-1}$.