Generalized additive model, Advanced Statistics

  • The linear component ηi, de?ned just in the traditional way: ηi = x'1
  • A monotone differentiable link function g that describes how E(Yi) = µi is related to the linear component:1590_Generalized additive model.png
  • Yi are independent (i = 1, 2, . . . , n) (but not identically distributed) and each has a distribution from the exponential family. For the purpose of Generalized Linear Models theory, the usual exponential family is parametrized in a bit more speci?c way. Let us focus for simplicity to the case where θ is one-dimensional. It is assumed that the probability density of the response Y can be expressed as

2165_Generalized additive model1.png

for some functions ai(Φ) = Φ/wi, b and ci = c(yi, Φ/wi), where wi is a known weight for each observation. Hereby, θi is called canonical parameter whereas Φ is called the dispersion parameter.

Note: For most models, the weight is the same for each i (that is, wi = 1 for all i) and then the scaling simpli?es to ai(Φ) = Φ), whereas ci(yi, Φ) becomes just c(yi, Φ) and (1) simpli?es to

1714_Generalized additive model2.png

It can be shown from a standard theory for these distributions that

1652_Generalized additive model3.png

Posted Date: 2/27/2013 12:45:32 AM | Location : United States

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