Extending the rank likelihood for semiparametric copula estimation

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 method of semiparametric inference for copula models via the construction of what we call a marginal set 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 marginal set 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 margins. Estimation and inference for parameters of the Gaussian copula are available via a straightforward Markov chain Monte Carlo algorithm based on Gibbs sampling.

[1]  Sufficient Statistics with Nuisance Parameters , 1956 .

[2]  G. Barnard.  Some Logical Aspects of the Fiducial Argument , 1963 .

[3]  P. Sen,et al.  Theory of rank tests , 1969 .

[4]  B. M. Hill,et al.  Posterior Distribution of Percentiles: Bayes' Theorem for Sampling From a Population , 1968 .

[5]  L. Waite,et al.  Early childbearing and educational attainment. , 1977, Family planning perspectives.

[6]  Ulf Olsson,et al.  Maximum likelihood estimation of the polychoric correlation coefficient , 1979 .

[7]  Anthony N. Pettitt,et al.  Inference for the Linear Model Using a Likelihood Based on Ranks , 1982 .

[8]  Marcel Rémon,et al.  On a concept of partial sufficiency : L-sufficiency , 1984 .

[9]  J. Blake.  Number of siblings and educational mobility , 1985 .

[10]  B. Arnold,et al.  Compatible Conditional Distributions , 1989 .

[11]  Nancy Reid,et al.  Parametric Statistical Models and Likelihood. , 1990 .

[12]  Kurt Weichselberger.  Axiomatic foundations of the theory of interval-probability , 1995 .

[13]  C. Genest,et al.  A semiparametric estimation procedure of dependence parameters in multivariate families of distributions , 1995 .

[14]  C. Klaassen,et al.  Efficient estimation in the bivariate normal copula model: normal margins are least favourable , 1997 .

[15]  L-Nonformation, L-Ancillarity, and L-Sufficiency , 2000 .

[16]  D. Dunson,et al.  Bayesian latent variable models for clustered mixed outcomes , 2000 .

[17]  T. Severini.  Likelihood Methods in Statistics , 2001 .

[18]  M. Francesconi,et al.  Family Matters: Impacts of Family Background on Educational Attainments , 2001 .

[19]  G. Heller,et al.  Pairwise Rank‐Based Likelihood for Estimation and Inference on the Mixture Proportion , 2001, Biometrics.

[20]  Eric C. Newburger,et al.  The Big Payoff: Educational Attainment and Synthetic Estimates of Work-Life Earnings. Special Studies. Current Population Reports. , 2002 .

[21]  John M. Olin.  Markov Chain Monte Carlo Analysis of Correlated Count Data , 2003 .

[22]  P. Müller,et al.  Nonparametric Bayesian Modeling for Multivariate Ordinal Data , 2005 .

[23]  M. Pitt,et al.  Efficient Bayesian inference for Gaussian copula regression models , 2006 .

[24]  R. Hopcroft.  Sex, status, and reproductive success in the contemporary United States , 2006 .

[25]  Peter Congdon,et al.  Wiley Series in Probability and Statistics , 2006 .