  Biomedical researchers usually study the effects of certain exposures on
disease risks among a well-defined population. To achieve this goal, the gold
standard is to design a trial with an appropriate sample from that population.
Due to the high cost of such trials, usually the sample size collected is
limited and is not enough to accurately estimate some exposures' effect. In
this paper, we discuss how to leverage the information from external `big data'
(data with much larger sample size) to improve the estimation accuracy at the
risk of introducing small bias. We proposed a family of weighted estimators to
balance the bias increase and variance reduction when including the big data.
We connect our proposed estimator to the established penalized regression
estimators. We derive the optimal weights using both second order and higher
order asymptotic expansions. Using extensive simulation studies, we showed that
the improvement in terms of mean square error (MSE) for the regression
coefficient can be substantial even with finite sample sizes and our weighted
method outperformed the existing methods such as penalized regression and James
Stein's approach. Also we provide theoretical guarantee that the proposed
estimators will never lead to asymptotic MSE larger than the maximum likelihood
estimator using small data only in general. We applied our proposed methods to
the Asia Cohort Consortium China cohort data to estimate the relationships
between age, BMI, smoking, alcohol use and mortality.
