Networks are often characterized by node heterogeneity for which nodes exhibit different degrees of interaction and link homophily for which nodes sharing common features tend to associate with each other. In this article, we rigorously study a directed network model that captures the former via node-specific parameterization and the latter by incorporating covariates. In particular, this model quantifies the extent of heterogeneity in terms of outgoingness and incomingness of each node by different parameters, thus allowing the number of heterogeneity parameters to be twice the number of nodes. We study the maximum likelihood estimation of the model and establish the uniform consistency and asymptotic normality of the resulting estimators. Numerical studies demonstrate our theoretical findings and two data analyses confirm the usefulness of our model. Supplementary materials for this article are available online.

In this paper, we extend the definitions of the net reclassification improvement (NRI) and the integrated discrimination improvement (IDI) in the context of multicategory classification. Both measures were proposed in Pencina and others (2008. Evaluating the added predictive ability of a new marker: from area under the receiver operating characteristic (ROC) curve to reclassification and beyond. Statistics in Medicine27, 157–172) as numeric characterizations of accuracy improvement for binary diagnostic tests and were shown to have certain advantage over analyses based on ROC curves or other regression approaches. Estimation and inference procedures for the multiclass NRI and IDI are provided in this paper along with necessary asymptotic distributional results. Simulations are conducted to study the finite-sample properties of the proposed estimators. Two medical examples are considered to illustrate our methodology.

This paper studies the dimension effect of the linear discriminant analysis (LDA) and the regularized linear discriminant analysis (RLDA) classifiers for large dimensional data where the observation dimension $p$ is of the same order as the sample size $n$. More specifically, built on properties of the Wishart distribution and recent results in random matrix theory, we derive explicit expressions for the asymptotic misclassification errors of LDA and RLDA respectively, from which we gain insights of how dimension affects the performance of classification and in what sense. Motivated by these results, we propose adjusted classifiers by correcting the bias brought by the unequal sample sizes. The bias-corrected LDA and RLDA classifiers are shown to have smaller misclassification rates than LDA and RLDA respectively. Several interesting examples are discussed in detail and the theoretical results on dimension effect are illustrated via extensive simulation studies.