Interaction Order Estimation in Tensor Curie-Weiss Models

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 the estimation of p is impossible if β is unknown. These impossibility results are also extended to the more general p-spin Erdös-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 allβ, consistent estimation of p is possible for β*(p)<β.