Exact information losses with respect to the maximum likelihood estimator(MLE) in various models are hardly explored. The aim of this paper is to reconsider a multivariate Efron's information curvature from the viewpoint of the regression and to investigate an exact Fisher information loss with respect to MLE in a parameterization in the multivariate gamma distribution. The exact information loss was explicitly calculated in detail and the results were also a natural extension for the Nile problem.