Question

4.
Let X_1…X_n be i.i.d. normal variable following the distribution N(u, τ) , where u is the mean and τ is the variance.
Denote by u and τ the maximum likelihood estimators of u and τ respectively based on the i.i.d. observations X_1….X_n.
(In our usual notation, τ=σ^2. We use τ in this problem to make clear that the parameter being estimated is σ^2 not σ .)

(a)
Is the estimator 2(ῡ)^2+Ť of 2(ῡ)^2+T asymptotically normal?
Y
N
Let g(u,T)=2u^2+T and let I be the Fisher information matrix of X_i ~ N(u,T) the asymptotic variance of 2(ῡ)^2 + T is:

1. ∇ (u, T)^Tr I (u, T)) ∇ g(u, T)
2. ∇ g(u,T)^Tr(I(u,T))^-1 ∇ g(u,T)
3. ∇ g(u,T)^Tr I (u,T)
4. ∇ (u,T)^Tr(I(u,T))^-1

Using the results from above and referring back to homework solutions if necessary, compute the asymptotic variance V(2(ῡ)^2+ Ť of the estimator (2(ῡ)^2+ Ť .
Hint: The inverse of a diagonal matrix [a,0:0,b] where a,b ≠0 is the diagonal matrix [1/a, 0 : 0,1/b]
V(2(ῡ)^2)+ Ť=?

Answer 1

a yes

in the nablas is the numer 2

the last is 16*mu^2*tau+2*tau^2