大样本Gamma回归的最优子抽样

Optimal Subsampling for Large-Sample Gamma Regression

摘要: 随着计算机行业的迅猛发展，人类社会逐渐迈入大数据时代。面对大规模右偏性和厚尾分布的数据，Gamma回归模型发挥着非常重要的作用。然而如何快速并准确估计出Gamma回归中感兴趣参数成为值得思考的热点问题。在本文中，我们提出两种两步算法分别有效地逼近Φ已知Gamma回归和Φ未知Gamma回归在全数据下的最大似然估计，从而解决了单参数与双参数大样本Gamma回归估计问题。首先在Φ已知情况下，可证明出在给定全数据下一般子抽样估计量渐近服从正态分布，推导出使估计量渐近均方误差最小的最优子抽样概率。为了进一步降低计算量，我们还提出了另一种最优子抽样概率。由于最优子抽样概率取决于未知参数，我们还提出了单参数两步算法。其次在Φ未知情况下，我们基于单参数两步算法提出了双参数两步算法。最后使用数值模拟表明两种算法的计算效率高，也证实了通过单参数两步算法得到的估计量与双参数两步算法得到的估计量差距不明显。

Abstract: With the rapid development of computer industry, human society is gradually moving into the era of big data. Gamma regression models play a very important role in the face of large-sample right-skewed and thick-tailed data. However, how to quickly and accurately estimate the parameters of interest in Gamma regression has become a hot issue to be considered. In this paper, we propose two two-step algorithms to efficiently approximate the maximum likelihood estimates of Φ known Gamma regression and Φ unknown Gamma regression under full data, respectively, thus solving the single-parameter and two-parameter large-sample Gamma regression estimation problems. Firstly, in the case where Φ is known, it can be shown that the general subsampling estimates asymptotically obey a normal distribution given the full data, and the optimal subsampling probability that minimizes the asymptotic mean square error of the estimates is derived. To further reduce the computational effort, we also propose an alternative optimal subsampling probability. Since the optimal subsampling probability depends on the unknown parameters, we also propose a single-parameter two-step algorithm. Secondly, in the case of Φ unknown, we propose a two-parameter two-step algorithm based on the one-parameter two-step algorithm. Finally, using numerical simulations, we show that the two algorithms are computationally efficient and also confirm that the difference between the estimates obtained by the one-parameter two-step algorithm and those obtained by the two-parameter two-step algorithm is not significant.