Ting Yan

Asymptotic normality in the maximum entropy models on graphs with an increasing number of parameters

Maximum entropy models, motivated by applications in neuron science, are natural generalizations of the β -model to weighted graphs. Similar to the β -model, each vertex in maximum entropy models is assigned a potential parameter, and the degree sequence is the natural sufficient statistic. Hillar and Wibisono (2013) have proved the consistency of the maximum likelihood estimators. In this paper, we further establish the asymptotic normality for any finite number of the maximum likelihood estimators in the maximum entropy models with three types of edge weights, when the total number of parameters goes to infinity. Simulation studies are provided to illustrate the asymptotic results.

Genetic Association With Multiple Traits in the Presence of Population Stratification

Testing association between a genetic marker and multiple‐dependent traits is a challenging task when both binary and quantitative traits are involved. The inverted regression model is a convenient method, in which the traits are treated as predictors although the genetic marker is an ordinal response. It is known that population stratification (PS) often affects population‐based association studies. However, how it would affect the inverted regression for pleiotropic association, especially with the mixed types of traits (binary and quantitative), is not examined and the performance of existing methods to correct for PS using the inverted regression analysis is unknown. In this paper, we focus on the methods based on genomic control and principal component analysis, and investigate type I error of pleiotropic association using the inverted regression model in the presence of PS with allele frequencies and the distributions (or disease prevalences) of multiple traits varying across the subpopulations. We focus on common alleles but simulation results for a rare variant are also reported. An application to the HapMap data is used for illustration.

Asymptotics in Undirected Random Graph Models Parameterized by the Strengths of Vertices

To capture the heterozygous of vertex degrees of networks and understand their distributions, a class of random graph models parameterized by the strengths of vertices, are proposed. These models are equipped in the framework of mutually independent edges, where the number of parameters matches with the size of networks. The asymptotic properties of the maximum likelihood estimator have been derived in some special models such as the β-model, but general results are lacking. In these models, the likelihood equations are identical to the moment equations. In this paper, we establish a unified asymptotic result including the consistency and asymptotic normality of the moment estimator instead of the maximum likelihood estimator, when the number of parameters goes to infinity. We apply it to the generalized β-model, maximum entropy models and Poisson models.

Limit theory of bivariate dual generalized order statistics with random index

ABSTRACT The class of limit distribution functions of bivariate extreme, intermediate and central dual generalized order statistics from independent and identically distributed random variables with random sample size is fully characterized. Two cases are considered. The first case is when the random sample size is assumed to be independent of all basic random variables. The second case is when the interrelation of the random size and the basic random variables is not restricted.

Ranking in the generalized Bradley–Terry models when the strong connection condition fails

ABSTRACT For non balanced paired comparisons, a wide variety of ranking methods have been proposed. One of the best popular methods is the Bradley–Terry model in which the ranking of a set of objects is decided by the maximum likelihood estimates (MLEs) of merits parameters. However, the existence of MLE for the Bradley–Terry model and its generalized models to allow for tied observation or home-field advantage or both to occur, crucially depends on the strong connection condition on the directed graph constructed by a win–loss matrix. When this condition fails, the MLE does not exist and hence there is no solution of ranking. In this paper, we propose an improved version of the ϵ singular perturbation proposed by Conner and Grant (2000), to address this problem and to extend it to the generalized Bradley–Terry models. Some necessary and sufficient conditions for the existence and uniqueness of the penalized MLEs for these generalized Bradley–Terry-ϵ models are derived. Numerical studies show that the ranking is robust to the different ϵ. We apply the proposed methods to the data of the 2008 NFL regular season.

A note on asymptotic distributions in directed exponential random graph models with bi-degree sequences

ABSTRACT The asymptotic normality of a fixed number of the maximum likelihood estimators (MLEs) in the directed exponential random graph models with an increasing bi-degree sequence has been established recently. In this article, we further derive a central limit theorem for a linear combination of all the MLEs with an increasing dimension. Simulation studies are provided to illustrate the asymptotic results.

Affiliation networks with an increasing degree sequence

ABSTRACT Affiliation network is one kind of two-mode social network with two different sets of nodes (namely, a set of actors and a set of social events) and edges representing the affiliation of the actors with the social events. Although a number of statistical models are proposed to analyze affiliation networks, the asymptotic behaviors of the estimator are still unknown or have not been properly explored. In this article, we study an affiliation model with the degree sequence as the exclusively natural sufficient statistic in the exponential family distributions. We establish the uniform consistency and asymptotic normality of the maximum likelihood estimator when the numbers of actors and events both go to infinity. Simulation studies and a real data example demonstrate our theoretical results.

Limit Theory of Bivariate Generalized Order Statistics with Random Sample Size

The class of limit distribution functions (df’s) of the random bivariate extreme, central and intermediate generalized order statistics (gos) from independent and identically distributed random variables (rv’s) is fully characterized. The cases, when the random sample size is independent of the basic variables and when the interrelation between the random sample size and the basic variables is not restricted, are considered.

Generalized-order statistics with random indices

ABSTRACT In this paper, we study the asymptotic behavior of general sequence of extreme, intermediate and central generalized-order statistics (gos), as well as dual generalized-order statistics (dgos), which are connected asymptotically with some regularly varying functions. Moreover, the limit distribution functions of gos, as well as dgos, with random indices, are obtained under general conditions.

Degree-based moment estimation for ordered networks

The edges between vertices in networks take not only the common binary values, but also the ordered values in some situations (e.g., the measurement of the relationship between people from worst to best in social networks). In this paper, the authors study the asymptotic property of the moment estimator based on the degrees of vertices in ordered networks whose edges are ordered random variables. In particular, the authors establish the uniform consistency and the asymptotic normality of the moment estimator when the number of parameters goes to infinity. Simulations and a real data example are provided to illustrate asymptotic results.