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We consider the problem of estimating the probability matrix governing atournament or linkage in graphs from incomplete observations, under theassumption that the probability matrix satisfies natural monotonicityconstraints after being permuted in both rows and columns by some latentpermutation. This condition is classical in the social sciences literature (seeFishburn(1973)) and has been studied in the statistics literature in recentwork (see Chatterjee (2015) and Shah et al. (2016)). In this paper, weinvestigate in detail a natural estimator which bypasses the need to searchover all possible latent permutations and hence is computationally tractable,and derive asymptotic risk bounds for our estimator. In addition, we prove anautomatic adaptation property of our estimator for several sub classes of ourparameter space which are of natural interest. These sub classes includegeneralizations of the popular Bradley Terry Model in the Tournament case, theBeta model and Stochastic Block Model in the Graph case, and Holder smoothmatrices in the tournament and graph settings.

On Estimation in Tournaments and Graphs under Monotonicity Constraints