Statistics of tensor forms appear in various contexts and provide a useful way to model dependence, which naturally arises in network data. In this thesis, we study the statistics of two different tensor forms.
In the first part of the thesis, we derive central limit theorem results which exhibit a fourth moment phenomenon. That is, the fourth moment converging to 3 implies the convergence of the statistic to a normal distribution. We also establish the other direction, which provides us with an if and only if condition for asymptotic normality. The settings and the results are very easily applied to the monochromatic subgraph count in the problem of graph coloring.
The second part of the thesis compares the relative efficiency of the maximum likelihood estimator (MLE) and the maximum pseudolikelihood estimator (MPLE) for particular p-tensor Ising models. Specifically, we show that in the graph case, i.e. when p = 2, the MLE and MPLE are equally Bahadur efficient. For high-order tensors, i.e. p ≥ 3, we show a two-layer phase transition in which the MPLE is less Bahadur efficient than the MLE in certain regimes of the parameter space, depending also on the magnitudes of the null and alternate parameters.
| Identifer | oai:union.ndltd.org:columbia.edu/oai:academiccommons.columbia.edu:10.7916/4jbm-f465 |
| Date | January 2025 |
| Creators | Son, Jaesung |
| Source Sets | Columbia University |
| Language | English |
| Detected Language | English |
| Type | Theses |
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