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.

翻译：大数据在官方统计及应用科学中的应用正在加速，但仅使用大数据计算的统计量往往存在显著的样本选择偏差，导致估计不准确和统计推断失效。我们通过构建将大数据与概率样本结合的估计方程，纠正了广泛线性与非线性统计量中的这一问题。在未知超总体的弱假设条件下，我们证明了积分估计量具有相合性、渐近无偏性及渐近正态分布。通过统一理论框架，可同时获得仅基于抽样设计以及联合超总体的方差估计量。一个令人意外的推论是：当总体与样本量较大时，最小化设计方差的策略几乎能同时最小化联合方差。若样本成员指标之间的依赖性较小且有限总体规模较大，则积分估计量比纯调查估计量更高效。我们以分层简单随机不放回抽样设计为例，展示了该方法在分位数、基尼系数、线性回归系数及最大似然估计中的应用，并通过澳大利亚个人收入模拟加以验证。