generalized linear model

Given a random vector, or the response variable, Y, a generalized linear model, or GLM for short, is a statistical model $\{f_{\text{𝐘}}(𝒚\mid 𝜽)\}$ such that

1. the components of Y are mutually independent of each other,

2. $f_{Y_i}(y_i\mid {\theta }_i)$ belongs to the exponential family of distributions and has the following canonical form:

   $$f_{Y_i}(y_i\mid {\theta }_i)=\exp [y{\theta }_i-b({\theta }_i)+c(y)],$$

   where the parameter ${\theta }_i$ is called the canonical parameter and $b({\theta }_i)$ is called the cumulant function.

3. for each component or variate $Y_i$, with a corresponding set of $p$ covariates $X_{ij}$, there exists a monotone differentiable function $g$, called the link function, such that

   $$g(\mathrm{E}[Y_i])=\text{𝐗}_i{}^{\mathrm{T}}𝜷,$$

   where $\text{𝐗}_i{}^{\mathrm{T}}=(X_{i1},\mathrm{\dots },X_{ip})$, and $𝜷={({\beta }_1,\mathrm{\dots },{\beta }_p)}^{\mathrm{T}}$ is a parameter vector.

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

$$f_{Y_i}(y_i\mid {\theta }_i)=\exp [\frac{y{\theta }_i-b({\theta }_i)}{a(\varphi )}+c(y,\varphi )]$$

Remarks

• 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