VGAM

VGAM stands for Vector Generalized Additive Models, a statistical framework that generalizes generalized linear and additive models to allow simultaneous modeling of multiple parameters of a distribution and, in some formulations, multiple response variables. In VGAM, a response distribution is specified, and linear predictors are linked to one or more of its parameters—such as location, scale, or shape—broadening the modeling scope beyond the mean alone. This framework unifies GLMs, GAMs, and more flexible location-scale-type models within a single coherent approach and supports modeling using both parametric and additive components, including smooth terms for continuous predictors.

Estimation in VGAM typically relies on maximum penalized likelihood, using iterative algorithms that extend generalized linear model techniques. Smoothing parameters for nonparametric terms are chosen through methods analogous to cross-validation or generalized cross-validation. The approach accommodates various distributions beyond the traditional exponential family and allows different link functions for different parameters, enabling flexible modeling of heteroscedasticity and distributional shape.

Software implementations are available in packages such as VGAM for R, developed to fit vector generalized additive models. The core fitting function, often designated to handle a range of families and link structures, supports a variety of distributions (for example gaussian, binomial, Poisson, beta, gamma, and more specialized forms) and can handle models where mean, dispersion, or other distributional parameters are modeled as functions of covariates. The VGAM framework is used in fields where both the location and scale or other distributional aspects vary with predictors, providing a unified approach to complex, multparameter modeling.

Limitations include computational intensity for large or highly parameterized models and the need for careful model selection to avoid overfitting or identifiability issues.