加权有向随机图模型中的渐近理论研究

发布时间：2019-07-04 11:00

【摘要】：近年来,对有向图模型统计性质的研究已经成为一个热门的主题。特别地,随着有向图中顶点个数趋于无穷时,基于顶点度序列参数的极大似然估计的渐近性质的研究是一个新的挑战。Yanetal.(2016a)对有向加权指数族随机图中边的权重取二元情形,无穷离散情形以及连续情形时参数的极大似然估计的渐近性质进行了研究。然而对于另外一种边的权重取有限离散情形还未涉及,我们进一步对这种情形进行研究,得到了类似的结果。在本文中,我们的目标是:当参数个数趋于无穷时,建立参数极大似然估计的一致相合性和渐近正态性。随着有限离散加权有向随机图中顶点个数趋于无穷且双度序列作为指数族分布的完全充分统计量的情况下,我们研究了参数极大似然估计的相合性和渐近正态性,并通过数值模拟和两个实际的数值例子论证了我们的理论结果。本文的主要结果以如下两个定理的形式给出:定理1(相合性)假设θ*∈ R2n-1且||θ*||∞≤τlogn,τ是一个满足0τ1/24的常数,A～Pθ*,其中Pθ*表示在参数θ*下有向图Gn的概率分布[详见(1.1)]。则当n趋于无穷大时,依概率为1,极大似然估计存在且满足||θ-θ*||∞=Op((logn)1/2e6||θ*||∞/n1/2)=Op(1).进一步地,如果极大似然估计存在必唯一。定理2(渐近正态性)假设A～Pθ*。如果||θ||∞≤τ logn,τ是满足τ∈(0,1/36)的常数,则对任意固定的kk ≥ 1,随着n→ ∞,向量θ-θ*的前k个元素渐近服从多元正态分布,其中参数均值为0,协方差阵由S*的左上角k×k的子矩阵组成,其中矩阵S*是通过真实值θ*替换矩阵S中的θ得到,S的定义见(4.6)。
文内图片：

图片说明图１：邋Ｋａｒａｔｅ数据集网络图逡逑
[Abstract]:In recent years, the research on the statistical properties of directed graph models has become a hot topic. In particular, as the number of vertices in the digraph tends to infinity, the study of the asymptotic behavior of the maximum likelihood estimation based on the vertex degree sequence parameters is a new challenge. Yanetal. (2016a) The asymptotic properties of the maximum likelihood estimation of the parameters in the case of infinite discrete and continuous cases are studied. However, for the weight of the other side, the finite discrete case is not involved, and we further study the situation to obtain similar results. In this paper, we aim to set up the consistent consistency and asymptotic normality of the maximum likelihood estimation of the parameters when the number of parameters tends to be infinite. With the fact that the number of vertices in the random graph tends to be infinite and the double-degree sequence is the full sufficient statistic of the exponential family distribution with the finite discrete weighting, we study the consistency and the asymptotic normality of the maximum likelihood estimation of the parameters. The theoretical results are demonstrated by numerical simulation and two practical numerical examples. The main results of this paper are given in the form of the following two theorems: Theorem 1 (Consistency) assumes that it is a constant of 0-1/24, A-P-*, Where P * * represents the probability distribution of the directed graph Gn under the parameter F *[see (1.1)]. Then when n tends to infinity, the probability is 1, the maximum likelihood estimate is present and satisfies the | | 1-1 * | | xt = Op ((logn)1/ 2e6 | | xt * | | 1/ n1/2) = Op (1). Further, if the maximum likelihood estimation is present, the presence of the maximum likelihood estimation is unique. Theorem 2 (Asymptotic Normality) assumes A-P-*. If | | 1 | | showcases (logn,1/36) is a constant, then any fixed kk = 1, with n = 0, the first k elements of the vector 1-1 * are asymptotically subordinate to the multivariate normal distribution, where the mean value of the parameter is 0, The covariance matrix consists of sub-matrices of the upper left-hand corner k-k of the S *, where the matrix S * is obtained by replacing the matrix S in the matrix S with the true value A *, and the definition of S is shown in (4.6). in-file picture: picture description fig.1: fig.1: kakrate data set network diagram
【学位授予单位】：华中师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212

【相似文献】