Asymptotically Optimal Sequential Design for Rank Aggregation

A sequential design problem for rank aggregation is commonly encountered in psychology, politics, marketing, sports, etc. In this problem, a decision maker is responsible for ranking K items by sequentially collecting noisy pairwise comparisons from judges. The decision maker needs to choose a pair of items for comparison in each step, decide when to stop data collection, and make a final decision after stopping based on a sequential flow of information. Because of the complex ranking structure, existing sequential analysis methods are not suitable. In this paper, we formulate the problem under a Bayesian decision framework and propose sequential procedures that are asymptotically optimal. These procedures achieve asymptotic optimality by seeking a balance between exploration (i.e., finding the most indistinguishable pair of items) and exploitation (i.e., comparing the most indistinguishable pair based on the current information). New analytical tools are developed for proving the asymptotic results, combining advanced change of measure techniques for handling the level crossing of likelihood ratios and classic large deviation results for martingales, which are of separate theoretical interest in solving complex sequential design problems. A mirror-descent algorithm is developed for the computation of the proposed sequential procedures.

Asymptotic limits for a nonlinear integro-differential equation modelling leukocytes’ rolling on arterial walls

We consider a nonlinear integro-differential model describing z, the position of the cell center on the real line presented in Grec et al (2018 J. Theor. Biol. 452 35–46). We introduce a new ɛ-scaling and we prove rigorously the asymptotics when ɛ goes to zero. We show that this scaling characterizes the long-time behavior of the solutions of our problem in the cinematic regime (i.e. the velocity z ˙ tends to a limit). The convergence results are first given when ψ, the elastic energy associated to linkages, is convex and regular (the second order derivative of ψ is bounded). In the absence of blood flow, when ψ, is quadratic, we compute the final position z ∞ to which we prove that z tends. We then build a rigorous mathematical framework for ψ being convex but only Lipschitz. We extend convergence results with respect to ɛ to the case when ψ′ admits a finite number of jumps. In the last part, we show that in the constant force case [see model 3 in Grec et al (2018 J. Theor. Biol. 452 35–46), i.e. ψ is the absolute value)] we solve explicitly the problem and recover the above asymptotic results.

Quantile estimation of semiparametric model with time-varying coefficients for panel count data

Panel count data frequently occurs in follow-up studies, such as medical research, social sciences, reliability studies, and tumorigenicity experiences. This type data has been extensively studied by various statistical models with time-invariant regression coefficients. However, the assumption of invariant coefficients may be violated in some reality, and the temporal covariate effects would be of great interest in research studies. This motivates us to consider a more flexible time-varying coefficient model. For statistical inference of the unknown functions, the quantile regression approach based on the B-spline approximation is developed. Asymptotic results on the convergence of the estimators are provided. Some simulation studies are presented to assess the finite-sample performance of the estimators. Finally, two applications of bladder cancer data and US flight delay data are analyzed by the proposed method.

Asymptotic Separation of Solutions to Fractional Stochastic Multi-Term Differential Equations

In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs). Our goal in this paper is to establish results of the global existence and uniqueness and continuity dependence of the initial values of the solutions to Caputo SMTDEs with non-permutable matrices of order α∈(12,1) and β∈(0,1) whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show the asymptotic separation property between two different solutions of Caputo SMTDEs with a more general condition based on λ. Furthermore, the asymptotic separation rate for the two distinct mild solutions reveals that our asymptotic results are general.

Randomness in classes of matroids

<p>This thesis is inspired by the observation that we have no good random model for matroids. That stands in contrast to graphs, which admit a number of elegant random models. As a result we have relatively little understanding of the properties of a "typical" matroid. Acknowledging the difficulty of the general case, we attempt to gain a grasp on randomness in some chosen classes of matroids. Firstly, we consider sparse paving matroids, which are conjectured to dominate the class of matroids (that is to say, asymptotically almost all matroids would be sparse paving). If this conjecture were true, then many results on properties of the random sparse paving matroid would also hold for the random matroid. Sparse paving matroids have limited richness of structure, making counting arguments in particular more feasible than for general matroids. This enables us to prove a number of asymptotic results, particularly with regards to minors. Secondly, we look at Graham-Sloane matroids, a special subset of sparse paving matroids with even more limited structure - in fact Graham-Sloane matroids on a labelled groundset can be randomly generated by a process as simple as independently including certain bases with probability 0.5. Notably, counting Graham-Sloane matroids has provided the best known lower bound on the total number of matroids, to log-log level. Despite the vast size of the class we are able to prove severe restrictions on what minors of Graham-Sloane matroids can look like. Finally we consider transversal matroids, which arise naturally in the context of other mathematical objects - in particular partial transversals of set systems and partial matchings in bipartite graphs. Although transversal matroids are not in one-to-one correspondence with bipartite graphs, we shall link the two closely enough to gain some useful results through exploiting the properties of random bipartite graphs. Returning to the theme of matroid minors, we prove that the class of transversal matroids of given rank is defined by finitely many excluded series-minors. Lastly we consider some other topics, including the axiomatisability of transversal matroids and related classes.</p>

Randomness in classes of matroids

<p>This thesis is inspired by the observation that we have no good random model for matroids. That stands in contrast to graphs, which admit a number of elegant random models. As a result we have relatively little understanding of the properties of a "typical" matroid. Acknowledging the difficulty of the general case, we attempt to gain a grasp on randomness in some chosen classes of matroids. Firstly, we consider sparse paving matroids, which are conjectured to dominate the class of matroids (that is to say, asymptotically almost all matroids would be sparse paving). If this conjecture were true, then many results on properties of the random sparse paving matroid would also hold for the random matroid. Sparse paving matroids have limited richness of structure, making counting arguments in particular more feasible than for general matroids. This enables us to prove a number of asymptotic results, particularly with regards to minors. Secondly, we look at Graham-Sloane matroids, a special subset of sparse paving matroids with even more limited structure - in fact Graham-Sloane matroids on a labelled groundset can be randomly generated by a process as simple as independently including certain bases with probability 0.5. Notably, counting Graham-Sloane matroids has provided the best known lower bound on the total number of matroids, to log-log level. Despite the vast size of the class we are able to prove severe restrictions on what minors of Graham-Sloane matroids can look like. Finally we consider transversal matroids, which arise naturally in the context of other mathematical objects - in particular partial transversals of set systems and partial matchings in bipartite graphs. Although transversal matroids are not in one-to-one correspondence with bipartite graphs, we shall link the two closely enough to gain some useful results through exploiting the properties of random bipartite graphs. Returning to the theme of matroid minors, we prove that the class of transversal matroids of given rank is defined by finitely many excluded series-minors. Lastly we consider some other topics, including the axiomatisability of transversal matroids and related classes.</p>

Asymptotic Results for Multinomial Models

In this work, we derived new asymptotic results for multinomial models. To obtain these results, we started by studying limit distributions in models with a compact parameter space. This restriction holds since the key parameter whose components are the probabilities of the possible outcomes have non-negative components that add up to 1. Based on these results, we obtained confidence ellipsoids and simultaneous confidence intervals for models with normal limit distributions. We then studied the covariance matrices of the limit normal distributions for the multinomial models. This was a transition between the previous general results and on the inference for multinomial models in which we considered the chi-square tests, confidence regions and non-linear statistics—namely log-linear models with two numerical applications to those models. Namely, our approach overcame the hierarchical restrictions assumed to analyse the multidimensional contingency table.

Greedy Matching in Bipartite Random Graphs

This paper studies the performance of greedy matching algorithms on bipartite graphs [Formula: see text]. We focus primarily on three classical algorithms: [Formula: see text], which sequentially selects random edges from [Formula: see text]; [Formula: see text], which sequentially matches random vertices in [Formula: see text] to random neighbors; and [Formula: see text], which generates a random priority order over vertices in [Formula: see text] and then sequentially matches random vertices in [Formula: see text] to their highest-priority remaining neighbor. Prior work has focused on identifying the worst-case approximation ratio for each algorithm. This guarantee is highest for [Formula: see text] and lowest for [Formula: see text]. Our work instead studies the average performance of these algorithms when the edge set [Formula: see text] is random. Our first result compares [Formula: see text] and [Formula: see text] and shows that on average, [Formula: see text] produces more matches. This result holds for finite graphs (in contrast to previous asymptotic results) and also applies to “many to one” matching in which each vertex in [Formula: see text] can match with multiple vertices in [Formula: see text]. Our second result compares [Formula: see text] and [Formula: see text] and shows that the better worst-case guarantee of [Formula: see text] does not translate into better average performance. In “one to one” settings where each vertex in [Formula: see text] can match with only one vertex in [Formula: see text], the algorithms result in the same number of matches. When each vertex in [Formula: see text] can match with two vertices in [Formula: see text] produces more matches than [Formula: see text].