Question

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) |

Answer #1

Confidence interval Concept Check
1 point possible (graded)
As in the previous section, let X1,…,Xn∼iidexp(λ). Let
λˆn:=n∑ni=1Xi
denote an estimator for λ. We know by now that λˆn is a
consistent and asymptotically
normal estimator for λ.
Recall qα/2 denote the 1−α/2 quantile of a standard Gaussian. By
the Delta method:
λ∈[λˆn−qα/2λn−−√,λˆn+qα/2λn−−√]=:I
with probability 1−α. However, I is still not a
confidence interval for λ.
Why is this the case?

Step 1: Moments Map for an Exponential Statistical Model
Let ?1,…,??∼Exp(?∗) denote a
statistical experiment where ?∗ is the true, unknown
parameter. You construct the associated statistical model
((0,∞),{Exp(?)}?∈(0,∞)). Since the parameter ? is
one-dimensional, we only consider the first moment with moment
map:
? : ℝ →ℝ
? ↦?1(?) := ?[?], (?∼Exp(?)).
What is ?(?)?
What is ?−1(?1)?

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