  The use of big data in official statistics and the applied sciences is
accelerating, but statistics computed using only big data often suffer from
substantial selection bias. This leads to inaccurate estimation and invalid
statistical inference. We rectify the issue for a broad class of linear and
nonlinear statistics by producing estimating equations that combine big data
with a probability sample. Under weak assumptions about an unknown
superpopulation, we show that our integrated estimator is consistent and
asymptotically unbiased with an asymptotic normal distribution. Variance
estimators with respect to both the sampling design alone and jointly with the
superpopulation are obtained at once using a single, unified theoretical
approach. A surprising corollary is that strategies minimising the design
variance almost minimise the joint variance when the population and sample
sizes are large. The integrated estimator is shown to be more efficient than
its survey-only counterpart if dependence between sample membership indicators
is small and the finite population is large. We illustrate our method for
quantiles, the Gini index, linear regression coefficients and maximum
likelihood estimators where the sampling design is stratified simple random
sampling without replacement. Our results are illustrated in a simulation of
individual Australian incomes.
