| The study in the statical properties of directed network models has become a hot problem in recent years.Especially,As the the number of parameters goes to infin-ity,studying asymptotic property of the the maximum likelihood estimators(MLEs)becomes open challenge.Yan et al.(2016a)have established asymptotic results in the cases of binary weights,continuous weights and infinite discrete weights.How-ever,the finite discrete weight is not considered in their work,we extend the work of to the case of finite discrete weights in the exponential family random graph mod-els(ERGMs)and derive the parallel results.In this paper,we aim at establishing the uniform consistency and asymptotic normality of the maximum likelihood estimators(MLEs)as the the number of pa-rameters goes to infinity.We study the distribution of the bi-degree sequence when it is the sufficient statistic in a directed random graph,then we establish asymptotic properties of MLEs in the exponential family distributions for the bi-degree sequence with finite discrete weights edges.A key step in our proofs to is to apply a highly accurate approximate inverse of the Fisher information matrix by a simple matrix in Yan et al.(2016a).Simulation studies and two real data examples are.provided to illustrate the asymptotic results.The main results in this paper are given through two theorem as follows:Theorem 1(Consistency)Assume that θ*E R2n-1 with ||θ*||∞≤τ log n,where 0<τ<1/24 is a constant,and that A～Pθ*,where Pθ*denotes the probability distribution(1.1)on A under the parameter 6*.Then as n goes to infinity,with probability approaching one,the MLE 6 exists and satisfies Further,if the MLE exists,it is unique.Theorem 2(Asymptotic normality)Assume that A～Pθ*.If ||θ*||∞≤τlot n,where τ∈(0,1/36)then for any fixed k≥1,as n → ∞,the vector consisting of the first k elements of θ-θ*is asymptotically multivariate normal with mean zero and covariance matrix given by the upper left,k × k block of S. |
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