Ranking problems based on pairwise comparisons, such as those arising in online gaming, often involve a large pool of items to order. In these situations, the gap in performance between any two items can be significant, and the smallest and largest winning probabilities can be very close to zero or one. Furthermore, each item may be compared only to a subset of all the items, so that not all pairwise comparisons are observed. In this paper, we study the performance of the Bradley-Terry-Luce model for ranking from pairwise comparison data under more realistic settings than those considered in the literature so far. In particular, we allow for near-degenerate winning probabilities and arbitrary comparison designs. We obtain novel results about the existence of the maximum likelihood estimator (MLE) and the corresponding $\ell_2$ estimation error without the bounded winning probability assumption commonly used in the literature and for arbitrary comparison graph topologies. Central to our approach is the reliance on the Fisher information matrix to express the dependence on the graph topologies and the impact of the values of the winning probabilities on the estimation risk and on the conditions for the existence of the MLE. Our bounds recover existing results as special cases but are more broadly applicable.

We propose a time-varying generalization of the Bradley-Terry model that allows for nonparametric modeling of dynamic global rankings of distinct teams. We develop a novel estimator that relies on kernel smoothing to pre-process the pairwise comparisons over time and is applicable in sparse settings where the Bradley-Terry may not be fit. We obtain necessary and sufficient conditions for the existence and uniqueness of our estimator. We also derive time-varying oracle bounds for both the estimation error and the excess risk in the model-agnostic setting where the Bradley-Terry model is not necessarily the true data generating process. We thoroughly test the practical effectiveness of our model using both simulated and real world data and suggest an efficient data-driven approach for bandwidth tuning.

High-dimensional neural recordings across multiple brain regions can be used to establish functional connectivity with good spatial and temporal resolution. We designed and implemented a novel method, Latent Dynamic Factor Analysis of High-dimensional time series (LDFA-H), which combines (a) a new approach to estimating the covariance structure among high-dimensional time series (for the observed variables) and (b) a new extension of probabilistic CCA to dynamic time series (for the latent variables). Our interest is in the cross-correlations among the latent variables which, in neural recordings, may capture the flow of information from one brain region to another. Simulations show that LDFA-H outperforms existing methods in the sense that it captures target factors even when within-region correlation due to noise dominates cross-region correlation. We applied our method to local field potential (LFP) recordings from 192 electrodes in Prefrontal Cortex (PFC) and visual area V4 during a memory-guided saccade task. The results capture time-varying lead-lag dependencies between PFC and V4, and display the associated spatial distribution of the signals.