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