The minimax rate for estimating d-th order moment tensors is sqrt(p/n) wedge 1, while low-degree evidence shows detection of vanishing cumulants is hard for n much less than p to the d/2, creating a reverse detection-estimation gap.
Concentration inequalities and moment bounds for sample covariance operators.Bernoulli, pages 110–133, 2017
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Detection Is Harder Than Estimation in Certain Regimes: Inference for Moment and Cumulant Tensors
The minimax rate for estimating d-th order moment tensors is sqrt(p/n) wedge 1, while low-degree evidence shows detection of vanishing cumulants is hard for n much less than p to the d/2, creating a reverse detection-estimation gap.