6.4. Multiparameter Case: Estimation 387

we have labeled these (R6)–(R9). In this section, when we say “under regularity
conditions,” we mean all of the conditions of (6.1.1), (6.2.1), (6.2.2), and (A.1.1)
that are relevant to the argument. The discrete case follows in the same way as the
continuous case, so in general we state material in terms of the continuous case.
Note that the proof of Theorem 6.1.1 does not depend on whether the parameter
is a scalar or a vector. Therefore, with probability going to 1,L(θ) is maximized at
thetruevalueofθ. Hence, as an estimate ofθwe consider the value that maximizes
L(θ) or equivalently solves the vector equation (∂/∂θ)l(θ)= 0. If it exists, this
value is called themaximum likelihood estimator(mle) and we denote it bŷθ.
Often we are interested in a function ofθ, say, the parameterη=g(θ). Because the
second part of the proof of Theorem 6.1.2 remains true forθas a vector,̂η=g(̂θ)
is the mle ofη.

Example 6.4.1(Maximum Likelihood Estimates Under the Normal Model).Sup-
poseX 1 ,...,Xnare iidN(μ, σ^2 ). In this case,θ=(μ, σ^2 )′and Ω is the product
space (−∞,∞)×(0,∞). The log of the likelihood simplifies to

l(μ, σ^2 )=−

n

2

log 2π−nlogσ−

1

2 σ^2

∑n

i=1

(xi−μ)^2. (6.4.2)

Taking partial derivatives of (6.4.2) with respect toμandσand setting them to 0,
we get the simultaneous equations

∂l

∂μ

=

1

σ^2

∑n

i=1

(xi−μ)=0

∂l

∂σ

= −

n

σ

+

1

σ^3

∑n

i=1

(xi−μ)^2 =0.

Solving these equations, we obtain̂μ=Xand̂σ=

√

(1/n)

∑n

i=1(Xi−X)

(^2) as
solutions. A check of the second partials shows that these maximizel(μ, σ^2 ), so
these are the mles. Also, by Theorem 6.1.2, (1/n)
∑n
i=1(Xi−X)
(^2) is the mle ofσ (^2).
We know from our discussion in Section 5.1 that these are consistent estimates of
μandσ^2 , respectively, thatμ̂is an unbiased estimate ofμ,andthatσ̂^2 is a biased
estimate ofσ^2 whose bias vanishes asn→∞.
Example 6.4.2(General Laplace pdf). LetX 1 ,X 2 ,...,Xnbe a random sample
from the Laplace pdffX(x)=(2b)−^1 exp{−|x−a|/b},−∞<x<∞,wherethe
parameters (a, b)areinthespaceΩ={(a, b):−∞<a<∞,b > 0 }. Recall in
Section 6.1 that we looked at the special case whereb= 1. As we now show, the
mle ofais the sample median, regardless of the value ofb. The log of the likelihood
function is
l(a, b)=−nlog 2−nlogb−
∑n
i=1
∣
∣
∣
∣
xi−a
b
∣
∣
∣
∣.
The partial ofl(a, b) with respect toais
∂l(a, b)
∂a

1
b
∑n
i=1
sgn
{
xi−a
b
}

1
b
∑n
i=1
sgn{xi−a},