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New consistent and asymptotically normal parameter estimates for random‐graph mixture models
Author(s) -
Ambroise Christophe,
Matias Catherine
Publication year - 2012
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/j.1467-9868.2011.01009.x
Subject(s) - estimator , mathematics , expectation–maximization algorithm , maximization , maximum a posteriori estimation , estimation theory , convergence (economics) , moment (physics) , mathematical optimization , maximum likelihood , statistics , physics , classical mechanics , economics , economic growth
Summary. Random‐graph mixture models are very popular for modelling real data networks. Parameter estimation procedures usually rely on variational approximations, either combined with the expectation–maximization ( EM ) algorithm or with Bayesian approaches. Despite good results on synthetic data, the validity of the variational approximation is, however, not established. Moreover, these variational approaches aim at approximating the maximum likelihood or the maximum a posteriori estimators, whose behaviour in an asymptotic framework (as the sample size increases to ∞) remains unknown for these models. In this work, we show that, in many different affiliation contexts (for binary or weighted graphs), parameter estimators based either on moment equations or on the maximization of some composite likelihood are strongly consistent and √ n convergent, when the number n of nodes increases to ∞. As a consequence, our result establishes that the overall structure of an affiliation model can be (asymptotically) caught by the description of the network in terms of its number of triads (order 3 structures) and edges (order 2 structures). Moreover, these parameter estimates are either explicit (as for the moment estimators) or may be approximated by using a simple EM algorithm, whose convergence properties are known. We illustrate the efficiency of our method on simulated data and compare its performances with other existing procedures. A data set of cross‐citations among economics journals is also analysed.