More Efficient Estimation for Logistic Regression with Optimal Subsamples
More Efficient Estimation for Logistic Regression with Optimal Subsamples
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发表时间：
2018-02
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通讯作者：
Haiying Wang
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作者：
Haiying Wang
在海量数据中，子采样是提取有用信息的一种实用技术。为此，Wang et al.（2017）开发了一种用于逻辑回归的A最优准则下的最优子采样方法（OSMAC），该方法以更高的概率采样更多信息的数据点。然而，原始OSMAC估计器使用最优子采样概率的倒数作为似然函数中的权重。这减少了更多信息数据点的贡献，并且所得到的估计器可能会失去效率。在本文中，我们提出了一个更有效的估计OSMAC子样本的基础上没有加权的似然函数。渐近结果和数值结果都表明新的估计是更有效的。此外，我们在本文中的重点是对真实参数的推断，而Wang et al.（2017）专注于近似完整数据估计。我们还开发了一个新的算法的基础上泊松抽样，它不需要近似的最佳子采样概率一次。当可用的随机存取存储器不足以保存全部数据时，这在计算上是有利的。有趣的是，渐近分布也表明，泊松抽样产生更有效的估计，如果抽样率，子样本大小的比例，全数据样本大小，不收敛到零。在Poisson抽样下，我们还得到了估计量的无条件渐近分布。
Facing large amounts of data, subsampling is a practical technique to extract useful information. For this purpose, Wang et al. (2017) developed an Optimal Subsampling Method under the A-optimality Criterion (OSMAC) for logistic regression that samples more informative data points with higher probabilities. However, the original OSMAC estimator use inverse of optimal subsampling probabilities as weights in the likelihood function. This reduces contributions of more informative data points and the resultant estimator may lose efficiency. In this paper, we propose a more efficient estimator based on OSMAC subsample without weighting the likelihood function. Both asymptotic results and numerical results show that the new estimator is more efficient. In addition, our focus in this paper is inference for the true parameter, while Wang et al. (2017) focuses on approximating the full data estimator. We also develop a new algorithm based on Poisson sampling, 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 sampling produces 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 sampling.