generalized linear model

Given a random vector, or the response variable, Y, a generalized linear model, or GLM for short, is a statistical model {f𝐘(𝒚𝜽)} such that

  1. 1.

    the componentsPlanetmathPlanetmath of Y are mutually independent of each other,

  2. 2.

    fYi(yiθi) belongs to the exponential family of distributions and has the following canonical form:


    where the parameter θi is called the canonical parameter and b(θi) is called the cumulant function.

  3. 3.

    for each component or variate Yi, with a corresponding set of p covariates Xij, there exists a monotoneMathworldPlanetmath differentiable function g, called the link function, such that


    where 𝐗iT=(Xi1,,Xip), and 𝜷=(β1,,βp)T is a parameter vector.

In practice, an extra parameter called the dispersion parameter, ϕ, is introducted to the model to lower a phenonmenon known as overdispersion. The GLM now looks like:



  • Below is a table of canonical parameters and cumulant functions for some well-known distributions from the exponential family:

    Title generalized linear model
    Canonical name GeneralizedLinearModel
    Date of creation 2013-03-22 14:30:11
    Last modified on 2013-03-22 14:30:11
    Owner CWoo (3771)
    Last modified by CWoo (3771)
    Numerical id 17
    Author CWoo (3771)
    Entry type Definition
    Classification msc 62J12
    Synonym GLM
    Defines link function
    Defines canonical parameter
    Defines cumulant function
    Defines variance function