  A preferential domain is a collection of sets of preferences which are linear
orders over a set of alternatives. These domains have been studied extensively
in social choice theory due to both its practical importance and theoretical
elegance. Examples of some extensively studied preferential domains include
single peaked, single crossing, Euclidean, etc. In this paper, we study the
sample complexity of testing whether a given preference profile is close to
some specific domain. We consider two notions of closeness: (a) closeness via
preferences, and (b) closeness via alternatives. We further explore the effect
of assuming that the {\em outlier} preferences/alternatives to be random
(instead of arbitrary) on the sample complexity of the testing problem. In most
cases, we show that the above testing problem can be solved with high
probability for all commonly used domains by observing only a small number of
samples (independent of the number of preferences, $n$, and often the number of
alternatives, $m$). In the remaining few cases, we prove either impossibility
results or $\Omega(n)$ lower bound on the sample complexity. We complement our
theoretical findings with extensive simulations to figure out the actual
constant factors of our asymptotic sample complexity bounds.
