Least squares for cardinal paired comparisons data

Least square estimators for graphical models for cardinal paired comparison data with and without covariates are rigorously analysed. Novel, graph-based, necessary, and sufficient conditions that guarantee strong consistency, asymptotic normality, and the exponential convergence of the estimated ranks are emphasized. A complete theory for models with covariates is laid out. In particular, conditions under which covariates can be safely omitted from the model are provided. The methodology is employed in the analysis of both finite and infinite sets of ranked items where the case of large sparse comparison graphs is addressed. The proposed methods are explored by simulation and applied to the ranking of teams in the National Basketball Association.