Estimation of nonparanormal graphical models based on ranked set sampling (RSS)

Abstract

Gaussian graphical model, which assumes that the variables of interest jointly follow a multivariate normal distribution with a sparse precision matrix, has been widely used to study the intrinsic dependence between the variables. However, we often encounter non-normal data, and we use variable transformation to achieve normality. Motivated by sparse additive models, the nonparanormal model extends the Gaussian graphical model to semiparametric Gaussian copula models, in which it is assumed that the variables follow a joint multivariate normal distribution after a set of unknown smooth monotone transformations. The basic methods of estimating undirected graphs in high-dimensional settings apply the simple random sampling (SRS) method to estimate the parameters of the model. We consider the ranked set sampling (RSS) approach for estimating the nonparanormal graphical model. Computationally, we show that using RSS leads to a better inference about the unknown features of the nonparanormal graphical model, and the proposed rank-based estimator performs as well as its counterparts estimators defined based on SRS. We study the numerical performance of the proposed method through a simulation study under both ideal and noisy settings and apply it to a genomic data set.