Bahadur Efficiency in Tensor Curie-Weiss Models

by   Somabha Mukherjee, et al.

The tensor Ising model is a discrete exponential family used for modeling binary data on networks with not just pairwise, but higher-order dependencies. In this exponential family, the sufficient statistic is a multi-linear form of degree p≥ 2, designed to capture p-fold interactions between the binary variables sitting on the nodes of a network. A particularly useful class of tensor Ising models are the tensor Curie-Weiss models, where one assumes that all p-tuples of nodes interact with the same intensity. Computing the maximum likelihood estimator (MLE) is computationally cumbersome in this model, due to the presence of an inexplicit normalizing constant in the likelihood, for which the standard alternative is to use the maximum pseudolikelihood estimator (MPLE). Both the MLE and the MPLE are consistent estimators of the natural parameter, provided the latter lies strictly above a certain threshold, which is slightly below log 2, and approaches log 2 as p increases. In this paper, we compute the Bahadur efficiencies of the MLE and the MPLE above the threshold, and derive the optimal sample size (number of nodes) needed for either of these tests to achieve significance. We show that the optimal sample size for the MPLE and the MLE agree if either p=2 or the null parameter is greater than or equal to log 2. On the other hand, if p≥ 3 and the null parameter lies strictly between the threshold and log 2, then the two differ for sufficiently large values of the alternative. In particular, for every fixed alternative above the threshold, the Bahadur asymptotic relative efficiency of the MLE with respect to the MPLE goes to ∞ as the null parameter approaches the threshold. We also provide graphical presentations of the exact numerical values of the theoretical optimal sample sizes in different settings.


Estimation in Tensor Ising Models

The p-tensor Ising model is a one-parameter discrete exponential family ...

Tensor Recovery in High-Dimensional Ising Models

The k-tensor Ising model is an exponential family on a p-dimensional bin...

Limit theorems for dependent combinatorial data, with applications in statistical inference

The Ising model is a celebrated example of a Markov random field, introd...

Adaptive exponential power distribution with moving estimator for nonstationary time series

While standard estimation assumes that all datapoints are from probabili...

Phase Transitions of the Maximum Likelihood Estimates in the Tensor Curie-Weiss Model

The p-tensor Curie-Weiss model is a two-parameter discrete exponential f...

Wilks' theorems in the β-model

Likelihood ratio tests and the Wilks theorems have been pivotal in stati...

Logistic-Regression with peer-group effects via inference in higher order Ising models

Spin glass models, such as the Sherrington-Kirkpatrick, Hopfield and Isi...

Please sign up or login with your details

Forgot password? Click here to reset