In statistics, the class of vector generalized linear models (VGLMs) was proposed to enlarge the scope of models catered for by generalized linear models (GLMs). In particular, VGLMs allow for response variables outside the classical exponential family and for more than one parameter. Each parameter (not necessarily a mean) can be transformed by a link function. The VGLM framework is also large enough to naturally accommodate multiple responses; these are several independent responses each coming from a particular statistical distribution with possibly different parameter values. Vector generalized linear models are described in detail in Yee (2015). The central algorithm adopted is the iteratively reweighted least squares method, for maximum likelihood estimation of usually all the model parameters. In particular, Fisher scoring is implemented by such, which, for most models, uses the first and expected second derivatives of the log-likelihood function.
Motivation GLMs essentially cover one-parameter models from the classical exponential family, and include 3 of the most important statistical regression models: the linear model, Poisson regression for counts, and logistic regression for binary responses. However, the exponential family is far too limiting for regular data analysis. For example, for counts, zero-inflation, zero-truncation and overdispersion are regularly encountered, and the makeshift adaptations made to the binomial and Poisson models in the form of quasi-binomial and quasi-Poisson can be argued as being ad hoc and unsatisfactory. But the VGLM framework readily handles models such as zero-inflated Poisson regression, zero-altered Poisson (hurdle) regression, positive-Poisson regression, and negative binomial regression. As another example, for the linear model, the variance of a normal distribution is relegated as a scale parameter and it is treated often as a nuisance parameter (if it is considered as a parameter at all). But the VGLM framework allows the variance to be modelled using covariates. As a whole, one can loosely think of VGLMs as GLMs that handle many models outside the classical exponential family and are not restricted to estimating a single mean. During estimation, rather than using weighted least squares during IRLS, one uses generalized least squares to handle the correlation between the M linear predictors.
Data and notation We suppose that the response or outcome or the dependent variable(s), y = ( y 1 , … , y Q 1 ) T {\displaystyle {\boldsymbol {y}}=(y_{1},\ldots ,y_{Q_{1}})^{T}} , are assumed to be generated from a particular distribution. Most distributions are univariate, so that Q 1 = 1 {\displaystyle Q_{1}=1} , and an example of Q 1 = 2 {\displaystyle Q_{1}=2} is the bivariate normal distribution. Sometimes we write our data as ( x i , w i , y i ) {\displaystyle ({\boldsymbol {x}}_{i},w_{i},{\boldsymbol {y}}_{i})}
for i = 1 , … , n {\displaystyle i=1,\ldots ,n} . Each of the n observations are considered to be independent. Then y i = ( y i 1 , … , y i Q 1 ) T {\displaystyle {\boldsymbol {y}}_{i}=(y_{i1},\ldots ,y_{iQ_{1}})^{T}} . The w i {\displaystyle w_{i}} are known positive prior weights, and often w i = 1 {\displaystyle w_{i}=1} . The explanatory or independent variables are written x = ( x 1 , … , x p ) T {\displaystyle {\boldsymbol {x}}=(x_{1},\ldots ,x_{p})^{T}} , or when i is needed, as x i = ( x i 1 , … , x i p ) T {\displaystyle {\boldsymbol {x}}_{i}=(x_{i1},\ldots ,x_{ip})^{T}} . Usually there is an intercept, in which case
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