Question

Weak Consistency of the MLE

1 point possible (graded)

Let (R,{Pθ}θ∈R) denote a statistical model associated to a statistical experiment X1,…,Xn∼iidPθ∗ for some true parameter θ∗ that we would like to estimate. You construct the maximum likelihood estimator θˆMLEn for θ∗. Which of the following conditions is not necessary for the MLE θˆMLEn 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 θˆMLEn is given by the sample average.

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

unanswered

Answer 1

The ML estimate of a parameter is defined as the point estimate at which likelihood function attains maxima. The MLEs may or may not be unbiased estimators of the parameter but MLE is always a consistent estimator. In other words, the consistent property of MLE states that " with probability approaching unity as n tends to infinity, the likelihood equation (del by del theta* of logL) =. (dlogL/d )=0, has solution which converges in probability to the true value theta* ". This property is also called cramer-rao theorem.MLE holds consistency property if it satisfies what we call as "regularity conditions". So the possible option among them which is not necessarily required to prove for consistency of MLE can be-

Since the range of integration should be independent of theta otherwise f(x,theta) will vanish at extreme points depending on theta, so 2nd property is important property.

Fisher's information function is expectation of second order derivative of logL so is a positive value, also it must be continuous function of theta in the range R. Thus 4th property is important property.

1st statement is important property for density function to hold .

So the right option is option 3 as not necessary sample average is mle of parameter ,for example in case of uniform distribution for (theta-1/2, theta +1/2), sample average is not mle, but the order statistic function is .

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