  Suppose we have available individual data from an internal study and various
types of summary statistics from relevant external studies. External summary
statistics have been used as constraints on the internal data distribution,
which promised to improve the statistical inference in the internal data;
however, the additional use of external summary data may lead to paradoxical
results: efficiency loss may occur if the uncertainty of summary statistics is
not negligible and large estimation bias can emerge even if the bias of
external summary statistics is small. We investigate these paradoxical results
in a semiparametric framework. We establish the semiparametric efficiency bound
for estimating a general functional of the internal data distribution, which is
shown to be no larger than that using only internal data. We propose a
data-fused efficient estimator that achieves this bound so that the efficiency
paradox is resolved. Besides, a debiased estimator is further proposed which
has selection consistency property by employing adaptive lasso penalty so that
the resultant estimator can achieve the same asymptotic distribution as the
oracle one that uses only unbiased summary statistics, which resolves the bias
paradox. Simulations and application to a Helicobacter pylori infection dataset
are used to illustrate the proposed methods.
