  In this paper we study a class of exponential family on permutations, which
includes some of the commonly studied Mallows models. We show that the
pseudo-likelihood estimator for the natural parameter in the exponential family
is asymptotically normal, with an explicit variance. Using this, we are able to
construct asymptotically valid confidence intervals. We also show that the MLE
for the same problem is consistent everywhere, and asymptotically normal at the
origin. In this special case, the asymptotic variance of the cost effective
pseudo-likelihood estimator turns out to be the same as the cost prohibitive
MLE. To the best of our knowledge, this is the first inference result on
permutation models including Mallows models, excluding the very special case of
Mallows model with Kendall's Tau.
