5L (Statistical Modelling)

a random component in which independent responses belong to an exponential dispersion family;
a systematic component $η_{i} = x_{i}^{T} β$ ;
a link function satisfying $g (μ_{i}) = η_{i}$ , where $μ_{i} = E Y_{i}$ .

b

Write the exponential-dispersion density as

f (y_{i}; θ_{i}, ϕ) = exp {\frac{y _{i} θ _{i} - b ( θ _{i} )}{ϕ} + c (y_{i}, ϕ)} .

With the canonical link,

θ_{i} = η_{i} = x_{i}^{T} β

. Therefore

E Y_{i} = b^{'} (x_{i}^{T} β), Var (Y_{i}) = ϕ b^{''} (x_{i}^{T} β) .

Known prior weights replace

ϕ

ϕ / w_{i}

In the Poisson GLM the responses are independent and

Y_{i} \sim Poisson (μ_{i}) .

The Poisson canonical link is logarithmic, so

lo g μ_{i} = x_{i}^{T} β, μ_{i} = e^{x_{i}^{T} β}, Var (Y_{i}) = μ_{i} .

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