Let (E,{Pθ}θ∈Θ) denote a statistical model associated to a statistical experiment X1,…,Xn∼iidPθ∗ where θ∗∈Θ is the true parameter. Assume that Θ⊂Rd for some d≥1 . Let mk(θ):=E[Xk] where X∼Pθ . mk(θ) is referred to as the k -th moment of Pθ . Also define the moments map: ψ:Θ →Rd
θ ↦(m1(θ),m2(θ),…,md(θ)).

Assume that ψ is one-to-one (and hence, invertible). What conditions on ψ do we have to assume so that the method of moments produces a consistent and asymptotically normal estimator?