Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

6.1. Maximum Likelihood Estimation 357

Definition 6.1.1(Maximum Likelihood Estimator).We say that̂θ=θ̂(X)is a
maximum likelihood estimator(mle) ofθif

θ̂=ArgmaxL(θ;X). (6.1.4)

The notation Argmax means thatL(θ;X)achieves its maximum value atθ̂.

As in Chapter 4, to determine the mle, we often take the log of the likelihood
and determine its critical value; that is, lettingl(θ)=logL(θ), the mle solves the
equation
∂l(θ)
∂θ

=0. (6.1.5)

This is an example of anestimating equation, which we often label as an EE.

This is the first of several EEs in the text.

Example 6.1.1(Laplace Distribution). LetX 1 ,...,Xnbe iid with density

f(x;θ)=

1

2

e−|x−θ|, −∞<x<∞,−∞<θ<∞. (6.1.6)

ThispdfisreferredtoaseithertheLaplaceor thedouble exponential distribution.
The log of the likelihood simplifies to

l(θ)=−nlog 2−

∑n

i=1

|xi−θ|.

The first partial derivative is

l′(θ)=

∑n

i=1

sgn(xi−θ), (6.1.7)

where sgn(t)=1, 0 ,or−1 depending on whethert> 0 ,t=0,ort<0. Note that

we have useddtd|t|=sgn(t), which is true unlesst= 0. Setting equation (6.1.7) to 0,

the solution forθis med{x 1 ,x 2 ,...,xn}, because the median makes half the terms

of the sum in expression (6.1.7) nonpositive and half nonnegative. Recall that we

defined the sample median in expression (4.4.4) and that we denote it byQ 2 (the

second quartile of the sample). Hence,̂θ=Q 2 is the mle ofθfor the Laplace pdf

(6.1.6).

There is no guarantee that the mle exists or, if it does, it is unique. This is often
clear from the application as in the next two examples. Other examples are given
in the exercises.

Example 6.1.2(Logistic Distribution).LetX 1 ,...,Xnbe iid with density

f(x;θ)=

exp{−(x−θ)}

(1 + exp{−(x−θ)})^2

, −∞<x<∞,−∞<θ<∞. (6.1.8)