  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.
