Abstract: This paper is concerned with various Mallows ranking models. We study the statistical properties of the MLE of Mallows' $\phi$ model. We also make connections of various Mallows ranking models, encompassing recent progress in mathematics. Motivated by the infinite top-$t$ ranking model, we propose an algorithm to select the model size $t$ automatically. The key idea relies on the renewal property of such an infinite random permutation. Our algorithm shows good performance on several data sets.