[Paper Review] A Bivariate Copula Additive Model for Location, Scale and Shape

Giampiero Marra, Rosalba Radice|arXiv (Cornell University)|May 24, 2016

Statistical Methods and Inference2 references3 citations

TL;DR

This paper introduces a bivariate copula additive model that extends GAMLSS to jointly model two continuous responses by allowing location, scale, and shape parameters of both margins and the copula dependence structure to be flexibly modeled via additive predictors. The method uses penalized likelihood estimation with automatic smoothing parameter selection, enabling simultaneous inference on marginal and dependence parameters using any parametric continuous distribution and copula family.

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.

Motivation & Objective

• To extend the GAMLSS framework to bivariate responses by incorporating copula-based dependence structures.
• To model location, scale, and shape parameters of both marginal distributions using additive predictors with flexible covariate effects.
• To estimate copula dependence parameters alongside marginal parameters in a unified, simultaneous framework.
• To enable the use of any parametric continuous marginal distribution and copula function within a unified estimation procedure.
• To provide a practical implementation via the copulaReg() function in the R package SemiParBIVProbit for applied researchers.

Proposed method

• Models the marginal distributions of two continuous responses using flexible parametric families, with location, scale, and shape parameters as additive predictors.
• Uses a copula function to model the dependence structure between the two responses, with dependence parameters also modeled via additive predictors.
• Employs a penalized likelihood approach with P-splines for smooth term estimation, allowing for non-linear and complex covariate effects.
• Applies a trust region algorithm for optimization, ensuring stable convergence during parameter estimation.
• Incorporates integrated automatic multiple smoothing parameter selection via generalized cross-validation or restricted maximum likelihood.
• Supports any parametric continuous marginal distribution and copula family (e.g., Gaussian, Clayton, Frank), enabling broad applicability.

Experimental results

Research questions

• RQ1 Can a unified model simultaneously estimate marginal distribution parameters and copula dependence parameters while allowing for flexible covariate effects?
• RQ2 How well can the proposed model capture non-linear and spatial effects in both marginal distributions and dependence structure?
• RQ3 What is the performance of the model in finite samples, particularly in comparison to standard bivariate models?
• RQ4 How effective is the automatic smoothing parameter selection in balancing model fit and complexity?
• RQ5 Can the model be practically applied to real-world data with complex dependence and non-linear covariate effects?

Key findings

• The proposed bivariate copula additive model successfully estimates all marginal and dependence parameters simultaneously using a unified penalized likelihood framework.
• The model achieves accurate estimation of non-linear and spatial effects in both margins and the copula dependence structure, as demonstrated in simulation studies.
• The automatic smoothing parameter selection procedure effectively balances model fit and complexity, reducing overfitting in finite samples.
• The method performs well on real data, as shown in two case studies: electricity price and demand, and birth records, where it captured complex dependence and covariate effects.
• The implementation via the copulaReg() function in the R package SemiParBIVProbit enables straightforward application by practitioners.
• The framework is flexible and general, supporting any parametric continuous marginal distribution and copula family, enhancing its practical utility.