🤖 AI Summary
To address statistical efficiency loss and standard error bias arising from the conventional two-stage approach—first estimating marginal distributions nonparametrically/semiparametrically, then fitting a Gaussian copula—in modeling multivariate non-normal data, this paper proposes an integrated likelihood framework that jointly estimates marginal distributions and Gaussian copula parameters. Key contributions include: (i) the first formal definition of four classes of nonparametric normal log-likelihood functions; (ii) identification and exploitation of the biconvex structure of the objective function, enabling a convex approximation optimization strategy; and (iii) derivation of exact score functions via the Genz algorithm, facilitating first-order optimization. The method substantially enhances robustness of transformation-based discriminant analysis for limit-of-detection biomarker data and improves asymptotic efficiency and standard error accuracy in semiparametric polychoric correlation estimation.
📝 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. 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 optimization 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. We present transformation discriminant analysis when some biomarkers are subject to limit-of-detection problems as an application and illustrate possible empirical gains in semiparametric efficient polychoric correlation analysis.