6.4. Multiparameter Case: Estimation 393

2. For any sequence that satisfies (1),
   √
   n(̂θn−θ)→DNp( 0 ,I−^1 (θ)).

The proof of this theorem can be found in more advanced books; see, for example,
Lehmann and Casella (1998). As in the scalar case, the theorem does not assure that
the maximum likelihood estimates are unique. But if the sequence of solutions are
unique, then they are both consistent and asymptotically normal. In applications,
we can often verify uniqueness.
We immediately have the following corollary,

Corollary 6.4.1.LetX 1 ,...,Xnbe iid with pdff(x;θ)forθ∈Ω. Assume the reg-

ularity conditions hold. Let̂θnbe a sequence of consistent solutions of the likelihood
equation. Then̂θnare asymptotically efficient estimates; that is, forj=1,...,p,

√

n(̂θn,j−θj)→DN(0,[I−^1 (θ)]jj).

Letgbe a transformationg(θ)=(g 1 (θ),...,gk(θ))′such that 1≤k≤pand
that thek×pmatrix of partial derivatives

B=

[

∂gi

∂θj

]

,i=1,...k, j=1,...,p,

has continuous elements and does not vanish in a neighborhood ofθ.Let̂η=g(̂θ).
Then̂ηis the mle ofη=g(θ). By Theorem 5.4.6,

√

n(̂η−η)→DNk( 0 ,BI−^1 (θ)B′). (6.4.23)

Hence the information matrix for

√

n(̂η−η)is

I(η)=

[

BI−^1 (θ)B′

]− 1

, (6.4.24)

provided that the inverse exists.
For a simple example of this result, reconsider Example 6.4.3.

Example 6.4.6(Information for the Variance of a Normal Distribution).Suppose
X 1 ,...,Xnare iidN(μ, σ^2 ). Recall from Example 6.4.3 that the information matrix
wasI(μ, σ)=diag{σ−^2 , 2 σ−^2 }.Consider the transformationg(μ, σ)=σ^2. Hence
the matrix of partialsBis the row vector [0 2σ]. Thus the information forσ^2 is

I(σ^2 )=

{

[ 02 σ]

[ 1

σ^20

(^0) σ^22
]− 1 [
0
2 σ
]}−^1

1
2 σ^4
.
The Rao-Cram ́er lower bound for the variance of an estimator ofσ^2 is (2σ^4 )/n.
Recall that the sample variance is unbiased forσ^2 , but its variance is (2σ^4 )/(n−1).
Hence, it is not efficient for finite samples, but it is asymptotically efficient.