Mathematics – Probability
Scientific paper
2006-04-28
Annals of Statistics 2007, Vol. 35, No. 5, 1931-1946
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/009053607000000109 the Annals of Statistics (http://www.imstat.org/aos/) by the Inst
Scientific paper
10.1214/009053607000000109
The Sherrington--Kirkpatrick model of spin glasses, the Hopfield model of neural networks and the Ising spin glass are all models of binary data belonging to the one-parameter exponential family with quadratic sufficient statistic. Under bare minimal conditions, we establish the $\sqrt{N}$-consistency of the maximum pseudolikelihood estimate of the natural parameter in this family, even at critical temperatures. Since very little is known about the low and critical temperature regimes of these extremely difficult models, the proof requires several new ideas. The author's version of Stein's method is a particularly useful tool. We aim to introduce these techniques into the realm of mathematical statistics through an example and present some open questions.
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