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Method of Moments Concept Question II 1 point possible (graded) Let (E,{Pθ}θ∈Θ) denote a statistical model...

Method of Moments Concept Question II

1 point possible (graded)

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)).

What conditions on ψ do we have to assume so that the method of moments produces a consistent and asymptotically normal estimator? (Choose all that apply.)

Recall that the method of moments estimator is

θˆMMn:=ψ−1(1n∑i=1nXi,1n∑i=1nX2i,…,1n∑i=1nXdi)

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