Finite and infinite Mallows ranking models, maximum likelihood estimator, and regeneration
Wenpin Tang · arXiv (Cornell University) · 2018
In this paper we are concerned with various Mallows ranking models. First we study the statistical properties of the MLE of Mallows' $\phi$ model: $\mathbb{P}_{\theta, \pi_0}(\pi) \propto \exp(-\theta \, inv(\pi \circ \pi_0^{-1}))$, where $\theta$ is the dispersion parameter and $\pi_0$ is the central ranking. We prove that (1). the MLE $\widehat{\theta}$ is biased upwards for both Mallows' $\phi$ model and the single parameter IGM model; (2). the MLE $\widehat{\pi}_0$ converges exponentially for Mallows' $\phi$ model. We also make connections of various Mallows ranking models, encompassing the work of Gnedin and Olshanski, and Pitman and Tang. Motivated by the infinite top-$t$ ranking model of Meilǎ and Bao, we propose an algorithm to select the model size $t$ automatically. The key idea relies on the regenerative property of such an infinite permutation. Finally, we apply our algorithm to several data sets including the APA election data and the University homepage search data.