Maximum likelihood estimation for incomplete multinomial data via the weaver algorithm

In a multinomial model, the sample space is partitioned into a disjoint union of cells. The partition is usually immutable during sampling of the cell counts. In this paper, we extend the multinomial model to the incomplete multinomial model by relaxing the constant partition assumption to allow the cells to be variable and the counts collected from non-disjoint cells to be modeled in an integrated manner for inference on the common underlying probability. The incomplete multinomial likelihood is parameterized by the complete-cell probabilities from the most refined partition. Its sufficient statistics include the variable-cell formation observed as an indicator matrix and all cell counts. With externally imposed structures on the cell formation process, it reduces to special models including the Bradley–Terry model, the Plackett–Luce model, etc. Since the conventional method, which solves for the zeros of the score functions, is unfruitful, we develop a new approach to establishing a simpler set of estimating equations to obtain the maximum likelihood estimate (MLE), which seeks the simultaneous maximization of all multiplicative components of the likelihood by fitting each component into an inequality. As a consequence, our estimation amounts to solving a system of the equality attainment conditions to the inequalities. The resultant MLE equations are simple and immediately invite a fixed-point iteration algorithm for solution, which is referred to as the weaver algorithm. The weaver algorithm is short and amenable to parallel implementation. We also derive the asymptotic covariance of the MLE, verify main results with simulations, and compare the weaver algorithm with an MM/EM algorithm based on fitting a Plackett–Luce model to a benchmark data set.