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Generalized Linear Model Material Type: Notes; Professor: Bloomfield; Class: Nonlinear Statistical Models for Univariate and Multivariate Response; Subject: Statistics; University: North Carolina State University; Term: Unknown 1989;
Typology: Study notes
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Generalized Linear Model
The (Scaled) Exponential Family
f (y; ξ, σ) = exp
{ yξ − b(ξ) σ^2
} .
var(Y ) = σ^2 bξξ
( b− ξ 1 (μ)
) = σ^2 g(μ)^2 ,
so the variance depends on the mean in a specific way.
Distribution b(ξ) ξ(μ) g(μ)^2
Normal, σ^2 = 1 ξ^2 / 2 μ 1 Poisson exp(ξ) log μ μ Gamma − log(−ξ) 1 /μ μ^2 Inverse Gaussian −
− 2 ξ 1 /μ^2 μ^3 Binomial log
( 1 + eξ
) log (^1) −μμ μ(1 − μ)
Sufficiency
log L =
∑^ n
j=
[ Yjξ − b(ξ) σ^2
]
σ^2
ξ
∑^ n
j=
Yj − nb(ξ)
(^) +
∑^ n
j=
c
( Yj, σ
)
so (if σ^2 is known)
∑ Yj is sufficient for ξ.
E
( Yj
∣∣
) = f
(
) ,
and it is still called a generalized linear model.
lose sufficiency–not a big deal.
chosen from a list.
Example: Six Cities Wheezing data
Generalized Nonlinear Model
( Yj
∣∣
) = f
(
) .
( ξj
) = f
(
) .
var
( Yj
∣∣
) = σ^2 g
{ E
( Yj
∣∣
)} 2 = σ^2 g
{ f
(
)} 2 .
