On Nonparanormal Likelihoods

Abstract

Nonparanormal models describe the joint distribution of multivariate responses via latent Gaussian, and thus parametric, copulae while allowing flexible nonparametric marginals. Some aspects of such distributions, for example conditional independence, are formulated parametrically. Other features, such as marginal distributions, can be formulated non- or semiparametrically. Such models are attractive when multivariate normality is questionable but interpretability paramount. Most estimation procedures perform two steps, first estimating the nonparametric part. The copula parameters come second, treating the marginal estimates as known. This is sufficient for some applications. For other applications, e.g. when a semiparametric margin features parameters of interest or when standard errors are important, a simultaneous estimation of all parameters might be more advantageous. We present suitable parameterisations of nonparanormal models, possibly including semiparametric effects, and define four novel nonparanormal log-likelihood functions. In general, the corresponding one-step optimisation problems are shown to be non-convex. In some cases, however, biconvex problems emerge. Several convex approximations are discussed. From a low-level computational point of view, the core contribution is the score function for multivariate normal log-probabilities computed via Genz procedure. As a demonstration for the versatility of the theoretical and computational framework, we present a series of nonparanormal models for transformation discriminant analysis when some biomarkers are subject to limit-of-detection problems. Possible empirical gains of full maximum likelihood estimation compared to two-step approaches are illustrated in a simulation study targeting semiparametric efficient polychoric correlation analysis where a theoretical benchmark is available.