an introduction to generalized linear models - GDM@FUDAN ...

|An| = det A are all positive.
4. The rank ofthe matrix A is also called the degrees offreedom ofthe
quadratic form Q = y T Ay.
5. Suppose Y1, ..., Yn are independent random variables each with the Normal
distribution N(0,σ2 ). Let Q = �n 2
i=1 Yi and let Q1, ..., Qk be quadratic
forms in the Yi’s such that
Q = Q1 + ... + Qk
where Qi has mi degrees offreedom (i =1,... ,k). Then
Q1, ..., Qk are independent random variables and
Q1/σ 2 ∼ χ 2 (m1), Q2/σ 2 ∼ χ 2 (m2), ···and Qk/σ 2 ∼ χ 2 (mk),
ifand only if,
m1 + m2 + ... + mk = n.
This is Cochran’s theorem; for a proof see, for example, Hogg and Craig
(1995). A similar result holds for non-central distributions; see Chapter 3
ofRao (1973).
6. A consequence ofCochran’s theorem is that the difference oftwo independent
random variables, X 2 1 ∼ χ 2 (m) and X 2 2 ∼ χ 2 (k), also has a chi-squared
distribution
provided that X 2 ≥ 0 and m>k.
1.6 Estimation
1.6.1 Maximum likelihood estimation
X 2 = X 2 1 − X 2 2 ∼ χ 2 (m − k)
Let y =[Y1, ..., Yn] T denote a random vector and let the joint probability
density function of the Yi’s be
f(y; θ)
which depends on the vector ofparameters θ =[θ1, ..., θp] T .
The likelihood function L(θ; y) is algebraically the same as the joint
probability density function f(y; θ) but the change in notation reflects a shift
ofemphasis from the random variables y, with θ fixed, to the parameters θ
with y fixed. Since L is defined in terms ofthe random vector y, it is itselfa
random variable. Let Ω denote the set ofall possible values ofthe parameter
vector θ; Ω is called the parameter space. The maximum likelihood
estimator of θ is the value � θ which maximizes the likelihood function, that
is
L( � θ; y) ≥ L(θ; y) for all θ in Ω.
Equivalently, � θ is the value which maximizes the log-likelihood function
© 2002 by Chapman & Hall/CRC