STAT 330 Lecture, July 12th, 2012
(a) θ and Θ;
(c) Estimator and estimate of θ;
(d) We use θ for both estimator and estimate.
2. Method of Moment (MM)
(b) MM estimator of τ(θ):
3. Maximum likelihood method
(a) Likelihood function:
1 Maximum likelihood estimate/estimator
The the maximum likelihood (ML) estimate of θ, denoted by θ = θ(x ,...,1 ) maxnmizes
L(θ). That is,
θ = θ(x 1...,x n = argmaxL(θ).
The corresponding ML estimator is θ = θ(X ,...,X ).
The log-likelihood function is deﬁned as l(θ) = logL(θ), where log is the natural logarithmic
function. The ML estimate of θ also maximizes l(θ). That is,
θ = θ(x1,...,x n = argmaxL(θ) = argmaxl(θ).
Estimating τ(θ) The ML estimator of τ(θ) is just τ(θ).ˆ
Example 2. Suppose X ,...1X are ind rvs from (a) Poi(θ) (b) Exp(θ) (try by yourself)
(c) N(µ,σ ) (d) Unif(0,θ), ﬁnd the ML estimator of θ in each case.