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Suppose we have available individual data from an internal study and varioustypes of summary statistics from relevant external studies. External summarystatistics 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 paradoxicalresults: efficiency loss may occur if the uncertainty of summary statistics isnot negligible and large estimation bias can emerge even if the bias ofexternal summary statistics is small. We investigate these paradoxical resultsin a semiparametric framework. We establish the semiparametric efficiency boundfor estimating a general functional of the internal data distribution, which isshown to be no larger than that using only internal data. We propose adata-fused efficient estimator that achieves this bound so that the efficiencyparadox is resolved. Besides, a debiased estimator is further proposed whichhas selection consistency property by employing adaptive lasso penalty so thatthe resultant estimator can achieve the same asymptotic distribution as theoracle one that uses only unbiased summary statistics, which resolves the biasparadox. Simulations and application to a Helicobacter pylori infection datasetare used to illustrate the proposed methods.

Paradoxes and resolutions for semiparametric fusion of individual and summary data