  Recently, there has been significant interest in linear regression in the
situation where predictors and responses are not observed in matching pairs
corresponding to the same statistical unit as a consequence of separate data
collection and uncertainty in data integration. Mismatched pairs can
considerably impact the model fit and disrupt the estimation of regression
parameters. In this paper, we present a method to adjust for such mismatches
under ``partial shuffling" in which a sufficiently large fraction of
(predictors, response)-pairs are observed in their correct correspondence. The
proposed approach is based on a pseudo-likelihood in which each term takes the
form of a two-component mixture density. Expectation-Maximization schemes are
proposed for optimization, which (i) scale favorably in the number of samples,
and (ii) achieve excellent statistical performance relative to an oracle that
has access to the correct pairings as certified by simulations and case
studies. In particular, the proposed approach can tolerate considerably larger
fraction of mismatches than existing approaches, and enables estimation of the
noise level as well as the fraction of mismatches. Inference for the resulting
estimator (standard errors, confidence intervals) can be based on established
theory for composite likelihood estimation. Along the way, we also propose a
statistical test for the presence of mismatches and establish its consistency
under suitable conditions.
