  In regression analysis of multivariate data, it is tacitly assumed that
response and predictor variables in each observed response-predictor pair
correspond to the same entity or unit. In this paper, we consider the situation
of "permuted data" in which this basic correspondence has been lost. Several
recent papers have considered this situation without further assumptions on the
underlying permutation. In applications, the latter is often to known to have
additional structure that can be leveraged. Specifically, we herein consider
the common scenario of "sparsely permuted data" in which only a small fraction
of the data is affected by a mismatch between response and predictors. However,
an adverse effect already observed for sparsely permuted data is that the least
squares estimator as well as other estimators not accounting for such partial
mismatch are inconsistent. One approach studied in detail herein is to treat
permuted data as outliers which motivates the use of robust regression
formulations to estimate the regression parameter. The resulting estimate can
subsequently be used to recover the permutation. A notable benefit of the
proposed approach is its computational simplicity given the general lack of
procedures for the above problem that are both statistically sound and
computationally appealing.
