A sparse p_0 model with covariates for directed networks
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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 ν, a 2n-dimensional node parameter η and a fixed dimensional regression coefficient γ of covariates. Previous studies focus on the restricted likelihood inference. When the number of nodes n goes to infinity, we derive the ℓ_∞-error between the maximum likelihood estimator (MLE) (η, γ) and its true value (η, γ). They are O_p( (log n/n)^1/2 ) for η and O_p( log n/n) for γ, up to an additional factor. This explains the asymptotic bias phenomenon in the asymptotic normality of γ in <cit.>. Further, we derive the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings.
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