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The mixtools package for R provides a set of functions for analyzing a variety of finite mixture models. These functions include both traditional methods, such as EM algorithms for univariate and multivariate normal mixtures, and newer methods that reflect some recent research in finite mixture models. In the latter category, mixtools provides algorithms for estimating parameters in a wide range of different mixture-of-regression contexts, in multinomial mixtures such as those arising from discretizing continuous multivariate data, in nonparametric situations where the multivariate component densities are completely unspecified, and in semiparametric situations such as a univariate location mixture of symmetric but otherwise unspecified densities. Many of the algorithms of the mixtools package are EM algorithms or are based on EM-like ideas, so this article includes an overview of EM algorithms for finite mixture models.

We describe some of the capabilities of the ergm package and the statistical theory underlying it. This package contains tools for accomplishing three important, and interrelated, tasks involving exponential-family random graph models (ERGMs): estimation, simulation, and goodness of fit. More precisely, ergm has the capability of approximating a maximum likelihood estimator for an ERGM given a network data set; simulating new network data sets from a fitted ERGM using Markov chain Monte Carlo; and assessing how well a fitted ERGM does at capturing characteristics of a particular network data set.

Abstractstatnet is a suite of software packages for statistical network analysis. The packages implement recent advances in network modeling based on exponential-family random graph models (ERGM). The components of the package provide a comprehensive framework for ERGM-based network modeling, including tools for model estimation, model evaluation, model-based network simulation, and network visualization. This broad functionality is powered by a central Markov chain Monte Carlo (MCMC) algorithm. The coding is optimized for speed and robustness.

We present a systematic examination of a real network dataset using maximum likelihood estimation for exponential random graph models as well as new procedures to evaluate how well the models fit the observed networks. These procedures compare structural statistics of the observed network with the corresponding statistics on networks simulated from the fitted model. We apply this approach to the study of friendship relations among high school students from the National Longitudinal Study of Adolescent Health (AddHealth). We focus primarily on one particular network of 205 nodes, though we also demonstrate that this method may be applied to the largest network in the AddHealth study, with 2209 nodes. We argue that several well-studied models in the networks literature do not fit these data well, and we demonstrate that the fit improves dramatically when the models include the recently-developed geometrically weighted edgewise shared partner (GWESP), geometrically weighted dyadic shared partner (GWDSP), and geometrically weighted degree (GWD) network statistics. We conclude that these models capture aspects of the social structure of adolescent friendship relations not represented by previous models.

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Variable selection is fundamental to high-dimensional statistical modeling. Many variable selection techniques may be implemented by maximum penalized likelihood using various penalty functions. Optimizing the penalized likelihood function is often challenging because it may be nondifferentiable and/or nonconcave. This article proposes a new class of algorithms for finding a maximizer of the penalized likelihood for a broad class of penalty functions. These algorithms operate by perturbing the penalty function slightly to render it differentiable, then optimizing this differentiable function using a minorize-maximize (MM) algorithm. MM algorithms are useful extensions of the well-known class of EM algorithms, a fact that allows us to analyze the local and global convergence of the proposed algorithm using some of the techniques employed for EM algorithms. In particular, we prove that when our MM algorithms converge, they must converge to a desirable point; we also discuss conditions under which this convergence may be guaranteed. We exploit the NewtonRaphson-like aspect of these algorithms to propose a sandwich estimator for the standard errors of the estimators. Our method performs well in numerical tests.