Information Criterion for the Gaussian and/or Laplace Distribution Models

The information criterion AIC has been used successfully in many areas of statistical modeling, and since it is derived based on the Taylor expansion of the log-likelihood function and the asymptotic distribution of the maximum likelihood estimator, it is not directly justified for likelihood functions that include non-differentiable points such as the Laplace distribution. In fact, it is known to work effectively in many such cases. In this paper, we attempt to evaluate the bias correction directly for the case where the true model or the model to be estimated is a simple Laplace distribution model. As a result, an approximate expression for the bias correction term was obtained. Numerical results show that the AIC approximations are relatively good except when the Gauss distribution model is fitted to data following the Laplace distribution.