A) Let x1....xn be i.i.d. normal variable following the distribution , where is the mean and is the variance.

Question

Denote by u and t the maximum likelihood estimators of u and t respectively based on the i.i.d. observations .
Is the stimator 2u^2+t asimptoticaly normal
1Yes
2No
3Ninfo

(In our usual notation, . We use in this problem to make clear that the parameter being estimated is not .)

B) Let g(u,t)=2u^2+t and let be the Fisher information matrix of xi~N(u,t).

The asymptotic variance of 2u^2+t is...
1) \nabla g(\mu ,\tau )^ T \mathbf{I}(\mu ,\tau ) \nabla g(\mu ,\tau )
2)\nabla g(\mu ,\tau )^ T \left(\mathbf{I}(\mu ,\tau )\right)^{-1} \nabla g(\mu ,\tau )
3) \nabla g(\mu ,\tau )^ T \mathbf{I}(\mu ,\tau )
4)\nabla g(\mu ,\tau )^ T \left(\mathbf{I}(\mu ,\tau )\right)^{-1}

C)
Using the results from above and referring back to homework solutions if necessary, compute the asymptotic variance V(2u^2+t) of the estimator 2u^2+t .

Hint: The inverse of a diagonal matrix a 0
0 a
where is the diagonal matrix

1/a 0
1/b 0