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This paper considers ranking inference of $n$ items based on the observeddata on the top choice among $M$ randomly selected items at each trial. This isa useful modification of the Plackett-Luce model for $M$-way ranking with onlythe top choice observed and is an extension of the celebratedBradley-Terry-Luce model that corresponds to $M=2$. Under a uniform samplingscheme in which any $M$ distinguished items are selected for comparisons withprobability $p$ and the selected $M$ items are compared $L$ times withmultinomial outcomes, we establish the statistical rates of convergence forunderlying $n$ preference scores using both $\ell_2$-norm and$\ell_\infty$-norm, with the minimum sampling complexity. In addition, weestablish the asymptotic normality of the maximum likelihood estimator thatallows us to construct confidence intervals for the underlying scores.Furthermore, we propose a novel inference framework for ranking items through asophisticated maximum pairwise difference statistic whose distribution isestimated via a valid Gaussian multiplier bootstrap. The estimated distributionis then used to construct simultaneous confidence intervals for the differencesin the preference scores and the ranks of individual items. They also enable usto address various inference questions on the ranks of these items. Extensivesimulation studies lend further support to our theoretical results. A real dataapplication illustrates the usefulness of the proposed methods convincingly.

Ranking Inferences Based on the Top Choice of Multiway Comparisons