388 Maximum Likelihood Methods
where the second equality follows becauseb>0. Setting this partial to 0, we obtain
the mle ofato beQ 2 =med{X 1 ,X 2 ,...,Xn}, just as in Example 6.1.1. Hence the
mle ofais invariant to the parameterb. Taking the partial ofl(a, b) with respect
tob,weobtain
∂l(a, b)
∂b
=−
n
b
+
1
b^2
∑n
i=1
|xi−a|.
Setting to 0 and solving the two equations simultaneously, we obtain, as the mle of
b,thestatistic
̂b=^1
n
∑n
i=1
|Xi−Q 2 |.
Recall that the Fisher information in the scalar case was the variance of the
random variable (∂/∂θ)logf(X;θ). The analog in the multiparameter case is the
variance-covariance matrix of the gradient of logf(X;θ), that is, the variance-
covariance matrix of the random vector given by
logf(X;θ)=
(
∂logf(X;θ)
∂θ 1
,...,
∂logf(X;θ)
∂θp
)′
. (6.4.3)
Fisher information is then defined by thep×pmatrix
I(θ)=Cov(logf(X;θ)). (6.4.4)
The (j, k)th entry ofI(θ)isgivenby
Ijk=cov
(
∂
∂θj
logf(X;θ),
∂
∂θk
logf(X;θ)
)
; j, k=1,...,p. (6.4.5)
As in the scalar case, we can simplify this by using the identity 1 =
∫
f(x;θ)dx.
Under the regularity conditions, as discussed in the second paragraph of this section,
the partial derivative of this identity with respect toθjresults in
0=
∫
∂
∂θj
f(x;θ)dx =
∫ [
∂
∂θj
logf(x;θ)
]
f(x;θ)dx
= E
[
∂
∂θj
logf(X;θ)
]
. (6.4.6)
Next, on both sides of the first equality above, take the partial derivative with
respect toθk. After simplification, this results in
0=
∫ (
∂^2
∂θj∂θk
logf(x;θ)
)
f(x;θ)dx
+
∫(
∂
∂θj
logf(x;θ)
∂
∂θk
logf(x;θ)
)
f(x;θ)dx;
that is,
E
[
∂
∂θj
logf(X;θ)
∂
∂θk
logf(X;θ)
]
=−E
[
∂^2
∂θj∂θk
logf(X;θ)
]
. (6.4.7)