  The Plackett--Luce model is a popular approach for ranking data analysis,
where a utility vector is employed to determine the probability of each outcome
based on Luce's choice axiom. In this paper, we investigate the asymptotic
theory of utility vector estimation by maximizing different types of
likelihood, such as the full-, marginal-, and quasi-likelihood. We provide a
rank-matching interpretation for the estimating equations of these estimators
and analyze their asymptotic behavior as the number of items being compared
tends to infinity. In particular, we establish the uniform consistency of these
estimators under conditions characterized by the topology of the underlying
comparison graph sequence and demonstrate that the proposed conditions are
sharp for common sampling scenarios such as the nonuniform random hypergraph
model and the hypergraph stochastic block model; we also obtain the asymptotic
normality of these estimators and discuss the trade-off between statistical
efficiency and computational complexity for practical uncertainty
quantification. Both results allow for nonuniform and inhomogeneous comparison
graphs with varying edge sizes and different asymptotic orders of edge
probabilities. We verify our theoretical findings by conducting detailed
numerical experiments.
