Statistical Estimation and Inference for Permutation Based Model

Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. People spend lots of time dealing with different kinds of data sets. The structure of the data plays an important role in statistics. Among different structures of data, one interesting structure is the permutation, which involves in different kinds of problems, such as recommender system, online gaming, decision making and sports tournament. This thesis is motivated by my interest in understanding the permutation in statistics. Comparing to the wide applications of permutation related model, little is known to the property of permutation in statistics. There are a variety challenges that arise and lots of problems waiting for us to explore in the permutation based model. This thesis aims to solve several interesting problems of the permutation based model in statistics, which may help us to understand more about the property and characteristic of permutation. As a result of the various topics explored, this thesis is split into three parts. In Chapter 2, we discuss the estimation problem of unimodal SST model in the pairwise comparison problem. We prove that the CLS estimator is rate optimal up to a poly(log log n) factor and propose the computational efficient interval sorting estimator, as a computational efficient algorithm to the estimation problem. In Chapter 3, we shift our attention to the inference problem of the permutation based model. We study different kinds of inference problem, including the hypothesis testing problem in noisy sorting model and confidence set construction problems in generalized permutation based model. Network analysis is another important topic related to the permutation. In Chapter 4, we study the optimality of local belief propagation algorithm in the partial recovery problem of stochastic block model. We prove that local BP algorithm can reach the optimality in a certain regime. Moreover, in the regime where local BP algorithm may not achieve the optimal misclassified fraction, we will prove that local BP algorithm can be used in correcting other algorithms and get optimal algorithm to the partial recovery problem