Interaction Order Estimation in Tensor Curie-Weiss Models

Abstract

In this paper, we consider the problem of estimating the interaction parameter $p$ of a $p$-spin Curie-Weiss model at inverse temperature $β$, given a single observation from this model. We show, by a contiguity argument, that joint estimation of the parameters $β$ and $p$ is impossible, which implies that estimation of $p$ is impossible if $β$ is unknown. These impossibility results are also extended to the more general $p$-spin Erdős-Rényi Ising model. The situation is more delicate when $β$ is known. In this case, we show that there exists an increasing threshold function $β^*(p)$, such that for all $β$, consistent estimation of $p$ is impossible when $β^*(p) > β$, and for almost all $β$, consistent estimation of $p$ is possible for $β^*(p)<β$.