4. Let X1…Xn be i.i.d. normal variable following the distribution N(u,t) where u is the mean and t is the variance.
Denote by u and t the maximum likelihood estimators of and respectively based on the i.i.d. observations .
(In our usual notation, t=o^2 . We use t in this problem to make clear that the parameter being estimated is o^2 not o.
a)
Is the estimator 2u^2+t de 2u^2+1 asimptotically normal.
Yes
No
Let
g(u,t)=2u^2+t
and let be the Fisher information matrix of X1~N(u,t)
The asymptotic variance of is...
• Nabla g(u,t)^T I(u,t) nabla(u,t)
•Nabla g(u,t)^T (I(u,t)^-1 Nabla g(u,t)
•Nabla g(u,t)^T I(u,t)
•Nabla g(u,t)^T(I(u,t))
c. Using the results from above and referring back to homework solutions if necessary, compute the asymptotic variance of the estimator .
Hint: The inverse of a diagonal matrix where is the diagonal matrix .
V(2u^2+t) or the estimator 2u^2+t
Usar la inversa de (a 0, 0 a) es (1/a 0, 0 1/a)
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