Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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)