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Algorithms for Discrete Data in Statistics and Operations Research
Title
Algorithms for Discrete Data in Statistics and Operations Research
Creator
Schwartz, William K.
Date
2021-11-19, 2021-12
Publisher
ProQuest, https://www.proquest.com/docview/2622985712
Description
Sponsorship: The Air Force Office of Scientific Research's grant FA9550-14-1-0141 supported Prof. Petrović's and my initial work on this project.
Algorithms for Discrete Data in Statistics and Operations Research
Title
Algorithms for Discrete Data in Statistics and Operations Research
Creator
Schwartz, William K.
Date
2021
Description
This thesis develops mathematical background for the design of algorithms for discrete-data problems, two in statistics and one in operations...
Show moreThis thesis develops mathematical background for the design of algorithms for discrete-data problems, two in statistics and one in operations research. Chapter 1 gives some background on what chapters 2 to 4 have in common. It also defines some basic terminology that the other chapters use.Chapter 2 offers a general approach to modeling longitudinal network data, including exponential random graph models (ERGMs), that vary according to certain discrete-time Markov chains (The abstract of chapter 2 borrows heavily from the abstract of Schwartz et al., 2021). It connects conditional and Markovian exponential families, permutation- uniform Markov chains, various (temporal) ERGMs, and statistical considerations such as dyadic independence and exchangeability. Markovian exponential families are explored in depth to prove that they and only they have exponential family finite sample distributions with the same parameter as that of the transition probabilities. Many new statistical and algebraic properties of permutation-uniform Markov chains are derived. We introduce exponential random ?-multigraph models, motivated by our result on replacing ? observations of a permutation-uniform Markov chain of graphs with a single observation of a corresponding multigraph. Our approach simplifies analysis of some network and autoregressive models from the literature. Removing models’ temporal dependence but not interpretability permitted us to offer closed-form expressions for maximum likelihood estimators that previously did not have closed-form expression available. Chapter 3 designs novel, exact, conditional tests of statistical goodness-of-fit for mixed membership stochastic block models (MMSBMs) of networks, both directed and undirected. The tests employ a ?²-like statistic from which we define p-values for the general null hypothesis that the observed network’s distribution is in the MMSBM as well as for the simple null hypothesis that the distribution is in the MMSBM with specified parameters. For both tests the alternative hypothesis is that the distribution is unconstrained, and they both assume we have observed the block assignments. As exact tests that avoid asymptotic arguments, they are suitable for both small and large networks. Further we provide and analyze a Monte Carlo algorithm to compute the p-value for the simple null hypothesis. In addition to our rigorous results, simulations demonstrate the validity of the test and the convergence of the algorithm. As a conditional test, it requires the algorithm sample the fiber of a sufficient statistic. In contrast to the Markov chain Monte Carlo samplers common in the literature, our algorithm is an exact simulation, so it is faster, more accurate, and easier to implement. Computing the p-value for the general null hypothesis remains an open problem because it depends on an intractable optimization problem. We discuss the two schools of thought evident in the literature on how to deal with such problems, and we recommend a future research program to bridge the gap those two schools. Chapter 4 investigates an auctioneer’s revenue maximization problem in combinatorial auctions. In combinatorial auctions bidders express demand for discrete packages of multiple units of multiple, indivisible goods. The auctioneer’s NP-complete winner determination problem (WDP) is to fit these packages together within the available supply to maximize the bids’ sum. To shorten the path practitioners traverse from from legalese auction rules to computer code, we offer a new wdp formalism to reflect how government auctioneers sell billions of dollars of radio-spectrum licenses in combinatorial auctions today. It models common tie-breaking rules by maximizing a sum of bid vectors lexicographically. After a novel pre-solving technique based on package bids’ marginal values, we develop an algorithm for the WDP. In developing the algorithm’s branch-and-bound part adapted to lexicographic maximization, we discover a partial explanation of why classical WDP has been successful in using the linear programming relaxation: it equals the Lagrangian dual. We adapt the relaxation to lexicographic maximization. The algorithm’s dynamic-programming part retrieves already computed partial solutions from a novel data structure suited specifically to our WDP formalism. Finally we show that the data structure can “warm start” a popular algorithm for solving for opportunity-cost prices.
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