Limit theorems for dependent combinatorial data, with applications in statistical inference / Somabha Mukherjee.

Format:

Book

Thesis/Dissertation

Author/Creator:

Mukherjee, Somabha, author.

Contributor:

Bhattacharya, Bhaswar B., degree supervisor.

University of Pennsylvania. Department of Statistics, degree granting institution.

Language:

English

Subjects (All):

Statistics.

Mathematics.

Statistical physics.

Statistics--Penn dissertations.

Penn dissertations--Statistics.

Local Subjects:

Statistics.

Mathematics.

Statistical physics.

Statistics--Penn dissertations.

Penn dissertations--Statistics.

Genre:

Academic theses.

Physical Description:

1 online resource (220 pages)

Contained In:

Dissertations Abstracts International 82-12B.

Place of Publication:

[Philadelphia, Pennsylvania] : University of Pennsylvania ; Ann Arbor : ProQuest Dissertations & Theses, 2021.

Language Note:

English

System Details:

Mode of access: World Wide Web.

text file

Summary:

The Ising model is a celebrated example of a Markov random field, which was introduced in statistical physics to model ferromagnetism. More recently, it has emerged as a useful model for understanding dependent binary data with an underlying network structure. This is a discrete exponential family with binary outcomes, where the sufficient statistic involves a quadratic term designed to capture correlations arising from pairwise interactions. However, in many situations the dependencies in a network arise not just from pairs, but from peer-group effects. A convenient mathematical framework for capturing higher-order dependencies, is the p-tensor Ising model, which is a discrete exponential family where the sufficient statistic consists of a multilinear polynomial of degree p. This thesis develops a framework for statistical inference of the natural parameters in p-tensor Ising models. We begin with the Curie-Weiss Ising model, where every p-tuple of nodes interact with equal strengths, where we unearth various non-standard phenomena in the asymptotics of the maximum-likelihood (ML) estimates of the parameters, such as the presence of a critical curve in the interior of the parameter space on which these estimates have a limiting mixture distribution, and a surprising superefficiency phenomenon at the boundary point(s) of this curve. However, ML estimation fails in more general p-tensor Ising models due to the presence of a computationally intractable normalizing constant. To overcome this issue, we use the popular maximum pseudo-likelihood (MPL) method, which avoids computing the inexplicit normalizing constant based on conditional distributions. We derive general conditions under which the MPL estimate is √N-consistent, where N is the size of the underlying network. Our conditions are robust enough to handle a variety of commonly used tensor Ising models, including spin glass models with random interactions and the hypergraph stochastic block model. Finally, we consider a more general Ising model, which incorporates high-dimensional covariates at the nodes of the network, that can also be viewed as a logistic regression model with dependent observations. In this model, we show that the parameters can be estimated consistently under sparsity assumptions on the true covariate vector.

Notes:

Source: Dissertations Abstracts International, Volume: 82-12, Section: B.

Advisors: Bhattacharya, Bhaswar B.; Committee members: Robin Pemantle; Nancy Zhang.

Department: Statistics.

Ph.D. University of Pennsylvania 2021.

Local Notes:

School code: 0175

ISBN:

9798738644702

Access Restriction:

Restricted for use by site license.

This item must not be sold to any third party vendors.