Abstract: Nonexistence of maximum likelihood estimation is a frequent phenomenon, typically occuring in nonnormal regression models. If the parameter vector cannot be estimated, it is of interest to estimate certain linear functions of the parameter vector. The definition of a generalized maximum likelihood estimation （G.M.L.E.） of the linear functions is given. The property of the G.M.L.E. is discussed. An approach of finding the linear functions having finite value of the G.M.L.E. is presented. Finally we give some examples, which are frequently encounted in quantal models, to demostrate how to get the G.M.L.E..