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Global testing against sparse alternatives under Ising models. (English) Zbl 1407.62160

This paper deals with Ising models of the form \[ \mathbb P_{\mathbf Q, \pmb \mu} \left( \mathbf X = \mathbf x\right) = \frac{1}{Z \left(\mathbf Q, \pmb \mu\right)} \exp\left(\frac12 \mathbf x^\intercal \mathbf Q \mathbf x + \pmb \mu^\intercal \mathbf x\right), \qquad \mathbf x \in \left\{\pm 1\right\}^n \] and the question how to test weather \( \pmb \mu = \mathbf 0\) or not from random observations \(\mathbf X = \left(X_1,...,X_n\right)^{\intercal} \in \left\{\pm 1\right\}^n\). The considered alternatives are assumed to be sparse in a suitable sense, and the authors study the impact of the dependency describing matrix \(\mathbf Q\) onto the detection threshold of the problem.
For different sub-classes of Ising models, the authors construct explicit testing procedures and prove that they are asymptotically rate optimal. Also a comparison between different models is provided.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G10 Nonparametric hypothesis testing
62P35 Applications of statistics to physics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
62C20 Minimax procedures in statistical decision theory
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