The uniqueness of extremum estimation

Let W denote a family of probability distributions with parameter space Γ, and WG be a subfamily of W depending on a mapping G:Θ→Γ. Extremum estimations of the parameter vector ϑ∈Θ are considered. Some sufficient conditions are presented to ensure the uniqueness with probability one. As important applications, the maximum likelihood estimation in curved exponential families and nonlinear regression models with independent disturbances as well as the maximum likelihood estimation of the location and scale parameters of Gumbel distributions are treated.