  This paper concerns with statistical estimation and inference for the ranking
problems based on pairwise comparisons with additional covariate information
such as the attributes of the compared items. Despite extensive studies, few
prior literatures investigate this problem under the more realistic setting
where covariate information exists. To tackle this issue, we propose a novel
model, Covariate-Assisted Ranking Estimation (CARE) model, that extends the
well-known Bradley-Terry-Luce (BTL) model, by incorporating the covariate
information. Specifically, instead of assuming every compared item has a fixed
latent score $\{\theta_i^*\}_{i=1}^n$, we assume the underlying scores are
given by $\{\alpha_i^*+{x}_i^\top\beta^*\}_{i=1}^n$, where $\alpha_i^*$ and
${x}_i^\top\beta^*$ represent latent baseline and covariate score of the $i$-th
item, respectively. We impose natural identifiability conditions and derive the
$\ell_{\infty}$- and $\ell_2$-optimal rates for the maximum likelihood
estimator of $\{\alpha_i^*\}_{i=1}^{n}$ and $\beta^*$ under a sparse comparison
graph, using a novel `leave-one-out' technique (Chen et al., 2019) . To conduct
statistical inferences, we further derive asymptotic distributions for the MLE
of $\{\alpha_i^*\}_{i=1}^n$ and $\beta^*$ with minimal sample complexity. This
allows us to answer the question whether some covariates have any explanation
power for latent scores and to threshold some sparse parameters to improve the
ranking performance. We improve the approximation method used in (Gao et al.,
2021) for the BLT model and generalize it to the CARE model. Moreover, we
validate our theoretical results through large-scale numerical studies and an
application to the mutual fund stock holding dataset.
