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B.2 Maximum Likelihood Estimation 387
x, the function L(θ) p(x; θ) on is called the likelihood function. A maximum
likelihood estimate ˆθ(x) of θ is a value of θ ∈ that maximizes the value of L(θ)
for the given observed value x, i.e.,
L ˆθ p x; ˆθ(x) max p(x; θ).
θ∈
Example B.2.1 If x (x1,...,xn) ′ is a vector of observations of independent N(µ, σ 2 ) random
variables, the likelihood function is
L µ, σ 2 1
2πσ2 n/2 exp
− 1
2σ 2
n
(xi − µ) 2
, −∞