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Register a new userOn the Existence of the MLE for a Directed Random Graph Network Model with Reciprocation
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Added by
Alessandro Rinaldo
Publication date
2010
fields
Mathematical Statistics
and research's language is
English
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Holland and Leinhardt (1981) proposed a directed random graph model, the p1 model, to describe dyadic interactions in a social network. In previous work (Petrovic et al., 2010), we studied the algebraic properties of the p1 model and showed that it is a toric model specified by a multi-homogeneous ideal. We conducted an extensive study of the Markov bases for p1 that incorporate explicitly the constraint arising from multi-homogeneity. Here we consider the properties of the corresponding toric variety and relate them to the conditions for the existence of the maximum likelihood and extended maximum likelihood estimators or the model parameters. Our results are directly relevant to the estimation and conditional goodness-of-fit testing problems in p1 models.
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The edges in networks are not only binary, either present or absent, but also take weighted values in many scenarios (e.g., the number of emails between two users). The covariate-$p_0$ model has been proposed to model binary directed networks with the degree heterogeneity and covariates. However, it may cause information loss when it is applied in weighted networks. In this paper, we propose to use the Poisson distribution to model weighted directed networks, which admits the sparsity of networks, the degree heterogeneity and the homophily caused by covariates of nodes. We call it the emph{network Poisson model}. The model contains a density parameter $mu$, a $2n$-dimensional node parameter ${theta}$ and a fixed dimensional regression coefficient ${gamma}$ of covariates. Since the number of parameters increases with $n$, asymptotic theory is nonstandard. When the number $n$ of nodes goes to infinity, we establish the $ell_infty$-errors for the maximum likelihood estimators (MLEs), $hat{theta}$ and $hat{{gamma}}$, which are $O_p( (log n/n)^{1/2} )$ for $hat{theta}$ and $O_p( log n/n)$ for $hat{{gamma}}$, up to an additional factor. We also obtain the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings. ) for b{theta} and Op(log n/n) for b{gamma}, up to an additional factor. We also obtain the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings.
This short note is to point the reader to notice that the proof of high dimensional asymptotic normality of MLE estimator for logistic regression under the regime $p_n=o(n)$ given in paper: Maximum likelihood estimation in logistic regression models with a diverging number of covariates. Electronic Journal of Statistics, 6, 1838-1846. is wrong.
We are concerned here with unrestricted maximum likelihood estimation in a sparse $p_0$ model with covariates for directed networks. The model has a density parameter $ u$, a $2n$-dimensional node parameter $bs{eta}$ and a fixed dimensional regression coefficient $bs{gamma}$ of covariates. Previous studies focus on the restricted likelihood inference. When the number of nodes $n$ goes to infinity, we derive the $ell_infty$-error between the maximum likelihood estimator (MLE) $(widehat{bs{eta}}, widehat{bs{gamma}})$ and its true value $(bs{eta}, bs{gamma})$. They are $O_p( (log n/n)^{1/2} )$ for $widehat{bs{eta}}$ and $O_p( log n/n)$ for $widehat{bs{gamma}}$, up to an additional factor. This explains the asymptotic bias phenomenon in the asymptotic normality of $widehat{bs{gamma}}$ in cite{Yan-Jiang-Fienberg-Leng2018}. Further, we derive the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings.
In this paper we are interested in the Maximum Likelihood Estimator (MLE) of the vector parameter of an autoregressive process of order $p$ with regular stationary Gaussian noise. We exhibit the large sample asymptotical properties of the MLE under very mild conditions. Simulations are done for fractional Gaussian noise (fGn), autoregressive noise (AR(1)) and moving average noise (MA(1)).
We study the problem of estimating the mean of a multivariatedistribution based on independent samples. The main result is the proof of existence of an estimator with a non-asymptotic sub-Gaussian performance for all distributions satisfying some mild moment assumptions.
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