Extending the rank likelihood for semiparametric copula estimation

Abstract

Quantitative studies in many fields involve the analysis of multivariate data of diverse types, including measurements that we may consider binary, ordinal and continuous. One approach to the analysis of such mixed data is to use a copula model, in which the associations among the variables are parameterized separately from their univariate marginal distributions. The purpose of this article is to provide a simple, general method of semiparametric inference for copula models via a type of rank likelihood function for the association parameters. The proposed method of inference can be viewed as a generalization of marginal likelihood estimation, in which inference for a parameter of interest is based on a summary statistic whose sampling distribution is not a function of any nuisance parameters. In the context of copula estimation, the extended rank likelihood is a function of the association parameters only and its applicability does not depend on any assumptions about the marginal distributions of the data, thus making it appropriate for the analysis of mixed continuous and discrete data with arbitrary marginal distributions. Estimation and inference for parameters of the Gaussian copula are available via a straightforward Markov chain Monte Carlo algorithm based on Gibbs sampling. Specification of prior distributions or a parametric form for the univariate marginal distributions of the data is not necessary. Some key words: Bayesian inference, latent variable model, marginal likelihood, Markov chainMonte Carlo, multivariate estimation, polychoric correlation, rank likelihood, sufficiency.

Copula Eigenfaces with Attributes: Semiparametric Principal Component Analysis for a Combined Color, Shape and Attribute Model

“…In order to resolve this problem, we follow the approach of the extended rank likelihood [8]. This provides us with an association-preserving mapping between measurement x ij and latent observationx ij .…”

Joint Bayesian Modeling of Binomial and Rank Data for Primate Cognition

“…Hoff (2007) used ranks to estimate copula, thereby allowing the marginal distributions in multivariate data to be unspecified while still modeling dependence. Murray et al (2013) built on Hoff (2007) by forming a Gaussian copula factor model to jointly handle rank response and other response types. While these approaches permit ties by considering the data to be only partially ordered, they do not provide a model for the probability that two outcomes will be tied.…”

Mixed Density Based Copula Likelihood

“…Both papers use Markov Chain Monte Carlo (MCMC) techniques. Hoff (2007) proposes a semi-parametric multivariate gaussian copula, where the marginal distributions are left unspecified. The only information he uses from the observed data to estimate the multivariate copula consists of the order statistics.…”

“…We benchmark our results against the approach of Hoff (2007) and the most commonly employed Pearson correlation measure. Over various sample sizes, we show how our method produces similar result to Hoff's method and yet does not require the computational burden of a Bayesian approach.…”