Bivariate copula additive models for location, scale and shape

Abstract

Rigby & Stasinopoulos (2005) introduced generalized additive models for location, scale and shape (GAMLSS) where the response distribution is not restricted to belong to the exponential family and its parameters can be specified as functions of additive predictors that allows for several types of covariate effects (e.g., linear, non-linear, random and spatial effects).In many empirical situations, however, modeling simultaneously two or more responses conditional on some covariates can be of considerable relevance. In this article, we extend the scope of GAMLSS by introducing a bivariate copula additive model with continuous margins for location, scale and shape. The framework permits the copula dependence and marginal distribution parameters to be estimated simultaneously and, like in GAMLSS, each parameter to be modeled using an additive predictor. Parameter estimation is achieved within a penalized likelihood framework using a trust region algorithm with integrated automatic multiple smoothing parameter selection. The proposed approach allows for straightforward inclusion of potentially any parametric continuous marginal distribution and copula function. The models can be easily used via the copulaReg() function in the R package SemiParBIVProbit.The usefulness of the proposal is illustrated on two case studies (which use electricity price and demand data, and birth records) and on simulated data.

A joint regression modeling framework for analyzing bivariate binary data in R

“…This approach follows the same rationale of [45], who introduced generalized additive models for location, scale and shape, where all the parameters characterizing a chosen distribution are related to predictors via suitable link functions. The copulae implemented in GJRM, corresponding ranges of θ and list of transformations m(·) are reported in Table 1 of [33]. Rotations by…”

“…. , Z νKν ) and [33] for other examples. The smoothing parameter λ νkν ∈ [ , ∞) controls the trade-o between t and smoothness, and plays a crucial role in determining the shape ofŝ νkν (z νkν i ).…”

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“…For instance, mixing the Clayton copula with its 90 degree (counter-clockwise) rotation allows one to model positive and negative tail dependence.The options available for the asymmetric copulae are: standard and rotated 90 degrees copulae, standard and rotated 270 degrees copulae, survival and rotated 90 degrees copulae and survival and rotated 270 degrees copulae (Marra and Radice, 2017). tions (Normal, logistic Figure 1 for some copula shapes 2 ). The marginal distribution parameters, π y and π x , are related to covariates and regression coefficients via link functions g:…”

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“…This paper, thus, reviews the available flexible regression models that meet these requirements. More specifically, in this manuscript, bivariate copula generalised additive models for location, scale, and shape (CGAMLSS) were considered, based on either frequentist or Bayesian inference principles. These types of models extend univariate generalised additive models for location, scale, and shape (GAMLSS), as well as univariate distributional regression, to the field of multivariate responses.…”

“…In addition, most of the existing multivariate distributions are simple extensions of corresponding univariate distributions and often suffer the restrictive property of all the marginal distributions being of the same type (by construction, all the marginal distributions of a multivariate normal are again normal). A major advantage of the copula approach is that the marginal distributions may belong to different nonstandard families . Moreover, copulas can address nonsymmetrical structures of dependencies rather than just those that are elliptical.…”