In this paper, we propose improved estimation method for logistic regression
based on subsamples taken according the optimal subsampling probabilities
developed in Wang et al. 2018 Both asymptotic results and numerical results
show that the new estimator has a higher estimation efficiency. We also develop
a new algorithm based on Poisson subsampling, which does not require to
approximate the optimal subsampling probabilities all at once. This is
computationally advantageous when available random-access memory is not enough
to hold the full data. Interestingly, asymptotic distributions also show that
Poisson subsampling produces a more efficient estimator if the sampling rate,
the ratio of the subsample size to the full data sample size, does not converge
to zero. We also obtain the unconditional asymptotic distribution for the
estimator based on Poisson subsampling. The proposed approach requires to use a
pilot estimator to correct biases of un-weighted estimators. We further show
that even if the pilot estimator is inconsistent, the resulting estimators are
still consistent and asymptotically normal if the model is correctly specified.

本文提出了一个基于子样本的改进逻辑回归估计方法，通过使用 2018 年 Wang 等人提出的最优子采样概率。无论是渐近结果还是数值结果都说明新的估计量具有更高的估计效率。我们还提出了一种基于泊松子采样的新算法，该算法不需要一次性近似最优子采样概率，避免了计算问题，并且当随机存储器的可用空间不足以存储全部数据时有计算优势。如果采样速率（子样本大小与全部数据样本大小的比例）不收敛于零，渐近分布表明泊松子采样可以产生更高效的估计量。我们进一步使用一个初始估计器来校正未加权估计器的偏差。我们还证明，即使初始估计器是不一致的，如果模型是正确的，则会产生一致和渐近正常的结果估计量。