  There are various parametric models for analyzing pairwise comparison data,
including the Bradley-Terry-Luce (BTL) and Thurstone models, but their reliance
on strong parametric assumptions is limiting. In this work, we study a flexible
model for pairwise comparisons, under which the probabilities of outcomes are
required only to satisfy a natural form of stochastic transitivity. This class
includes parametric models including the BTL and Thurstone models as special
cases, but is considerably more general. We provide various examples of models
in this broader stochastically transitive class for which classical parametric
models provide poor fits. Despite this greater flexibility, we show that the
matrix of probabilities can be estimated at the same rate as in standard
parametric models. On the other hand, unlike in the BTL and Thurstone models,
computing the minimax-optimal estimator in the stochastically transitive model
is non-trivial, and we explore various computationally tractable alternatives.
We show that a simple singular value thresholding algorithm is statistically
consistent but does not achieve the minimax rate. We then propose and study
algorithms that achieve the minimax rate over interesting sub-classes of the
full stochastically transitive class. We complement our theoretical results
with thorough numerical simulations.
