When the degree sequence is a sufficient statistic

Abstract

There is a uniquely defined random graph model with independent adjacencies in which the degree sequence is a sufficient statistic. The model was recently discovered independently by several authors. Here we join to the statistical investigation of the model, proving that if the degree sequence is in the interior of the polytope defined by the Erdős-Gallai conditions, then a unique maximum likelihood estimate exists.

The $\beta$-Model—Maximum Likelihood, Cramér–Rao Bounds, and Hypothesis Testing

“…According to the authors' experience, the fixed-point method presented in this section has more favorable convergence properties. This could also have been expected given the convergence theorems, earlier mentioned in Section II-A, that were presented for iteration (5) in [16] and [25].…”

“…APPENDIX B Here, we derive the FIM in (24), given for the directed β-model. In addition, we also derive the inverse FIM in the special case considered in (25).…”

“…The FIM in (24) now follows directly from (39) and (40). Given the assumptions preceding (25), it follows that the submatrices in (24) are…”

“…Assuming that a unique ML solution exists and that N ij = 1 for all i = j, geometrically fast convergence to the fixed point was shown in [16] and [25]. Conditions for the asymptotic normality of the ML estimator when n tends to infinity and N ij = 1 for all i = j were presented in [26].…”

“…Conditions for the asymptotic normality of the ML estimator when n tends to infinity and N ij = 1 for all i = j were presented in [26]. Necessary and sufficient conditions for the existence of a finite ML estimate were presented in [14], [25], and [27]. As an example, it should be clear that there is no finiteθ satisfying (4) whenever Fig.…”