weak consistency of the mle 1 point possible graded let rpr denote a statistical model associate

Step 1: The MLE (Maximum Likelihood Estimator) is a consistent estimator if it satisfies regular

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Weak Consistency of the MLE Let (R, {Pθ}θ∈R) denote a statistical model associated with a statistical experiment X1,...,Xn ∼ iid Pθ* for some true parameter θ* that we would like to estimate. You construct the maximum likelihood estimator θ̂MLEn for θ. Which of the following conditions is not necessary for the MLE θ̂MLEn to converge to θ in probability? The model (R, {Pθ}θ∈R) is identified. (Recall that the parameter θ is identified if the map θ ↦ Pθ is injective.) For all θ ∈ Θ, the support of Pθ does not depend on θ. The MLE θ̂MLEn is given by the sample average. The Fisher information I(θ) is non-zero in an interval containing the true parameter θ*. (Note that this is what it means for a 1x1 matrix, a scalar, to be invertible.)

          Weak Consistency of the MLE

Let (R, {Pθ}θ∈R) denote a statistical model associated with a statistical experiment X1,...,Xn ∼ iid Pθ* for some true parameter θ* that we would like to estimate. You construct the maximum likelihood estimator θ̂MLEn for θ*. Which of the following conditions is not necessary for the MLE θ̂MLEn to converge to θ* in probability?

The model (R, {Pθ}θ∈R) is identified. (Recall that the parameter θ is identified if the map θ ↦ Pθ is injective.)

For all θ ∈ Θ, the support of Pθ does not depend on θ.

The MLE θ̂MLEn is given by the sample average.

The Fisher information I(θ) is non-zero in an interval containing the true parameter θ*. (Note that this is what it means for a 1x1 matrix, a scalar, to be invertible.)

Added by Anthony C.

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