On asymptotic behavior of the powers of the tests for the case of Laplace distribution

A formula for the limit of the normalized difference between the power of the asymptotically most powerful test and the power of the asymptotically optimal test for the case of Laplace distribution was proved. Due to the nonregularity of the Laplace distribution, the logarithm of the likelihood ratio admits nonregular stochastic expansion, and an analog of Cram‚er condition is not valid for the sign statistic which is the basis of the asymptotically optimal test. Then direct use of theorem 3.2.1 from [1] or theorem 2.1 from [2] is difficult, and in the present paper, their proofs for the case of Laplace distribution are revisited.