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.

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